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Lectures on Dynamical Systems Anatoly Neishtadt
Lectures for "Mathematics Access Grid Instruction and Collaboration" (MAGIC) consortium, Loughborough University, 2007 Part 2


LECTURE 10


LOCAL INVARIANT MANIFOLDS


The center manifold theorem
Consider an ODE x = Ax + O (|x |2 ), x R
n

with the right hand side of smoothness C r , r < . Assume that the matrix A has ns , nu and nc eigenvalues in the left complex half-plane, right complex half-plane and on imaginary axis respectively, ns + nu + nc = n. Denote T s , T u and T c the corresponding invariant planes of A . (Note: "s" is for "stable", "u" is for "unstable", "c" is for "center ").

Theorem (The center manifold theorem: Pliss-Kelley-Hirsch-Pugh-Shub)
In some neighborhood U of the origin this ODE has C r -smooth invariant manifolds W s , W u and C r -1 -smooth invariant manifold W c , which are tangent at the origin to the planes T s , T u and T c respectively. Trajectories in the manifolds W s and W u exponentially fast tend to the origin as t + and t - respectively. Trajectories which remain in U for all t 0 (t 0) tend to W c as t + (t -). W s , W u and W c are called the stable, the unstable and a center manifolds of the equilibrium 0 respectively.

Remark
Behavior of trajectories on W c is determined by nonlinear terms.


The center manifold theorem, continued


The center manifold theorem, continued

Remark
If the original equation has smoothness C or C , then W s and W u also have smoothness C or C . However W c in general has only a finite smoothness.

Remark
If ns = 0 or nu = 0 and the original equation has smoothness C r , r < , then W c has smoothness C r .


The center manifold theorem, examples

Example (A center manifold need not be unique)
x = x 2 , y = -y


The center manifold theorem, examples

Example (A center manifold in general has only finite smoothness)
x = xz - x 3 , y = y + x 4 , z = 0


Center manifold reduction

Consider an ODE x = Ax + O (|x |2 ), x R
n

with the right hand side of smoothness C 2 . Assume that the matrix A has ns , nu and nc eigenvalues in the left complex half-plane, right complex half-plane and on imaginary axis respectively, ns + nu + nc = n.

Theorem ( Center manifold reduction: Pliss-Kelley-Hirsch-Pugh-Shub)
In a neighborhood of the coordinate origin this ODE is topologically equivalent to the direct product of restriction of this equation to the center manifold and the "standard saddle": = w (), W c , = - , Rns , = , R
nu

The Tailor expansion for a center manifold can be computed by the method of undetermined coefficients.


Center manifold reduction, continued
Consider an ODE x = Ax + O (|x |2 ), x R
n

with the right hand side of smoothness C r , r > 2. Assume that the matrix A is block-diagonal with blocks B and C , where B is nc â nc -matrix with all eigenvalues on imaginary axis and B is ns â ns -matrix with all eigenvalues in the left complex half-plane, nc + ns = n.


Center manifold reduction, continued
Consider an ODE x = Ax + O (|x |2 ), x R
n

with the right hand side of smoothness C r , r > 2. Assume that the matrix A is block-diagonal with blocks B and C , where B is nc â nc -matrix with all eigenvalues on imaginary axis and B is ns â ns -matrix with all eigenvalues in the left complex half-plane, nc + ns = n.

Theorem ( Reduction near a center manifold)
In a neighborhood of the coordinate origin this ODE by a C r -1 -smooth transformation of variables x , which is C 1 -close to the identity near the origin the system can be reduced to the form where G Cr , F C
r -1

= =

B + G (), Rnc , (C + F (, )) , Rns ,

, G (0) = 0, G (0)/ = 0, F (0, 0) = 0.

Remark
The surface { = 0} is a center manifold.


Center manifold reduction, continued
Consider an ODE x = Ax + O (|x |2 ), x R
n

with the right hand side of smoothness C r , r > 2. Assume that the matrix A is block-diagonal with blocks B and C , where B is nc â nc -matrix with all eigenvalues on imaginary axis and B is ns â ns -matrix with all eigenvalues in the left complex half-plane, nc + ns = n.

Theorem ( Reduction near a center manifold)
In a neighborhood of the coordinate origin this ODE by a C r -1 -smooth transformation of variables x , which is C 1 -close to the identity near the origin the system can be reduced to the form where G Cr , F C
r -1

= =

B + G (), Rnc , (C + F (, )) , Rns ,

, G (0) = 0, G (0)/ = 0, F (0, 0) = 0.

Remark
The surface { = 0} is a center manifold. The Tailor expansion for the transformation x , can be computed by the methods of normal forms theory.


Center manifold reduction for systems with parameters
Consider an ODE (actually, k -parametric family of ODE's) x = v (x , ), v = A()x + O (|x |2 ), x Rn , R
k

with the right hand side of smoothness C 2 . Assume that the matrix A(0) has ns , nu and nc eigenvalues in the left complex half-plane, right complex half-plane and on imaginary axis respectively, ns + nu + nc = n. Consider the extended system x = v (x , ), = 0 This system has in a neighborhood of the origin of the coordinates (x , ) a center manifold of dimension nc + k .

Theorem ( Shoshitaishvili reduction principle)
In a neighborhood of the coordinates' origin this ODE is topologically equivalent to the direct product of restriction of this equation to the center manifold and the "standard saddle": = w (, ), Rnc , = 0, Rk , = - , Rns , = , R The homeomorphism which realizes equivalence does not change .
nu


Some definitions in bifurcation theory


Some definitions in bifurcation theory
The phase portrait of a dynamical system is a partitioning of the state space into orbits.


Some definitions in bifurcation theory
The phase portrait of a dynamical system is a partitioning of the state space into orbits. Consider a dynamical system that depends on parameters (actually, family of dynamical systems).


Some definitions in bifurcation theory
The phase portrait of a dynamical system is a partitioning of the state space into orbits. Consider a dynamical system that depends on parameters (actually, family of dynamical systems). The appearance of a topologically nonequivalent phase portraits under variation of parameters is called a bifurcation.


Some definitions in bifurcation theory
The phase portrait of a dynamical system is a partitioning of the state space into orbits. Consider a dynamical system that depends on parameters (actually, family of dynamical systems). The appearance of a topologically nonequivalent phase portraits under variation of parameters is called a bifurcation. A bifurcation diagram of the dynamical system is a stratification of its parameter space induced by the topological equivalence, together with representative phase portraits for each stratum.


Some definitions in bifurcation theory
The phase portrait of a dynamical system is a partitioning of the state space into orbits. Consider a dynamical system that depends on parameters (actually, family of dynamical systems). The appearance of a topologically nonequivalent phase portraits under variation of parameters is called a bifurcation. A bifurcation diagram of the dynamical system is a stratification of its parameter space induced by the topological equivalence, together with representative phase portraits for each stratum. A bifurcation boundary is a surface in the parameter space on which a bifurcation occurs.


Some definitions in bifurcation theory
The phase portrait of a dynamical system is a partitioning of the state space into orbits. Consider a dynamical system that depends on parameters (actually, family of dynamical systems). The appearance of a topologically nonequivalent phase portraits under variation of parameters is called a bifurcation. A bifurcation diagram of the dynamical system is a stratification of its parameter space induced by the topological equivalence, together with representative phase portraits for each stratum. A bifurcation boundary is a surface in the parameter space on which a bifurcation occurs. If family of dynamical systems is generic, then the codimension of a bifurcation is the difference between the dimension of the parameter space and the dimension of the corresponding bifurcation boundary.


Some definitions in bifurcation theory
The phase portrait of a dynamical system is a partitioning of the state space into orbits. Consider a dynamical system that depends on parameters (actually, family of dynamical systems). The appearance of a topologically nonequivalent phase portraits under variation of parameters is called a bifurcation. A bifurcation diagram of the dynamical system is a stratification of its parameter space induced by the topological equivalence, together with representative phase portraits for each stratum. A bifurcation boundary is a surface in the parameter space on which a bifurcation occurs. If family of dynamical systems is generic, then the codimension of a bifurcation is the difference between the dimension of the parameter space and the dimension of the corresponding bifurcation boundary. The codimension of the bifurcation of a given type is the minimal number of parameters of families in which that bifurcation type occurs. Equivalently, the codimension is the number of equality conditions that characterize a bifurcation.


Example: Saddle-node bifurcation
Example (Saddle-node bifurcation; also called fold, tangent, limit point, turning point bifurcation)
The saddle-node bifurcation is a local bifurcation which takes place in generic ODEs when at some value of a parameter there is an equilibrium with the eigenvalue 0. In this case as the parameter changes two equilibria collide and disappear.


Example: Saddle-node bifurcation
Example (Saddle-node bifurcation; also called fold, tangent, limit point, turning point bifurcation)
The saddle-node bifurcation is a local bifurcation which takes place in generic ODEs when at some value of a parameter there is an equilibrium with the eigenvalue 0. In this case as the parameter changes two equilibria collide and disappear. Consider a one-parametric family of ODEs x = ax 2 + + O (|x |3 + |x | + 2 + y 2 + . . .), y = -y + O (y 2 + || + x 2 + . . .) Here x R1 , y R1 , R1 , a = const = 0.


Example: Saddle-node bifurcation
Example (Saddle-node bifurcation; also called fold, tangent, limit point, turning point bifurcation)
The saddle-node bifurcation is a local bifurcation which takes place in generic ODEs when at some value of a parameter there is an equilibrium with the eigenvalue 0. In this case as the parameter changes two equilibria collide and disappear. Consider a one-parametric family of ODEs x = ax 2 + + O (|x |3 + |x | + 2 + y 2 + . . .), y = -y + O (y 2 + || + x 2 + . . .) Here x R1 , y R1 , R1 , a = const = 0.
2

The extended system is

x = ax + + O (. . .), y = -y + O (. . .), = 0


Example: Saddle-node bifurcation
Example (Saddle-node bifurcation; also called fold, tangent, limit point, turning point bifurcation)
The saddle-node bifurcation is a local bifurcation which takes place in generic ODEs when at some value of a parameter there is an equilibrium with the eigenvalue 0. In this case as the parameter changes two equilibria collide and disappear. Consider a one-parametric family of ODEs x = ax 2 + + O (|x |3 + |x | + 2 + y 2 + . . .), y = -y + O (y 2 + || + x 2 + . . .) Here x R1 , y R1 , R1 , a = const = 0.
2

The extended system is

x = ax + + O (. . .), y = -y + O (. . .), = 0 On the local center manifold for the extended system y = O (|| + x 2 ).


