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

Fixed points and Hartman theorem
Consider the following initial value problem:

with $$x(0)=0$$. It defines a flow: $$\phi_t:R^n\rightarrow R^n$$ defined by $$\phi_t(x_0)=x(t,x_0)$$.

By Linearization around a fixed point such that $$f(\bar x)=0$$:

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?

When $$Df(\bar x)$$ has no eigen values with zero real part, $$\bar x$$ 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 $$\pm i$$. If $$\epsilon>0$$: a spiral sink, if $$\epsilon<0$$: a repelling source, if $$\epsilon=0$$ a center (hamiltonian system).

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

For continuous time system, to draw the unstable manifold, one has just to integrate forward in time from $$x_\alpha$$. For discrete time system, one has to integrate forward in time the dynamics for points in the segment $$\mathrel{]}\Phi^{-1}(x_\alpha),x_\alpha\mathrel{]}$$ where $$Phi$$ is the application.

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

Periodic orbits
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 $$U=0$$ as one sided Poincare section. (The 'side' of the section is here defined by $$U$$ 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 $$H(X)=P(X)-X$$ where $$P(X)$$ is the Poincare map associated to our system which can be written as :

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

where $$DP_{X^k}$$ is the Jacobian of the Poincare map $$P(X)$$ evaluated in $$X^k$$.

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

where $$DF_{X,\epsilon}$$ is the Jacobian of $$F_\epsilon$$ in $$X$$, and $$X_0$$ 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 $$0.003$$.

We have the relation:

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