# 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:

eqnl

$\dot x=f(x), x\in R^n$

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 $f(\bar x)=0$:

eql

$\dot \xi=Df(\bar x)\xi, \xi\in R^n$

The linearized flow obeys:

$D\Phi_t(\bar x)\xi=e^{tDf(\bar x)}\xi$

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 $Df(\bar x)$ has no zero or purely imaginary eigenvalues, then there is a homeomorphism $h$ defined on some neighborhood $U$ of $\bar x$ in $R^n$ locally taking orbits of the nonlinear flow $\phi_t$ to those of the linear flow $e^{tDf(\bar x)}$. The homeomorphism preserves the sense of orbits and can be chosen to preserve parametrization by time.

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:

$\left( \begin{array}{c} \dot x_1\\ \dot x_2 \end{array} \right) = \left( \begin{array}{cc} 0&1\\ 1&0 \end{array} \right) \left( \begin{array}{c} x_1\\ x_2 \end{array} \right) - \epsilon \left( \begin{array}{c} 0\\ x_1^2x_2 \end{array} \right)$

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

Definition:

The local stable and unstable manifolds $W^s_{loc}$ and $W^u_{loc}$ of a fixed point $x^*$ are

$W^s_{loc}=\{x\in U \| \phi_t(x)\rightarrow x^* \mbox{ as } t\rightarrow +\infty, \mbox{ and } \phi_t(x)\in U \mbox{ for all } t\geq 0\}$

$W^u_{loc}=\{x\in U \| \phi_t(x)\rightarrow x^* \mbox{ as } t\rightarrow -\infty, \mbox{ and } \phi_t(x)\in U \mbox{ for all } t\leq 0\}$

where $U$ is a neighborhood of the fixed point $X^*$.

Theorem:

(Stable manifold theorem for a fixed point). Let $x^*$ be a hyperbolic fixed point. There exist local stable and unstable manifold $W^s_{loc}$ and $W^u_{loc}$ of the same dimesnion $n_s$ and $n_u$ as those of the eigenspaces $E^s$, and $E^u$ of the linearized system, and tangent to $E^u$ and $E^s$ at $x^*$. $W^s_{loc}$ and $W^u_{loc}$ are as smooth as the function $f$.

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:

eqalphchoose

$x_\alpha=x^*+\alpha e_u.$

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 :

$\frac{dX}{dt}=F_\epsilon(X)$

$X(0)=X_0$

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

eqnewton

$X^{k+1}=X^k-(DP_{X^k}-I)^{-1}(P(X^k)-X^k)$

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:

$\frac{dX}{dt}=F_\epsilon(X)$

$X(0)=X_0$

$\frac{d\Phi^t}{dt}=DF_{X,\epsilon}.\Phi^t$

$\Phi^0=Id$

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:

$DP_X=\left( I-\frac{F(P(X)).h^+}{F(P(X))^+.h}\right)\Phi^T$

where $T$ 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 $\Phi^T$ which is one.