Example: Saddle-node bifurcation
Example (Saddle-node bifurcation; also called fold, tangent, limit point, turning point bifurcation)
The saddle-node bifurcation is a local bifurcation which takes place in generic ODEs when at some value of a parameter there is an equilibrium with the eigenvalue 0. In this case as the parameter changes two equilibria collide and disappear. Consider a one-parametric family of ODEs x = ax 2 + + O (|x |3 + |x | + 2 + y 2 + . . .), y = -y + O (y 2 + || + x 2 + . . .) Here x R1 , y R1 , R1 , a = const = 0.
2

The extended system is

x = ax + + O (. . .), y = -y + O (. . .), = 0 On the local center manifold for the extended system y = O (|| + x 2 ). reduced family is x = ax 2 + + O (|x |3 + |x | + 2 ), = 0 The


Example: Saddle-node bifurcation
Example (Saddle-node bifurcation; also called fold, tangent, limit point, turning point bifurcation)
The saddle-node bifurcation is a local bifurcation which takes place in generic ODEs when at some value of a parameter there is an equilibrium with the eigenvalue 0. In this case as the parameter changes two equilibria collide and disappear. Consider a one-parametric family of ODEs x = ax 2 + + O (|x |3 + |x | + 2 + y 2 + . . .), y = -y + O (y 2 + || + x 2 + . . .) Here x R1 , y R1 , R1 , a = const = 0.
2

The extended system is

x = ax + + O (. . .), y = -y + O (. . .), = 0 On the local center manifold for the extended system y = O (|| + x 2 ). reduced family is x = ax 2 + + O (|x |3 + |x | + 2 ), = 0 The truncated system is z = az 2 + , = 0 The


Example: Saddle-node bifurcation, continued
The phase portrait of the truncated system for a > 0 looks like this:


Example: Saddle-node bifurcation, continued
The phase portrait of the truncated system for a > 0 looks like this:

In the reduced on the central manifold family as the parameter groves and passes trough 0 two equilibria, stable and unstable ones, collide and disappear. One can drove the bifurcation diagram for this codimension 1 bifurcation:


Example: Saddle-node bifurcation, continued
The phase portrait of the truncated system for a > 0 looks like this:

In the reduced on the central manifold family as the parameter groves and passes trough 0 two equilibria, stable and unstable ones, collide and disappear. One can drove the bifurcation diagram for this codimension 1 bifurcation:

Such bifurcation diagram for reduced on a central manifold family typically app ears for bifurcation at which one eigenvalue vanishes and all other eigenvalues have non-zero real parts.


Example: Saddle-node bifurcation, continued
The phase portrait of the truncated system for a > 0 looks like this:

In the reduced on the central manifold family as the parameter groves and passes trough 0 two equilibria, stable and unstable ones, collide and disappear. One can drove the bifurcation diagram for this codimension 1 bifurcation:

Such bifurcation diagram for reduced on a central manifold family typically app ears for bifurcation at which one eigenvalue vanishes and all other eigenvalues have non-zero real parts. For the the original system the bifurcation diagram looks as follows:


Example: Saddle-node bifurcation, continued

Analogous bifurcation, also called the saddle-node bifurcation, takes place in generic ODEs when at some value of a parameter there is a periodic trajectory with the multiplier 1 (and in generic maps when at some value of a parameter there is a fixed point with the multiplier 1). As parameter changes two periodic trajectory (respectively, two fixed points) collide and disappear. The bifurcation diagram looks as follows (for periodic trajectories this is a picture on Poincar´ e section):


Example: Saddle-node bifurcation, continued

The bifurcation diagram for a planar system looks as follows:


Example of non-local bifurcation: The bifurcation of a limit cycle from the homoclinic loop of the saddle-node
Consider an ODE x = v (x , ), x Rn , R
k

Let for = 0 this equation have a fixed point with all eigenvalues in the left half-plane but one equal to 0 (a saddle-node).


Example of non-local bifurcation: The bifurcation of a limit cycle from the homoclinic loop of the saddle-node
Consider an ODE x = v (x , ), x Rn , R
k

Let for = 0 this equation have a fixed point with all eigenvalues in the left half-plane but one equal to 0 (a saddle-node). Assume that for = 0 there is a homoclinic trajectory to this saddle-node.


Example of non-local bifurcation: The bifurcation of a limit cycle from the homoclinic loop of the saddle-node
Consider an ODE x = v (x , ), x Rn , R
k

Let for = 0 this equation have a fixed point with all eigenvalues in the left half-plane but one equal to 0 (a saddle-node). Assume that for = 0 there is a homoclinic trajectory to this saddle-node. Under some generality assumptions the bifurcation diagram looks as follows (Andronov-Vitt-Leontovich-Shilnikov):

This is a codimension 1 bifurcation.


The bifurcation of a limit cycle from the homoclinic loop of the saddle-node, continued

The metamorphose of a phase portrait in 2D with such bifurcation may look like this (example from A.A.Andronov, A.A. Vitt, S.E. Khajkin, Theory of Oscillations, 1966):


NORMAL FORMS


Preliminary transformation: shift of the origin

Consider an ODE depending on parameters (actually, a family of ODEs) x = v (x , ), x D Rn , U Rk , v C 2 (D â U ) Let for = 0 this ODE has an equilibrium x = x0 . Therefore, x= v (x0 , 0 ) (x - x0 ) + O (|x - x0 |2 + | - 0 |) x

Assume that the equilibrium is non-degenerate, i.e. matrix A0 = v (x0x,0 ) is non-degenerate (does not have the eigenvalue 0). Then by the implicit function theorem for each value of close enough to 0 the equation has the equilibrium x = X () such that X (0 ) = x0 . Introduce x = x - X (). We get ~ the ODE whose equilibrium is x = 0 for all values of under consideration. ~


Preliminary transformation: shift of the origin

Consider an ODE depending on parameters (actually, a family of ODEs) x = v (x , ), x D Rn , U Rk , v C 2 (D â U ) Let for = 0 this ODE has an equilibrium x = x0 . Therefore, x= v (x0 , 0 ) (x - x0 ) + O (|x - x0 |2 + | - 0 |) x

Assume that the equilibrium is non-degenerate, i.e. matrix A0 = v (x0x,0 ) is non-degenerate (does not have the eigenvalue 0). Then by the implicit function theorem for each value of close enough to 0 the equation has the equilibrium x = X () such that X (0 ) = x0 . Introduce x = x - X (). We get ~ the ODE whose equilibrium is x = 0 for all values of under consideration. ~ In the following we will assume that there is no eigenvalue 0. So, without loss of generality we may assume that the equilibrium is at the coordinate origin.


Preliminary transformation: shift of the origin

Consider an ODE depending on parameters (actually, a family of ODEs) x = v (x , ), x D Rn , U Rk , v C 2 (D â U ) Let for = 0 this ODE has an equilibrium x = x0 . Therefore, x= v (x0 , 0 ) (x - x0 ) + O (|x - x0 |2 + | - 0 |) x

Assume that the equilibrium is non-degenerate, i.e. matrix A0 = v (x0x,0 ) is non-degenerate (does not have the eigenvalue 0). Then by the implicit function theorem for each value of close enough to 0 the equation has the equilibrium x = X () such that X (0 ) = x0 . Introduce x = x - X (). We get ~ the ODE whose equilibrium is x = 0 for all values of under consideration. ~ In the following we will assume that there is no eigenvalue 0. So, without loss of generality we may assume that the equilibrium is at the coordinate origin. Recall that if there is the eigenvalue 0, then typically there is saddle-node bifurcation of equilibria.


Preliminary transformation: shift of the origin, continued

Consider a map depending on parameters (actually, a family of maps) x P (x , ), x D Rn , U Rk , P C 2 (D â U ) Let for = 0 this map has a fixed point x = x0 . Therefore, x x0 + P (x0 , 0 ) (x - x0 ) + O (|x - x0 |2 + | - 0 |) x

Assume that the fixed point is non-degenerate, i.e. it does not have the x multiplier 1 (i.e. matrix A0 = P (0x,0 ) does not have the eigenvalue 1). Then by the implicit function theorem for each value of close enough to 0 the map has the fixed point x = X () such that X (0 ) = x0 . Introduce x = x - X (). We get the map whose fixed point is x = 0 for all values of ~ ~ under consideration.


Preliminary transformation: shift of the origin, continued

Consider a map depending on parameters (actually, a family of maps) x P (x , ), x D Rn , U Rk , P C 2 (D â U ) Let for = 0 this map has a fixed point x = x0 . Therefore, x x0 + P (x0 , 0 ) (x - x0 ) + O (|x - x0 |2 + | - 0 |) x

Assume that the fixed point is non-degenerate, i.e. it does not have the x multiplier 1 (i.e. matrix A0 = P (0x,0 ) does not have the eigenvalue 1). Then by the implicit function theorem for each value of close enough to 0 the map has the fixed point x = X () such that X (0 ) = x0 . Introduce x = x - X (). We get the map whose fixed point is x = 0 for all values of ~ ~ under consideration. In the following we will assume that there is no multiplier 1. So, without loss of generality we may assume that the fixed point is at the coordinate origin.


Preliminary transformation: shift of the origin, continued

Consider a map depending on parameters (actually, a family of maps) x P (x , ), x D Rn , U Rk , P C 2 (D â U ) Let for = 0 this map has a fixed point x = x0 . Therefore, x x0 + P (x0 , 0 ) (x - x0 ) + O (|x - x0 |2 + | - 0 |) x

Assume that the fixed point is non-degenerate, i.e. it does not have the x multiplier 1 (i.e. matrix A0 = P (0x,0 ) does not have the eigenvalue 1). Then by the implicit function theorem for each value of close enough to 0 the map has the fixed point x = X () such that X (0 ) = x0 . Introduce x = x - X (). We get the map whose fixed point is x = 0 for all values of ~ ~ under consideration. In the following we will assume that there is no multiplier 1. So, without loss of generality we may assume that the fixed point is at the coordinate origin. Recall that if there is the multiplier 1, then typically there is saddle-node bifurcation of fixed points.


Preliminary transformation: shift of the origin, continued
Consider an ODE depending on parameters (actually, a family of ODEs) x = v (x , ), x D Rn , U Rk , v C 2 (D â U ) Let for = 0 this ODE has a periodic trajectory.


Preliminary transformation: shift of the origin, continued
Consider an ODE depending on parameters (actually, a family of ODEs) x = v (x , ), x D Rn , U Rk , v C 2 (D â U ) Let for = 0 this ODE has a periodic trajectory.

In the normal coordinates near this trajectory the equation has the form dy = w (y , , ), w (0, , 0 ) 0, y R d
n -1

, S1


Preliminary transformation: shift of the origin, continued
Consider an ODE depending on parameters (actually, a family of ODEs) x = v (x , ), x D Rn , U Rk , v C 2 (D â U ) Let for = 0 this ODE has a periodic trajectory.

