Introduction to Mathematical Physics/Some mathematical problems and their solution/Particular trajectories and geometry in space phase

Fixed points and Hartman theoremEdit

Consider the following initial value problem:

eqnl

 

with  . It defines a flow:   defined by  .


By Linearization around a fixed point such that  :


eql

 

The linearized flow obeys:

 


It is natural to ask the following question: What can we say about the solutions of eqnl based on our knowledge of eql?

Theorem:

If   has no zero or purely imaginary eigenvalues, then there is a homeomorphism   defined on some neighborhood   of   in   locally taking orbits of the nonlinear flow   to those of the linear flow  . The homeomorphism preserves the sense of orbits and can be chosen to preserve parametrization by time.

When   has no eigen values with zero real part,   is called a hyperbolic or nondegenerate fixed point and the asymptotic behaviour near it is determined by the linearization.

In the degenerate case, stability cannot be determined by linearization.

Consider for example:

 

Eigenvalues of the linear part are  . If  : a spiral sink, if  : a repelling source, if   a center (hamiltonian system).

Stable and unstable manifoldsEdit

Definition:

The local stable and unstable manifolds   and   of a fixed point   are

 

 

where   is a neighborhood of the fixed point  .

Theorem:

(Stable manifold theorem for a fixed point). Let   be a hyperbolic fixed point. There exist local stable and unstable manifold   and   of the same dimesnion   and   as those of the eigenspaces  , and   of the linearized system, and tangent to   and   at  .   and   are as smooth as the function  .

An algorithm to get unstable and stable manifolds is given in ([#References|references]). It basically consists in finding an point   sufficiently close to the fixed point  , belonging to an unstable linear eigenvector space:

eqalphchoose

 

For continuous time system, to draw the unstable manifold, one has just to integrate forward in time from  . For discrete time system, one has to integrate forward in time the dynamics for points in the segment   where   is the application.

The number   in equation eqalphchoose has to be small enough for the linear approximation to be accurate. Typically, to choose   one compares the distant between the images of   given by the linearized dynamics and the exact dynamics. If it is too large, then   is divided by 2. The process is iterated untill an acceptable accuracy is reached.

Periodic orbitsEdit

It is well known ([#References|references]) that there exist periodic (unstable) orbits in a chaotic system. We will first detect some of them. A periodic orbit in the 3-D phase space corresponds to a fixed point of the Poincar\'e map.

The method we choosed to locate periodic orbits is "the Poincare map" method ([#References|references]). It uses the fact that periodic orbits correspond to fixed points of Poincare maps. We chose the plane   as one sided Poincare section. (The 'side' of the section is here defined by   becoming positive)

Let us recall the main steps in locating periodic orbits by using the Poincare map method : we apply the Newton-Raphson algorithm to the application   where   is the Poincare map associated to our system which can be written as :

 

 

where   denotes the set of the control parameters. Namely, the Newton-Raphson algorithm is here:

eqnewton

 

where   is the Jacobian of the Poincare map   evaluated in  .


The jacobian of poincare map   needed in the scheme of equation eqnewton is computed via the integration of the dynamical system:

 

 

 

 


where   is the Jacobian of   in  , and   is a Point of the Poincare section. We chose a Runge--Kutta scheme, fourth order ([#References|references]) for the time integration of the whole previous system. The time step was  .

We have the relation:

 

where   is the time needed at which the trajectory crosses le Poincare section again.

Remark:

Note that a good test for the accuracy of the integration is to check that on a periodic orbit, there is one eigenvalue of   which is one.