In the normal coordinates near this trajectory the equation has the form dy = w (y , , ), w (0, , 0 ) 0, y Rn-1 , S1 d The monodromy map for the section { = 0} has for = 0 the fixed point at y = 0. Assume that the periodic trajectory is non-degenerate, i.e. the fixed point does not have multiplier 1. Then for each value of close enough to 0 the map has the fixed point y = y () such that y (0 ) = 0. The equation has periodic solution Y (, ), S1 with the initial condition Y (0, ) = y ().


Preliminary transformation: shift of the origin, continued

Introduce y = y - Y (, ). We get the time-periodic ODE which has ~ equilibrium y = 0 for all values of under consideration. ~ In the following we will assume that there is no multiplier 1. So, without loss of generality we may assume that the system in normal coordinates has equilibrium at the coordinate origin.


Preliminary transformation: shift of the origin, continued

Introduce y = y - Y (, ). We get the time-periodic ODE which has ~ equilibrium y = 0 for all values of under consideration. ~ In the following we will assume that there is no multiplier 1. So, without loss of generality we may assume that the system in normal coordinates has equilibrium at the coordinate origin. If all multipliers are different, then according to Floquet-Lyapunov theory without loss of generality we may assume that the linearised near the equilibrium system has constant coefficients.


Bibliography
Arnold, V.I. Ordinary differential equations. Springer-Verlag, Berlin, 2006. Arnold, V. I. Geometrical methods in the theory of ordinary differential equations. Springer-Verlag, New York, 1988. Arnold, V. I. Mathematical methods of classical mechanics. Springer-Verlag, New York, 1989. Arrowsmith, D. , Place, C. An introduction to dynamical systems, Cambridge University Press, Cambridge, 1990. Dynamical systems. I, Encyclopaedia Math. Sci., 1, Springer-Verlag, Berlin, 1998. Guckenheimer, J. Holmes, P . Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer-Verlag, Berlin, 1990. Ilyashenko, Yu., Weigu Li, Nonlocal Bifurcations, AMS, 1999. Hartman, P. Ordinary differential equations. (SIAM), Philadelphia, PA, 2002. Kuznetsov, Yu.A. Elements of applied bifurcation theory. Springer-Verlag, New York, 2004. Shilnikov, L. P.; Shilnikov, A. L.; Turaev, D. V.; Chua, L. O. Methods of qualitative theory in nonlinear dynamics. Part I. World Scientific, Inc., River Edge, NJ, 1998.


LECTURE 11


NORMAL FORMS


Resonances near equilibria
Consider an ODE x = Ax + O (|x |2 ), x R
n

where A is a linear operator. Assume that right hand side of this ODE is analytic in some neighborhood of 0. Denote 1 , 2 , . . . , n the eigenvalues of A.

Definition
The set of eigenvalues of the operator A is called a resonant one if a relation of the form s = m1 1 + m2 2 + . . . + mn n , n P with integer non-negative m1 , m2 , . . . , mn such that mj 2 is satisfied. This
j =1

relation is called a resonance relation or just a resonance. The value n P |m| = mj is called an order of the resonance.
j =1

Denote (m, ) = m1 1 + m2 2 + . . . + mn n .

def


Resonances near equilibria
Consider an ODE x = Ax + O (|x |2 ), x R
n

where A is a linear operator. Assume that right hand side of this ODE is analytic in some neighborhood of 0. Denote 1 , 2 , . . . , n the eigenvalues of A.

Definition
The set of eigenvalues of the operator A is called a resonant one if a relation of the form s = m1 1 + m2 2 + . . . + mn n , n P with integer non-negative m1 , m2 , . . . , mn such that mj 2 is satisfied. This
j =1

relation is called a resonance relation or just a resonance. The value n P |m| = mj is called an order of the resonance.
j =1

Denote (m, ) = m1 1 + m2 2 + . . . + mn n .

def

Example
1 = 2 + 3 is the resonance of order 2. 21 = 32 is not a resonance. If 1 = -2 then there is infinite number of resonances s = s + k (1 + 2 ), k = 1, 2, 3, . . ..


Reduction to a linear system, the Poincar´ theorem e

Theorem (H. Poincar´) e
If eigenvalues of an equilibrium do not satisfy resonance relations up to an order N inclusively, then by a polynomial real close to the identical transformation of variables x = y + O (|y |2 ) the system is reducible to the form y = Ay + O (|y |N
+1

)

Corollary
If there are no resonances of any order, then a formal transformation of variables reduces original nonlinear system to the linear one.


Reduction to a linear system, the Poincar´ theorem e

Theorem (H. Poincar´) e
If eigenvalues of an equilibrium do not satisfy resonance relations up to an order N inclusively, then by a polynomial real close to the identical transformation of variables x = y + O (|y |2 ) the system is reducible to the form y = Ay + O (|y |N
+1

)

Corollary
If there are no resonances of any order, then a formal transformation of variables reduces original nonlinear system to the linear one. If all eigenvalues are situated in one complex half-plane, either in the left or in the right one, then the formal series for the transformation of variables converges in some neighborhood of 0. So, by an analytic transformation of variables the system is reducible to the linear one (H. Poincar´ e).


Proof of the Poincar´ theorem e
The system under consideration has the form x = Ax + V (x ), V (x ) = v2 (x ) + v3 (x ) + . . . + vN (x ) + O (|x |N where vr (x ) is the homogeneous vector polynomial of x of degree r .
+1

)


Proof of the Poincar´ theorem e
The system under consideration has the form x = Ax + V (x ), V (x ) = v2 (x ) + v3 (x ) + . . . + vN (x ) + O (|x |N where vr (x ) is the homogeneous vector polynomial of x of degree r . We are looking for a transformation of variables x y of the form x = y + h(y ), h(y ) = h2 (y ) + h3 (y ) + . . . + hN (y ) where hr (y ) is a homogeneous vector polynomial of y of degree r .
+1

)


Proof of the Poincar´ theorem e
The system under consideration has the form x = Ax + V (x ), V (x ) = v2 (x ) + v3 (x ) + . . . + vN (x ) + O (|x |N where vr (x ) is the homogeneous vector polynomial of x of degree r . We are looking for a transformation of variables x y of the form x = y + h(y ), h(y ) = h2 (y ) + h3 (y ) + . . . + hN (y ) where hr (y ) is a homogeneous vector polynomial of y of degree r . h y = A(y + h(y )) + V (y + h(y )) + O (|y |N y+ y
+1 +1

)

We have

)


Proof of the Poincar´ theorem e
The system under consideration has the form x = Ax + V (x ), V (x ) = v2 (x ) + v3 (x ) + . . . + vN (x ) + O (|x |N where vr (x ) is the homogeneous vector polynomial of x of degree r . We are looking for a transformation of variables x y of the form x = y + h(y ), h(y ) = h2 (y ) + h3 (y ) + . . . + hN (y ) where hr (y ) is a homogeneous vector polynomial of y of degree r . We have h y = A(y + h(y )) + V (y + h(y )) + O (|y |N +1 ) y+ y Assume that the transformation reduces the system to the required form. Equating terms of order r we get a homological equation (called also a co-homological equation) hr Ay - Ahr (y ) = Vr (y ) y where Vr is the homogeneous vector polynomial of degree r whose coefficients are expressed through coefficients of v2 , . . . , vr , h2 , . . . , hr -1 .
+1

)


Proof of the Poincar´ theorem e
The system under consideration has the form x = Ax + V (x ), V (x ) = v2 (x ) + v3 (x ) + . . . + vN (x ) + O (|x |N where vr (x ) is the homogeneous vector polynomial of x of degree r . We are looking for a transformation of variables x y of the form x = y + h(y ), h(y ) = h2 (y ) + h3 (y ) + . . . + hN (y ) where hr (y ) is a homogeneous vector polynomial of y of degree r . We have h y = A(y + h(y )) + V (y + h(y )) + O (|y |N +1 ) y+ y Assume that the transformation reduces the system to the required form. Equating terms of order r we get a homological equation (called also a co-homological equation) hr Ay - Ahr (y ) = Vr (y ) y where Vr is the homogeneous vector polynomial of degree r whose coefficients are expressed through coefficients of v2 , . . . , vr , h2 , . . . , hr -1 .
+1

)

Lemma
In absence of resonances of order r for any Vr the homological equation has a unique solution hr .


Proof of the Poincar´ theorem e
The system under consideration has the form x = Ax + V (x ), V (x ) = v2 (x ) + v3 (x ) + . . . + vN (x ) + O (|x |N where vr (x ) is the homogeneous vector polynomial of x of degree r . We are looking for a transformation of variables x y of the form x = y + h(y ), h(y ) = h2 (y ) + h3 (y ) + . . . + hN (y ) where hr (y ) is a homogeneous vector polynomial of y of degree r . We have h y = A(y + h(y )) + V (y + h(y )) + O (|y |N +1 ) y+ y Assume that the transformation reduces the system to the required form. Equating terms of order r we get a homological equation (called also a co-homological equation) hr Ay - Ahr (y ) = Vr (y ) y where Vr is the homogeneous vector polynomial of degree r whose coefficients are expressed through coefficients of v2 , . . . , vr , h2 , . . . , hr -1 .
+1

)

Lemma
In absence of resonances of order r for any Vr the homological equation has a unique solution hr . Induction in r completes the proof.


Proof of the Lemma about homological equation
To simplify the reasoning assume that eigenvalues of A are all different (the result is valid in the general case). The homological equation has the form h(y ) Ay - Ah(y ) = U (y ) y (Note that Here U (y ), Let e1 , e2 , . correspond Cn . Let y1 , ( h/ y )Ay - Ah is the commutator of the vector fields Ay and h.) h(y ) are the homogeneous vector polynomials of y of degree r . . . , en be eigenvectors of the complexified operator A, that to the eigenvalues 1 , 2 , . . . , n . The eigenvectors form a basis in y2 , . . . , yn be coordinates of y in this basis. Then X X U= Us ,m y m es , h = hs ,m y m es
s =1,...,n; |m|=r s =1,...,n; |m|=r

Here def def m m m m = (m1 , . . . , mn ) Zn , mi 0, |m| = m1 + . . . + mn , y m = y1 1 y2 2 . . . yn n . m Equating in the homological equation the coefficients in front of y es , we get (m1 1 + m2 2 + . . . + mn n - s )hs
,m

= Us

,m

Thus, hs ,m = Us ,m /((m, ) - s ). If y is real, then h(y ) is real. This completes the proof.


Resonant monomials, resonant normal form near equilibria
For simplicity of formulations assume that eigenvalues 1 , 2 , . . . , n of the operator A are all different. So, the the eigenvectors e1 , e2 , . . . , en of the complexified operator A form a basis in Cn .


Resonant monomials, resonant normal form near equilibria
For simplicity of formulations assume that eigenvalues 1 , 2 , . . . , n of the operator A are all different. So, the the eigenvectors e1 , e2 , . . . , en of the complexified operator A form a basis in Cn . Let some system S of resonance relations be given. We will assume that S contains all resonance relations which can be derived from any subsystem of S .


Resonant monomials, resonant normal form near equilibria
For simplicity of formulations assume that eigenvalues 1 , 2 , . . . , n of the operator A are all different. So, the the eigenvectors e1 , e2 , . . . , en of the complexified operator A form a basis in Cn . Let some system S of resonance relations be given. We will assume that S contains all resonance relations which can be derived from any subsystem of S .

Definition
A vector monomial x m es is called a resonant one for resonances in S if the resonance relation s = (m, ) is presented in the system S .


Resonant monomials, resonant normal form near equilibria
For simplicity of formulations assume that eigenvalues 1 , 2 , . . . , n of the operator A are all different. So, the the eigenvectors e1 , e2 , . . . , en of the complexified operator A form a basis in Cn . Let some system S of resonance relations be given. We will assume that S contains all resonance relations which can be derived from any subsystem of S .

Definition
A vector monomial x m es is called a resonant one for resonances in S if the resonance relation s = (m, ) is presented in the system S .

Example
If S includes relation 1 = 2 + 3 , then the vector monomial x2 x3 e1 is a resonant one. If S includes relation 1 = 21 + 2 , then all vector monomials (x1 x2 )k xs es are resonant ones.


Resonant monomials, resonant normal form near equilibria
For simplicity of formulations assume that eigenvalues 1 , 2 , . . . , n of the operator A are all different. So, the the eigenvectors e1 , e2 , . . . , en of the complexified operator A form a basis in Cn . Let some system S of resonance relations be given. We will assume that S contains all resonance relations which can be derived from any subsystem of S .

Definition
A vector monomial x m es is called a resonant one for resonances in S if the resonance relation s = (m, ) is presented in the system S .

Example
If S includes relation 1 = 2 + 3 , then the vector monomial x2 x3 e1 is a resonant one. If S includes relation 1 = 21 + 2 , then all vector monomials (x1 x2 )k xs es are resonant ones.

Definition
A system x = Ax + . . . is said to be in the resonant normal form for resonances from S if the nonlinear part of its right hand side is a sum of resonant vector monomials.


Reduction to resonant normal form, the Poincar´-Dulac theorem e
Theorem (H. Poincar´-H.Dulac) e
If eigenvalues of an equilibrium do not satisfy resonance relations up to an order N inclusively except, may be, resonances from S , then by a polynomial real close to the identical transformation of variables x = y + O (|y |2 ) the system is reducible to the form y = Ay + w (y ) + O (|y |N
+1

)

were w is a sum of resonant vector monomials of degrees not exceeding N .


Reduction to resonant normal form, the Poincar´-Dulac theorem e
Theorem (H. Poincar´-H.Dulac) e
If eigenvalues of an equilibrium do not satisfy resonance relations up to an order N inclusively except, may be, resonances from S , then by a polynomial real close to the identical transformation of variables x = y + O (|y |2 ) the system is reducible to the form y = Ay + w (y ) + O (|y |N
+1

)

were w is a sum of resonant vector monomials of degrees not exceeding N . Thus, the system without the term O (|y |N is in a resonant normal form.
+1

) (also called a truncated system )


Reduction to resonant normal form, the Poincar´-Dulac theorem e
Theorem (H. Poincar´-H.Dulac) e
If eigenvalues of an equilibrium do not satisfy resonance relations up to an order N inclusively except, may be, resonances from S , then by a polynomial real close to the identical transformation of variables x = y + O (|y |2 ) the system is reducible to the form y = Ay + w (y ) + O (|y |N
+1

)

were w is a sum of resonant vector monomials of degrees not exceeding N . Thus, the system without the term O (|y |N is in a resonant normal form.
+1

) (also called a truncated system )

Corollary
If there are no resonances of any order, except, may be, resonances from S , then a formal transformation of variables reduces original system to a system in a formal resonant normal form.


Reduction to resonant normal form, the Poincar´-Dulac theorem e
Theorem (H. Poincar´-H.Dulac) e
If eigenvalues of an equilibrium do not satisfy resonance relations up to an order N inclusively except, may be, resonances from S , then by a polynomial real close to the identical transformation of variables x = y + O (|y |2 ) the system is reducible to the form y = Ay + w (y ) + O (|y |N
+1

)

were w is a sum of resonant vector monomials of degrees not exceeding N . Thus, the system without the term O (|y |N is in a resonant normal form.
+1

) (also called a truncated system )

Corollary
If there are no resonances of any order, except, may be, resonances from S , then a formal transformation of variables reduces original system to a system in a formal resonant normal form.

Example
If n = 2 and the only possible resonance is 1 = 22 , then the system in formal 2 normal form is x1 = 1 x1 + cx2 , x2 = 2 x2 , c =const.


Proof of the Poincar´-Dulac theorem e
The system under consideration has the form x = Ax + V (x ), V (x ) = v2 (x ) + v3 (x ) + . . . + vN (x ) + O (|x |N where vr (x ) is the homogeneous vector polynomial of x of degree r .
+1

)


Proof of the Poincar´-Dulac theorem e
The system under consideration has the form x = Ax + V (x ), V (x ) = v2 (x ) + v3 (x ) + . . . + vN (x ) + O (|x |N where vr (x ) is the homogeneous vector polynomial of x of degree r . We are looking for a transformation of variables x y of the form x = y + h(y ), h(y ) = h2 (y ) + h3 (y ) + . . . + hN (y ) which reduces the system to the form y = Ay + w (y ) + O (|y |N
+1 +1

)

), w (y ) = w2 (y ) + w3 (y ) + . . . + wN (y )

where hr (y ), wr (y ) are homogeneous vector polynomials of y of degree r , and wr (y ) contains only resonant monomials.


Proof of the Poincar´-Dulac theorem e
The system under consideration has the form x = Ax + V (x ), V (x ) = v2 (x ) + v3 (x ) + . . . + vN (x ) + O (|x |N where vr (x ) is the homogeneous vector polynomial of x of degree r . We are looking for a transformation of variables x y of the form x = y + h(y ), h(y ) = h2 (y ) + h3 (y ) + . . . + hN (y ) which reduces the system to the form y = Ay + w (y ) + O (|y |N
+1 +1

)

), w (y ) = w2 (y ) + w3 (y ) + . . . + wN (y )

where hr (y ), wr (y ) are homogeneous vector polynomials of y of degree r , and wr (y ) contains only resonant monomials. We have y+ h y = A(y + h(y )) + V (y + h(y )) + O (|y |N y
+1

)


Proof of the Poincar´-Dulac theorem e
The system under consideration has the form x = Ax + V (x ), V (x ) = v2 (x ) + v3 (x ) + . . . + vN (x ) + O (|x |N where vr (x ) is the homogeneous vector polynomial of x of degree r . We are looking for a transformation of variables x y of the form x = y + h(y ), h(y ) = h2 (y ) + h3 (y ) + . . . + hN (y ) which reduces the system to the form y = Ay + w (y ) + O (|y |N
+1 +1

)

), w (y ) = w2 (y ) + w3 (y ) + . . . + wN (y )

where hr (y ), wr (y ) are homogeneous vector polynomials of y of degree r , and wr (y ) contains only resonant monomials. We have y+ h y = A(y + h(y )) + V (y + h(y )) + O (|y |N y
+1

)

Assume that the transformation reduces the system to the required form. Equating terms of order r we get a homological equation hr Ay - Ahr (y ) = Vr (y ) - wr (y ) y where Vr is the homogeneous vector polynomial of degree r whose coefficients are expressed through coefficients of v2 , . . . , vr , h2 , . . . , hr -1 , w2 , . . . , wr -1 .


Proof of the Poincar´-Dulac theorem e
The system under consideration has the form x = Ax + V (x ), V (x ) = v2 (x ) + v3 (x ) + . . . + vN (x ) + O (|x |N where vr (x ) is the homogeneous vector polynomial of x of degree r . We are looking for a transformation of variables x y of the form x = y + h(y ), h(y ) = h2 (y ) + h3 (y ) + . . . + hN (y ) which reduces the system to the form y = Ay + w (y ) + O (|y |N
+1 +1

)

), w (y ) = w2 (y ) + w3 (y ) + . . . + wN (y )

where hr (y ), wr (y ) are homogeneous vector polynomials of y of degree r , and wr (y ) contains only resonant monomials. We have y+ h y = A(y + h(y )) + V (y + h(y )) + O (|y |N y
+1

)

Assume that the transformation reduces the system to the required form. Equating terms of order r we get a homological equation hr Ay - Ahr (y ) = Vr (y ) - wr (y ) y where Vr is the homogeneous vector polynomial of degree r whose coefficients are expressed through coefficients of v2 , . . . , vr , h2 , . . . , hr -1 , w2 , . . . , wr -1 . Take as wr (y ) the sum of resonant monomials in Vr (y ).


Proof of the Poincar´-Dulac theorem, continued e

Lemma
For this choice of wr the homological equation has a solution hr in the form of the sum of non-resonant monomials. The solution in such form is a unique.


Proof of the Poincar´-Dulac theorem, continued e

Lemma
For this choice of wr the homological equation has a solution hr in the form of the sum of non-resonant monomials. The solution in such form is a unique. Induction in r completes the proof of the theorem.


Proof of the Poincar´-Dulac theorem, continued e

Lemma
For this choice of wr the homological equation has a solution hr in the form of the sum of non-resonant monomials. The solution in such form is a unique. Induction in r completes the proof of the theorem.

Proof of the Lemma about the homological equation.
The solution is constructed by the method of undetermined coefficients exactly as in the proof of the Poincar´ theorem. Denominators in the formulas do not e vanish because the right hand side of the homological equation does not contain resonant monomials.


Exercises

Exercises
1. Check that the vector field ( h/ y )Ay - Ah is the commutator of the vector fields Ay and h 2. The operator ( (·)/ y )Ay - A(·) is a linear operator in the space of homogeneous vector polynomials of any given degree. Find eigenvalues of this operator. 3. Find formal normal form for a system of 3 equations in the case of the resonance 1 = 2 + 3 . 4.Prove that the phase flow of any system in normal form for resonances in S commutes with the phase flow of its linear part provided that all resonance relation from S are indeed satisfied.


LECTURE 12


NORMAL FORMS


Resonances near equilibria
Consider an ODE x = Ax + O (|x |2 ), x R
n

where A is a linear operator. Assume that right hand side of this ODE is analytic in some neighborhood of 0. Denote 1 , 2 , . . . , n the eigenvalues of A.

Definition
The set of eigenvalues of the operator A is called a resonant one if a relation of the form s = m1 1 + m2 2 + . . . + mn n , n P with integer non-negative m1 , m2 , . . . , mn such that mj 2 is satisfied. This
j =1

relation is called a resonance relation or just a resonance. The value n P |m| = mj is called an order of the resonance.
j =1

Denote (m, ) = m1 1 + m2 2 + . . . + mn n .

def

Example
1 = 2 + 3 is the resonance of order 2. 21 = 32 is not a resonance. If 1 = -2 then there is infinite number of resonances s = s + k (1 + 2 ), k = 1, 2, 3, . . ..


Resonant monomials, resonant normal form near equilibria
For simplicity of formulations assume that eigenvalues 1 , 2 , . . . , n of the operator A are all different. So, the the eigenvectors e1 , e2 , . . . , en of the complexified operator A form a basis in Cn . Let some system S of resonance relations be given. We will assume that S contains all resonance relations which can be derived from any subsystem of S .

Definition
A vector monomial x m es is called a resonant one for resonances in S if the resonance relation s = (m, ) is presented in the system S .

Example
If S includes relation 1 = 2 + 3 , then the vector monomial x2 x3 e1 is a resonant one. If S includes relation 1 = 21 + 2 , then all vector monomials (x1 x2 )k xs es are resonant ones.

Definition
A system x = Ax + . . . is said to be in the resonant normal form for resonances from S if the nonlinear part of its right hand side is a sum of resonant vector monomials.


Reduction to resonant normal form, the Poincar´-Dulac theorem e
Theorem (H. Poincar´-H.Dulac) e
If eigenvalues of an equilibrium do not satisfy resonance relations up to an order N inclusively except, may be, resonances from S , then by a polynomial real close to the identical transformation of variables x = y + O (|y |2 ) the system is reducible to the form y = Ay + w (y ) + O (|y |N
+1

)

were w is a sum of resonant vector monomials of degrees not exceeding N . Thus, the system without the term O (|y |N is in a resonant normal form.
+1

) (also called a truncated system )

Corollary
If there are no resonances of any order, except, may be, resonances from S , then a formal transformation of variables reduces original system to a system in a formal resonant normal form.

Example
If n = 2 and the only possible resonance is 1 = 22 , then the system in formal 2 normal form is x1 = 1 x1 + cx2 , x2 = 2 x2 , c =const.


Example: normal form for Poincar´ Andronov-Hopf bifurcation eThe Poincar´ Andronov-Hopf bifurcation is a local bifurcation which takes eplace in generic ODEs when an equilibrium loses stability as a pair of complex conjugate eigenvalues cross the imaginary axis of the complex plane.


Example: normal form for Poincar´ Andronov-Hopf bifurcation eThe Poincar´ Andronov-Hopf bifurcation is a local bifurcation which takes eplace in generic ODEs when an equilibrium loses stability as a pair of complex conjugate eigenvalues cross the imaginary axis of the complex plane. Assume that 1,2 = ± i , where is small and = 0. Assume that all other eigenvalues have negative real parts.


Example: normal form for Poincar´ Andronov-Hopf bifurcation eThe Poincar´ Andronov-Hopf bifurcation is a local bifurcation which takes eplace in generic ODEs when an equilibrium loses stability as a pair of complex conjugate eigenvalues cross the imaginary axis of the complex plane. Assume that 1,2 = ± i , where is small and = 0. Assume that all other eigenvalues have negative real parts. Assume that possible resonances up to certain order N are only ones created by the relation 1 + 2 = 0.


Example: normal form for Poincar´ Andronov-Hopf bifurcation eThe Poincar´ Andronov-Hopf bifurcation is a local bifurcation which takes eplace in generic ODEs when an equilibrium loses stability as a pair of complex conjugate eigenvalues cross the imaginary axis of the complex plane. Assume that 1,2 = ± i , where is small and = 0. Assume that all other eigenvalues have negative real parts. Assume that possible resonances up to certain order N are only ones created by the relation 1 + 2 = 0. According to Poincar´ e-Dulac's theorem the system can be transformed to form z1 z2 zj j = = = = 1 z1 + (c0 (z1 z2 ) + c1 (z1 z2 )2 + . . . + cN 2 z2 + (c0 (z1 z2 ) + c1 (z1 z2 )2 + . . . + cN ¯ ¯ ¯
2 -2 -2

(z1 z2 )( (z1 z2 )(

N -1)/2 N -1)/2

)z1 + O (|z |N ¯ )z2 + O (|z |N

+1 +1

) ) )

j zj + (dj ,0 (z1 z2 ) + dj ,1 (z1 z2 ) + . . . + dj 3, 4, . . . , n

,N -2

(z1 z2 )

(N -1)/2

)zj + O (|z |

N +1


Example: normal form for Poincar´ Andronov-Hopf bifurcation eThe Poincar´ Andronov-Hopf bifurcation is a local bifurcation which takes eplace in generic ODEs when an equilibrium loses stability as a pair of complex conjugate eigenvalues cross the imaginary axis of the complex plane. Assume that 1,2 = ± i , where is small and = 0. Assume that all other eigenvalues have negative real parts. Assume that possible resonances up to certain order N are only ones created by the relation 1 + 2 = 0. According to Poincar´ e-Dulac's theorem the system can be transformed to form z1 z2 zj j = = = = 1 z1 + (c0 (z1 z2 ) + c1 (z1 z2 )2 + . . . + cN 2 z2 + (c0 (z1 z2 ) + c1 (z1 z2 )2 + . . . + cN ¯ ¯ ¯
2 -2 -2

(z1 z2 )( (z1 z2 )(

N -1)/2 N -1)/2

)z1 + O (|z |N ¯ )z2 + O (|z |N

+1 +1

) ) )

j zj + (dj ,0 (z1 z2 ) + dj ,1 (z1 z2 ) + . . . + dj 3, 4, . . . , n

,N -2

(z1 z2 )

(N -1)/2

)zj + O (|z |

N +1

The center manifold is approximated by the plane of variables z1 , z2 .


Example: normal form for Poincar´ Andronov-Hopf bifurcation eThe Poincar´ Andronov-Hopf bifurcation is a local bifurcation which takes eplace in generic ODEs when an equilibrium loses stability as a pair of complex conjugate eigenvalues cross the imaginary axis of the complex plane. Assume that 1,2 = ± i , where is small and = 0. Assume that all other eigenvalues have negative real parts. Assume that possible resonances up to certain order N are only ones created by the relation 1 + 2 = 0. According to Poincar´ e-Dulac's theorem the system can be transformed to form z1 z2 zj j = = = = 1 z1 + (c0 (z1 z2 ) + c1 (z1 z2 )2 + . . . + cN 2 z2 + (c0 (z1 z2 ) + c1 (z1 z2 )2 + . . . + cN ¯ ¯ ¯
2 -2 -2

(z1 z2 )( (z1 z2 )(

N -1)/2 N -1)/2

)z1 + O (|z |N ¯ )z2 + O (|z |N

+1 +1

) ) )

j zj + (dj ,0 (z1 z2 ) + dj ,1 (z1 z2 ) + . . . + dj 3, 4, . . . , n

,N -2

(z1 z2 )

(N -1)/2

)zj + O (|z |

N +1

The center manifold is approximated by the plane of variables z1 , z2 . For real initial data z2 = z1 along solutions. Denote z = z1 , = 1 . Truncated at ¯ the terms of the 3rd order equation for z is z = ( + c0 |z |2 )z


Example: normal form for Poincar´ Andronov-Hopf bifurcation eThe Poincar´ Andronov-Hopf bifurcation is a local bifurcation which takes eplace in generic ODEs when an equilibrium loses stability as a pair of complex conjugate eigenvalues cross the imaginary axis of the complex plane. Assume that 1,2 = ± i , where is small and = 0. Assume that all other eigenvalues have negative real parts. Assume that possible resonances up to certain order N are only ones created by the relation 1 + 2 = 0. According to Poincar´ e-Dulac's theorem the system can be transformed to form z1 z2 zj j = = = = 1 z1 + (c0 (z1 z2 ) + c1 (z1 z2 )2 + . . . + cN 2 z2 + (c0 (z1 z2 ) + c1 (z1 z2 )2 + . . . + cN ¯ ¯ ¯
2 -2 -2

(z1 z2 )( (z1 z2 )(

N -1)/2 N -1)/2

)z1 + O (|z |N ¯ )z2 + O (|z |N

+1 +1

) ) )

j zj + (dj ,0 (z1 z2 ) + dj ,1 (z1 z2 ) + . . . + dj 3, 4, . . . , n

,N -2

(z1 z2 )

(N -1)/2

)zj + O (|z |

N +1

The center manifold is approximated by the plane of variables z1 , z2 . For real initial data z2 = z1 along solutions. Denote z = z1 , = 1 . Truncated at ¯ the terms of the 3rd order equation for z is z = ( + c0 |z |2 )z Introduce polar coordinates r , : z = re i . We get equations: r = ( + ar 2 )r , = + br 2 , were a = Re c0 , b = Im c0


Example: normal form for Poincar´ Andronov-Hopf bifurcation, continued eThe bifurcation diagram for the case a < 0 (so called supercritical, or soft, or non-catastrophic bifurcation) looks as follows (image by Yuri Kuznetsov at Scholarpedia, = Re ):


Example: normal form for Poincar´ Andronov-Hopf bifurcation, continued eThe bifurcation diagram for the case a < 0 (so called supercritical, or soft, or non-catastrophic bifurcation) looks as follows (image by Yuri Kuznetsov at Scholarpedia, = Re ):

The bifurcation diagram for the case a > 0 (so called subcritical, or sharp, or catastrophic bifurcation) looks as follows (image by Yuri Kuznetsov at Scholarpedia, = Re ):


Example: normal form for Poincar´ Andronov-Hopf bifurcation, continued e The phase portraits for the extended system (we add equation = 0) looks as follows


Exercises

Exercises
1. Consider the system x = -y + x + xy , y = x + y + xy + x
2

The parameter grows and passing through the value = 0. For which values of parameters , , the stability loss of the equilibrium x = y = 0 will be a "soft" one?


Resonances near periodic trajectory

Consider an ODE x = Ax + V (x , t ), V (x , t + 2 ) = V (x , t ), V = O (|x |2 ), x R
n

where A is a linear operator. Assume that function V is analytic in some neighborhood of {0} â S1 . We use previous notation: j , j = 1, 2, . . . , n are eigenvalues of A, = (1 , 2 , . . . , n ) , m = (m1 , m2 , . . . , mn ), |m| = |m1 | + |m2 | + . . . + |mn |, (m, ) = m1 1 + m2 2 + . . . + mn n .

Definition
The set of eigenvalues of the operator A is called a resonant one if a relation of the form s = (m, ) + ik , is satisfied, where components of m are integer non-negative, |m| 2, k is integer. This relation is called a resonance relation or just a resonance. The value |m| is called an order of the resonance. Note that number of resonances of given order |m| is finite.


Resonant monomials, resonant normal form near periodic trajectory

Assume that eigenvalues 1 , 2 , . . . , n of the operator A are all different. So, the the eigenvectors e1 , e2 , . . . , en of the complexified operator A form a basis in Cn .


Resonant monomials, resonant normal form near periodic trajectory

Assume that eigenvalues 1 , 2 , . . . , n of the operator A are all different. So, the the eigenvectors e1 , e2 , . . . , en of the complexified operator A form a basis in Cn . Let some system S of resonance relations be given. We will assume that S contains all resonance relations which can be derived from any subsystem of S .


Resonant monomials, resonant normal form near periodic trajectory

Assume that eigenvalues 1 , 2 , . . . , n of the operator A are all different. So, the the eigenvectors e1 , e2 , . . . , en of the complexified operator A form a basis in Cn . Let some system S of resonance relations be given. We will assume that S contains all resonance relations which can be derived from any subsystem of S .

Definition
A vector monomial e ikt x m es is called a resonant one for resonances in S if the resonance relation s = (m, ) + ik is presented in the system S .


Resonant monomials, resonant normal form near periodic trajectory

Assume that eigenvalues 1 , 2 , . . . , n of the operator A are all different. So, the the eigenvectors e1 , e2 , . . . , en of the complexified operator A form a basis in Cn . Let some system S of resonance relations be given. We will assume that S contains all resonance relations which can be derived from any subsystem of S .

Definition
A vector monomial e ikt x m es is called a resonant one for resonances in S if the resonance relation s = (m, ) + ik is presented in the system S .

Definition
A system x = Ax + . . . is said to be in the resonant normal form for resonances from S if the nonlinear part of its right hand side is a sum of resonant vector monomials.


Reduction to resonant normal form near periodic trajectory
Theorem
If eigenvalues of an equilibrium of time-periodic system do not satisfy resonance relations up to an order N inclusively except, may be, resonances from S , then by a polynomial in space coordinates and periodic in time real close to the identical transformation of variables x = y + O (|y |2 ) the system is reducible to the form y = Ay + w (y , t ) + O (|y |N
+1

)

were w is a sum of resonant vector monomials of degrees not exceeding N .


Reduction to resonant normal form near periodic trajectory
Theorem
If eigenvalues of an equilibrium of time-periodic system do not satisfy resonance relations up to an order N inclusively except, may be, resonances from S , then by a polynomial in space coordinates and periodic in time real close to the identical transformation of variables x = y + O (|y |2 ) the system is reducible to the form y = Ay + w (y , t ) + O (|y |N
+1

)

were w is a sum of resonant vector monomials of degrees not exceeding N . Thus, the system without the term O (|y |N is in a resonant normal form.
+1

) (also called a truncated system )


Reduction to resonant normal form near periodic trajectory
Theorem
If eigenvalues of an equilibrium of time-periodic system do not satisfy resonance relations up to an order N inclusively except, may be, resonances from S , then by a polynomial in space coordinates and periodic in time real close to the identical transformation of variables x = y + O (|y |2 ) the system is reducible to the form y = Ay + w (y , t ) + O (|y |N
+1

)

were w is a sum of resonant vector monomials of degrees not exceeding N . Thus, the system without the term O (|y |N is in a resonant normal form.
+1

) (also called a truncated system )

Corollary
If there are no resonances of any order, except, may be, resonances from S , then a formal transformation of variables reduces the original system to a system in a formal resonant normal form.


Reduction to resonant normal form near periodic trajectory, continued
Procedure of reduction to resonant normal form near a periodic trajectory is analogous to that near an equilibrium The system under consideration has the form x = Ax + V (x , t ), V (x , t ) = v2 (x , t ) + v3 (x , t ) + . . . + vN (x , t ) + O (|x |N where vr (x , t ) is the homogeneous vector polynomial of x of degree r .
+1

)


Reduction to resonant normal form near periodic trajectory, continued
Procedure of reduction to resonant normal form near a periodic trajectory is analogous to that near an equilibrium The system under consideration has the form x = Ax + V (x , t ), V (x , t ) = v2 (x , t ) + v3 (x , t ) + . . . + vN (x , t ) + O (|x |N where vr (x , t ) is the homogeneous vector polynomial of x of degree r . We are looking for a transformation of variables x , t y , t of the form x = y + h(y , t ), h(y , t ) = h2 (y , t ) + h3 (y , t ) + . . . + hN (y , t ) which reduces the system to the form y = Ay + w (y , t ) + O (|y |N
+1 +1

)

), w (y , t ) = w2 (y , t ) + w3 (y , t ) + . . . + wN (y , t )

where hr (y , t ), wr (y , t ) are homogeneous vector polynomials of y of degree r , and wr (y , t ) contains only resonant monomials.


Reduction to resonant normal form near periodic trajectory, continued
Procedure of reduction to resonant normal form near a periodic trajectory is analogous to that near an equilibrium The system under consideration has the form x = Ax + V (x , t ), V (x , t ) = v2 (x , t ) + v3 (x , t ) + . . . + vN (x , t ) + O (|x |N where vr (x , t ) is the homogeneous vector polynomial of x of degree r . We are looking for a transformation of variables x , t y , t of the form x = y + h(y , t ), h(y , t ) = h2 (y , t ) + h3 (y , t ) + . . . + hN (y , t ) which reduces the system to the form y = Ay + w (y , t ) + O (|y |N
+1 +1

)

), w (y , t ) = w2 (y , t ) + w3 (y , t ) + . . . + wN (y , t )

where hr (y , t ), wr (y , t ) are homogeneous vector polynomials of y of degree r , and wr (y , t ) contains only resonant monomials. Plugging the transformation of variables into original differential equation, assuming that the transformed equation has required form and equating terms of order r we get a homological equation hr hr + Ay - Ahr (y , t ) = Vr (y , t ) - wr (y , t ) t y where Vr is the homogeneous vector polynomial of degree r whose coefficients are expressed through coefficients of v2 , . . . , vr , h2 , . . . , hr -1 , w2 , . . . , wr -1 .


Reduction to resonant normal form near periodic trajectory, continued
Procedure of reduction to resonant normal form near a periodic trajectory is analogous to that near an equilibrium The system under consideration has the form x = Ax + V (x , t ), V (x , t ) = v2 (x , t ) + v3 (x , t ) + . . . + vN (x , t ) + O (|x |N where vr (x , t ) is the homogeneous vector polynomial of x of degree r . We are looking for a transformation of variables x , t y , t of the form x = y + h(y , t ), h(y , t ) = h2 (y , t ) + h3 (y , t ) + . . . + hN (y , t ) which reduces the system to the form y = Ay + w (y , t ) + O (|y |N
+1 +1

)

), w (y , t ) = w2 (y , t ) + w3 (y , t ) + . . . + wN (y , t )

where hr (y , t ), wr (y , t ) are homogeneous vector polynomials of y of degree r , and wr (y , t ) contains only resonant monomials. Plugging the transformation of variables into original differential equation, assuming that the transformed equation has required form and equating terms of order r we get a homological equation hr hr + Ay - Ahr (y , t ) = Vr (y , t ) - wr (y , t ) t y where Vr is the homogeneous vector polynomial of degree r whose coefficients are expressed through coefficients of v2 , . . . , vr , h2 , . . . , hr -1 , w2 , . . . , wr -1 . Take as wr (y , t ) the sum of resonant monomials in Vr (y , t ).


Reduction to resonant normal form near periodic trajectory, continued
Lemma
For this choice of wr the homological equation has a solution hr in the form of a sum of non-resonant monomials. The solution in such form is a unique.

Proof.
Let e1 , e2 , . . . , en be eigenvectors of the complexified operator A, correspond to the eigenvalues 1 , 2 , . . . , n . The eigenvalues of different, and so the eigenvectors form a basis in Cn . Let y1 , y2 , . coordinates of y in this basis. Denote Ur (y , t ) = Vr (y , t ) - wr (y Ur = X
k Z; s =1,...,n; |m|=r

that A are all . . , yn be , t ). Then e ikt y m es

Uk

,s ,m

e ikt y m es , h =

X
k Z; s =1,...,n; |m|=r

hk

,s ,m

Equating in the homological equation the coefficients in front of e ik y m es , we get (ik + m1 1 + m2 2 + . . . + mn n - s )hs
,m

= Us

,m

Thus, hk ,s ,m = Uk ,s ,m /(ik + (m, ) - s ). If y is real, then h(y ) is real. This completes the proof.


Example: normal form for Neimark-Sacker bifurcation
The Neimark-Sacker bifurcation is a local bifurcation which takes place in generic ODEs when a periodic trajectory loses stability as a pair of complex conjugate multipliers cross the unit circle in the complex plane not close to points 1, -1, e ±i 2/3 , e ±i /2 .


Example: normal form for Neimark-Sacker bifurcation
The Neimark-Sacker bifurcation is a local bifurcation which takes place in generic ODEs when a periodic trajectory loses stability as a pair of complex conjugate multipliers cross the unit circle in the complex plane not close to points 1, -1, e ±i 2/3 , e ±i /2 . Consider the case n = 2. General case reduces to this one by means of Shoshitaisvili reducton principle. Assume that 1,2 = ± i , where is small and = 0.


Example: normal form for Neimark-Sacker bifurcation
The Neimark-Sacker bifurcation is a local bifurcation which takes place in generic ODEs when a periodic trajectory loses stability as a pair of complex conjugate multipliers cross the unit circle in the complex plane not close to points 1, -1, e ±i 2/3 , e ±i /2 . Consider the case n = 2. General case reduces to this one by means of Shoshitaisvili reducton principle. Assume that 1,2 = ± i , where is small and = 0. The resonance relation 1 = m1 1 + m2 2 + ik for 1,2 = ±i reduces to (m1 - m2 - 1) + k = 0 The correspoding multiplier is = e 2i . Enumerate possible resonances of the 2nd and 3rd order: m1 = 2, m2 = 0, = -k , = 1 m1 = 1, m2 = 1, = k , = 1 2 i m1 = 0, m2 = 2, 3 = k , = e 3 k m m m m
1 1 1 1

= = = =

3, 2, 1, 0,

m m m m

2 2 2 2

= = = =

0, 1, 2, 3,

2 = -k 0 = -k , 2 = k , 4 = k ,

, = e -i k = ±1 any = e 2i = e i k = ±1 i =e 2k


Example: normal form for Neimark-Sacker bifurcation
The Neimark-Sacker bifurcation is a local bifurcation which takes place in generic ODEs when a periodic trajectory loses stability as a pair of complex conjugate multipliers cross the unit circle in the complex plane not close to points 1, -1, e ±i 2/3 , e ±i /2 . Consider the case n = 2. General case reduces to this one by means of Shoshitaisvili reducton principle. Assume that 1,2 = ± i , where is small and = 0. The resonance relation 1 = m1 1 + m2 2 + ik for 1,2 = ±i reduces to (m1 - m2 - 1) + k = 0 The correspoding multiplier is = e 2i . Enumerate possible resonances of the 2nd and 3rd order: m1 = 2, m2 = 0, = -k , = 1 m1 = 1, m2 = 1, = k , = 1 2 i m1 = 0, m2 = 2, 3 = k , = e 3 k m1 = 3, m2 = 0, m1 = 2, m2 = 1, m1 = 1, m2 = 2, m1 = 0, m2 = 3, Our assumptions 2 = -k , = e -i k = ±1 0 = -k , any = e 2i 2 = k , = e i k = ±1 i 4 = k , = e 2 k about multipliers exclude all cases but m1 = 2, m2 = 1.


Example: normal form for Neimark-Sacker bifurcation, continued

So, the only possible resonance relation with |m| 3 is 1 = 21 + 2 , or 1 + 2 = 0 like for Poincar´ Andronov-Hopf bifurcation. e-


Example: normal form for Neimark-Sacker bifurcation, continued

So, the only possible resonance relation with |m| 3 is 1 = 21 + 2 , or 1 + 2 = 0 like for Poincar´ Andronov-Hopf bifurcation. eThe system can be transformed to the form ¯ z1 = 1 z1 + c0 (z1 z2 )z1 + O (|z |4 ), z2 = 2 z2 + c0 (z1 z2 )z2 + O (|z |4 ) ¯


Example: normal form for Neimark-Sacker bifurcation, continued

So, the only possible resonance relation with |m| 3 is 1 = 21 + 2 , or 1 + 2 = 0 like for Poincar´ Andronov-Hopf bifurcation. eThe system can be transformed to the form ¯ z1 = 1 z1 + c0 (z1 z2 )z1 + O (|z |4 ), z2 = 2 z2 + c0 (z1 z2 )z2 + O (|z |4 ) ¯ For real initial data along solutions z2 = z1 . Denote z = z1 , = 1 . Truncated ¯ at the terms of the 3rd order equation for z is z = ( + c0 |z |2 )z


Example: normal form for Neimark-Sacker bifurcation, continued

So, the only possible resonance relation with |m| 3 is 1 = 21 + 2 , or 1 + 2 = 0 like for Poincar´ Andronov-Hopf bifurcation. eThe system can be transformed to the form ¯ z1 = 1 z1 + c0 (z1 z2 )z1 + O (|z |4 ), z2 = 2 z2 + c0 (z1 z2 )z2 + O (|z |4 ) ¯ For real initial data along solutions z2 = z1 . Denote z = z1 , = 1 . Truncated ¯ at the terms of the 3rd order equation for z is z = ( + c0 |z |2 )z Introduce polar coordinates r , : z = re i . We get equations: r = ( + ar 2 )r , = + br 2 , were a = Re c , b = Im c


Example: normal form for Neimark-Sacker bifurcation, continued

So, the only possible resonance relation with |m| 3 is 1 = 21 + 2 , or 1 + 2 = 0 like for Poincar´ Andronov-Hopf bifurcation. eThe system can be transformed to the form ¯ z1 = 1 z1 + c0 (z1 z2 )z1 + O (|z |4 ), z2 = 2 z2 + c0 (z1 z2 )z2 + O (|z |4 ) ¯ For real initial data along solutions z2 = z1 . Denote z = z1 , = 1 . Truncated ¯ at the terms of the 3rd order equation for z is z = ( + c0 |z |2 )z Introduce polar coordinates r , : z = re i . We get equations: r = ( + ar 2 )r , = + br 2 , were a = Re c , b = Im c The bifurcation diagram for truncated equation is exactly the same as for the Poincar´ ndronov-Hopf bifurcation. The bifurcation for a < 0 is called e-A supercritical, or soft, or non-catastrophic bifurcation), while for a > 0 it is called subcritical, or sharp, or catastrophic bifurcation. In the original system a two-dimensional invariant torus branches off the periodic solution either at > 0 ( supercritical case) or at < 0 ( subcritical case).


Example: normal form for Neimark-Sacker bifurcation, continued
The bifurcation diagrams look like this: Supercritical case:


Example: normal form for Neimark-Sacker bifurcation, continued
The bifurcation diagrams look like this: Supercritical case:

Subcritical case:


Example: normal form for Neimark-Sacker bifurcation, continued
There is nevertheless essential difference in behavior of trajectories in exact and truncated system. The Poincar´ section for exact system may look like this. e


Example: normal form for Neimark-Sacker bifurcation, continued
There is nevertheless essential difference in behavior of trajectories in exact and truncated system. The Poincar´ section for exact system may look like this. e

In the original system as a parameter changes on the invariant torus appear an disappear isolated periodic trajectories. Invariant torus in general has only finite smoothness.


On stability loss of periodic trajectory near the resonance
What happens if a pair of complex-conjugated multipliers , cross the unit ¯ 2 i circle near the points e ± 3 ?


On stability loss of periodic trajectory near the resonance
What happens if a pair of complex-conjugated multipliers , cross the unit ¯ 2 i circle near the points e ± 3 ? This is described by the following bifurcation diagram ("clock-face"), and the similar diagram with the reverse of all arrows, V.I.Arnold, Geometrical methods in the theory of ordinary differential equations", = ln - 2i . 3 A two-parametric bifurcation diagram is needed here.


On stability loss of periodic trajectory near the resonance
What happens if a pair of complex-conjugated multipliers , cross the unit ¯ i circle near the points e ± 2 ?


On stability loss of periodic trajectory near the resonance
What happens if a pair of complex-conjugated multipliers , cross the unit ¯ i circle near the points e ± 2 ? Complete answer is not known. Here is one of scenarios, V.I.Arnold, Geometrical methods in the theory of ordinary differential equations", = ln - i . 2


LECTURE 13


NORMAL FORMS


Resonances near periodic trajectory

Consider an ODE x = Ax + V (x , t ), V (x , t + 2 ) = V (x , t ), V = O (|x |2 ), x R
n

where A is a linear operator. Assume that function V is analytic in some neighborhood of {0} â S1 . We use previous notation: j , j = 1, 2, . . . , n are eigenvalues of A, = (1 , 2 , . . . , n ) , m = (m1 , m2 , . . . , mn ), |m| = |m1 | + |m2 | + . . . + |mn |, (m, ) = m1 1 + m2 2 + . . . + mn n .

Definition
The set of eigenvalues of the operator A is called a resonant one if a relation of the form s = (m, ) + ik , is satisfied, where components of m are integer non-negative, |m| 2, k is integer. This relation is called a resonance relation or just a resonance. The value |m| is called an order of the resonance. Note that number of resonances of given order |m| is finite.


Resonant monomials, resonant normal form near periodic trajectory

Assume that eigenvalues 1 , 2 , . . . , n of the operator A are all different. So, the the eigenvectors e1 , e2 , . . . , en of the complexified operator A form a basis in Cn . Let some system S of resonance relations be given. We will assume that S contains all resonance relations which can be derived from any subsystem of S .

Definition
A vector monomial e ikt x m es is called a resonant one for resonances in S if the resonance relation s = (m, ) + ik is presented in the system S .

Definition
A system x = Ax + . . . is said to be in the resonant normal form for resonances from S if the nonlinear part of its right hand side is a sum of resonant vector monomials.


Reduction to resonant normal form near periodic trajectory
Theorem
If eigenvalues of an equilibrium of time-periodic system do not satisfy resonance relations up to an order N inclusively except, may be, resonances from S , then by a polynomial in space coordinates and periodic in time real close to the identical transformation of variables x = y + O (|y |2 ) the system is reducible to the form y = Ay + w (y , t ) + O (|y |N
+1

)

were w is a sum of resonant vector monomials of degrees not exceeding N . Thus, the system without the term O (|y |N is in a resonant normal form.
+1

) (also called a truncated system )

Corollary
If there are no resonances of any order, except, may be, resonances from S , then a formal transformation of variables reduces the original system to a system in a formal resonant normal form.


Example: normal form for Neimark-Sacker bifurcation
The Neimark-Sacker bifurcation is a local bifurcation which takes place in generic ODEs when a periodic trajectory loses stability as a pair of complex conjugate multipliers cross the unit circle in the complex plane not close to points 1, -1, e ±i 2/3 , e ±i /2 . Consider the case n = 2. General case reduces to this one by means of Shoshitaisvili reducton principle. Assume that 1,2 = ± i , where is small and = 0. The only possible resonance relation with |m| 3 is 1 = 21 + 2 , or 1 + 2 = 0 like for Poincar´ Andronov-Hopf bifurcation. eThe system in a polar coordinates can be transformed to the form r = ( + ar 2 )r , = + br 2 , were a = Re c , b = Im c The bifurcation diagram for truncated equation is exactly the same as for the Poincar´ ndronov-Hopf bifurcation. The bifurcation for a < 0 is called e-A supercritical, or soft, or non-catastrophic bifurcation), while for a > 0 it is called subcritical, or sharp, or catastrophic bifurcation. In the original system a two-dimensional invariant torus branches off the periodic solution either at > 0 ( supercritical case) or at < 0 ( subcritical case).


Example: normal form for Neimark-Sacker bifurcation, continued
The bifurcation diagrams look like this: Supercritical case:

Subcritical case:


Example: normal form for Neimark-Sacker bifurcation, continued
There is nevertheless essential difference in behavior of trajectories in exact and truncated system. The Poincar´ section for exact system may look like this. e

In the original system as a parameter changes on the invariant torus appear an disappear isolated periodic trajectories.


On stability loss of periodic trajectory near the resonance 1:3
What happens if a pair of complex-conjugated multipliers , cross the unit ¯ 2 i circle near the points e ± 3 ? This is described by the following bifurcation diagram ("clock-face"), and the similar diagram with the reverse of all arrows, V.I.Arnold, Geometrical methods in the theory of ordinary differential equations", = ln - 2i . 3 A two-parametric bifurcation diagram is needed here.


On stability loss of periodic trajectory near the resonance 1:4
What happens if a pair of complex-conjugated multipliers , cross the unit ¯ i circle near the points e ± 2 ? Complete answer is not known. Here is one of scenarios, V.I.Arnold, Geometrical methods in the theory of ordinary differential equations", = ln - i . 2


On stability loss of periodic trajectory near the resonance 1 : q , q 5
What happens if a pair of complex-conjugated multipliers , cross the unit ¯ ± 2 ki q , q 5, k and q are co-prime? circle near the points e The bifurcation diagrams are all similar, here is the diagram for q = 5, k = 1, V.I.Arnold, Geometrical methods in the theory of ordinary differential equations", = ln - 2i . 5


On period-doubling bifurcation for periodic trajectory
The period-doubling bifurcation is a local bifurcation which takes place in generic ODEs and maps when a periodic trajectory (or a fixed point, for a map) loses stability as a real multiplier crosses the unit circle in the complex plane in the point -1 .


On period-doubling bifurcation for periodic trajectory
The period-doubling bifurcation is a local bifurcation which takes place in generic ODEs and maps when a periodic trajectory (or a fixed point, for a map) loses stability as a real multiplier crosses the unit circle in the complex plane in the point -1 . For the analysis of this bifurcation in ODEs we will use a normal form for a Poincar´ return map. So, the detailed analysis is postponed till the section e about maps.


On period-doubling bifurcation for periodic trajectory
The period-doubling bifurcation is a local bifurcation which takes place in generic ODEs and maps when a periodic trajectory (or a fixed point, for a map) loses stability as a real multiplier crosses the unit circle in the complex plane in the point -1 . For the analysis of this bifurcation in ODEs we will use a normal form for a Poincar´ return map. So, the detailed analysis is postponed till the section e about maps. Here is the bifurcation diagram for ODEs for the supercritical case, = -1 + .


On period-doubling cascade and Feigenbaum's universality
There were observed many cases when in a one parametric family of systems there is an infinite sequence of period-doublings.


On period-doubling cascade and Feigenbaum's universality
There were observed many cases when in a one parametric family of systems there is an infinite sequence of period-doublings. Let be a parameter of the family. For (1 , 2 ) there is a stable periodic trajectory of a period T . At = 2 a real multiplier of this trajectory passes through -1, the trajectory loses its stability, and a new stable periodic trajectory of the period 2T branches off. This trajectory remains stable for (2 , 3 ). At = 3 a real multiplier of this trajectory passes through -1, the trajectory loses its stablity, and a new stable periodic trajectory of the period 4T branches off, and so on. For (n , n+1 ) there is a a stable periodic trajectory of the period 2n T . The sequence {n } has a limit as n .


On period-doubling cascade and Feigenbaum's universality
There were observed many cases when in a one parametric family of systems there is an infinite sequence of period-doublings. Let be a parameter of the family. For (1 , 2 ) there is a stable periodic trajectory of a period T . At = 2 a real multiplier of this trajectory passes through -1, the trajectory loses its stability, and a new stable periodic trajectory of the period 2T branches off. This trajectory remains stable for (2 , 3 ). At = 3 a real multiplier of this trajectory passes through -1, the trajectory loses its stablity, and a new stable periodic trajectory of the period 4T branches off, and so on. For (n , n+1 ) there is a a stable periodic trajectory of the period 2n T . The sequence {n } has a limit as n . Moreover, the distance between successive moments of bifurcation decay about as in a geometric progression with universal common ratio: k - k -1 lim = µF = 4.6692 . . . k k +1 - k


On period-doubling cascade and Feigenbaum's universality
There were observed many cases when in a one parametric family of systems there is an infinite sequence of period-doublings. Let be a parameter of the family. For (1 , 2 ) there is a stable periodic trajectory of a period T . At = 2 a real multiplier of this trajectory passes through -1, the trajectory loses its stability, and a new stable periodic trajectory of the period 2T branches off. This trajectory remains stable for (2 , 3 ). At = 3 a real multiplier of this trajectory passes through -1, the trajectory loses its stablity, and a new stable periodic trajectory of the period 4T branches off, and so on. For (n , n+1 ) there is a a stable periodic trajectory of the period 2n T . The sequence {n } has a limit as n . Moreover, the distance between successive moments of bifurcation decay about as in a geometric progression with universal common ratio: k - k -1 lim = µF = 4.6692 . . . k k +1 - k The discussion of explanation of this phenomenon is postponed till the section about period doubling for maps. This phenomenon is called Feigenbaum's universality in period-doubling cascade. The constant µF is called the Feigenbaum constant. It is a new mathematical constant like e or .


Resonances near fixed points of maps

Consider a map x Ax + V (x ), V = O (|x |2 ), x R
n

where A is a linear operator. Assume that V is analytic in some neighborhood of 0. We use the notation: j , j = 1, 2, . . . , n are eigenvalues of A (multipliers of fixed point 0), = (1 , 2 , . . . , n ) , m = (m1 , m2 , . . . , mn ), |m| = |m1 | + |m2 | + . . . + |mn |, m = m1 m2 · · · mn . n 1 2

Definition
The set of eigenvalues of the operator A is called a resonant one if a relation of the form s = m , is satisfied, where components of m are integer non-negative, |m| 2. This relation is called a resonance relation or just a resonance. The value |m| is called an order of the resonance.


Resonant monomials, resonant normal form near fixed point

Assume that eigenvalues 1 , 2 , . . . , n of the operator A are all different. So, the the eigenvectors e1 , e2 , . . . , en of the complexified operator A form a basis in Cn . Let some system S of resonance relations be given. We will assume that S contains all resonance relations which can be derived from any subsystem of S .

Definition
A vector monomial x m es is called a resonant one for resonances in S if the resonance relation s = m is presented in the system S .

Definition
A map x Ax + . . . is said to be in the resonant normal form for resonances from S if the nonlinear part of its right hand side is a sum of resonant vector monomials.


Reduction to resonant normal form near fixed point

Theorem
If multipliers of a fixed point do not satisfy resonance relations up to an order N inclusively except, may be, resonances from S , then by a polynomial real close to the identical transformation of variables x = y + O (|y |2 ) the system is reducible to the form y Ay + w (y ) + O (|y |N
+1

)

were w is a sum of resonant vector monomials of degrees not exceeding N . Thus, the system without the term O (|y |N is in a resonant normal form.
+1

) (also called a truncated system )

Corollary
If there are no resonances of any order, except, may be, resonances from S , then a formal transformation of variables reduces the original system to a system in a formal resonant normal form.


Reduction to resonant normal form near fixed point
Procedure of reduction to resonant normal form near fixed point is analogous to that near an equilibrium The map under consideration has the form x Ax + V (x ), V (x ) = v2 (x ) + v3 (x ) + . . . + vN (x ) + O (|x |N
+1

)

where vr (x ) is the homogeneous vector polynomial of x of degree r .


Reduction to resonant normal form near fixed point
Procedure of reduction to resonant normal form near fixed point is analogous to that near an equilibrium The map under consideration has the form x Ax + V (x ), V (x ) = v2 (x ) + v3 (x ) + . . . + vN (x ) + O (|x |N
+1

)

where vr (x ) is the homogeneous vector polynomial of x of degree r . We are looking for a transformation of variables x y of the form x = y + h(y ), h(y ) = h2 (y ) + h3 (y ) + . . . + hN (y ) which reduces the map to the form y Ay + w (y ) + O (|y |N
+1

), w (y ) = w2 (y ) + w3 (y ) + . . . + wN (y )

where hr (y ), wr (y ) are homogeneous vector polynomials of y of degree r , and wr (y ) contains only resonant monomials.


Reduction to resonant normal form near fixed point
Procedure of reduction to resonant normal form near fixed point is analogous to that near an equilibrium The map under consideration has the form x Ax + V (x ), V (x ) = v2 (x ) + v3 (x ) + . . . + vN (x ) + O (|x |N
+1

)

where vr (x ) is the homogeneous vector polynomial of x of degree r . We are looking for a transformation of variables x y of the form x = y + h(y ), h(y ) = h2 (y ) + h3 (y ) + . . . + hN (y ) which reduces the map to the form y Ay + w (y ) + O (|y |N
+1

), w (y ) = w2 (y ) + w3 (y ) + . . . + wN (y )

where hr (y ), wr (y ) are homogeneous vector polynomials of y of degree r , and wr (y ) contains only resonant monomials. Plugging the transformation of variables into original map, assuming that the transformed map has required form and equating terms of order r we get a homological equation hr (Ay ) - Ahr (y ) = Vr (y ) - wr (y ) where Vr is the homogeneous vector polynomial of degree r whose coefficients are expressed through coefficients of v2 , . . . , vr , h2 , . . . , hr -1 , w2 , . . . , wr -1 .


Reduction to resonant normal form near fixed point
Procedure of reduction to resonant normal form near fixed point is analogous to that near an equilibrium The map under consideration has the form x Ax + V (x ), V (x ) = v2 (x ) + v3 (x ) + . . . + vN (x ) + O (|x |N
+1

)

where vr (x ) is the homogeneous vector polynomial of x of degree r . We are looking for a transformation of variables x y of the form x = y + h(y ), h(y ) = h2 (y ) + h3 (y ) + . . . + hN (y ) which reduces the map to the form y Ay + w (y ) + O (|y |N
+1

), w (y ) = w2 (y ) + w3 (y ) + . . . + wN (y )

where hr (y ), wr (y ) are homogeneous vector polynomials of y of degree r , and wr (y ) contains only resonant monomials. Plugging the transformation of variables into original map, assuming that the transformed map has required form and equating terms of order r we get a homological equation hr (Ay ) - Ahr (y ) = Vr (y ) - wr (y ) where Vr is the homogeneous vector polynomial of degree r whose coefficients are expressed through coefficients of v2 , . . . , vr , h2 , . . . , hr -1 , w2 , . . . , wr -1 . Take as wr (y , t ) the sum of resonant monomials in Vr (y , t ).


Reduction to resonant normal form near near fixed point, continued

Lemma
For this choice of wr the homological equation has a solution hr in the form of a sum of non-resonant monomials. The solution in such form is a unique.

Proof.
Let e1 , e2 , . . . , en be eigenvectors of the complexified operator A, that correspond to the eigenvalues 1 , 2 , . . . , n . The eigenvalues of A are all different, and so the eigenvectors form a basis in Cn . Let y1 , y2 , . . . , yn be coordinates of y in this basis. Denote Ur (y ) = Vr (y ) - wr (y ). Then X X Ur = Us ,m y m es , h = hs ,m y m es
s =1,...,n; |m|=r s =1,...,n; |m|=r

Equating in the homological equation the coefficients in front of y m es , we get (m - s )hs Thus, hs proof.
,m ,m

= Us

,m

= Us ,m /(m - s ). If y is real, then h(y ) is real. This completes the