# 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

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

with ${\displaystyle x(0)=0}$. It defines a flow: ${\displaystyle \phi _{t}:R^{n}\rightarrow R^{n}}$ defined by ${\displaystyle \phi _{t}(x_{0})=x(t,x_{0})}$.

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

eql

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

The linearized flow obeys:

${\displaystyle D\Phi _{t}({\bar {x}})\xi =e^{tDf({\bar {x}})}\xi }$

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

When ${\displaystyle Df({\bar {x}})}$ has no eigen values with zero real part, ${\displaystyle {\bar {x}}}$ 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:

${\displaystyle \left({\begin{array}{c}{\dot {x}}_{1}\\{\dot {x}}_{2}\end{array}}\right)=\left({\begin{array}{cc}0&1\\1&0\end{array}}\right)\left({\begin{array}{c}x_{1}\\x_{2}\end{array}}\right)-\epsilon \left({\begin{array}{c}0\\x_{1}^{2}x_{2}\end{array}}\right)}$

Eigenvalues of the linear part are ${\displaystyle \pm i}$. If ${\displaystyle \epsilon >0}$: a spiral sink, if ${\displaystyle \epsilon <0}$: a repelling source, if ${\displaystyle \epsilon =0}$ a center (hamiltonian system).

## Stable and unstable manifolds

Definition:

The local stable and unstable manifolds ${\displaystyle W_{loc}^{s}}$ and ${\displaystyle W_{loc}^{u}}$ of a fixed point ${\displaystyle x^{*}}$ are

${\displaystyle W_{loc}^{s}=\{x\in U\|\phi _{t}(x)\rightarrow x^{*}{\mbox{ as }}t\rightarrow +\infty ,{\mbox{ and }}\phi _{t}(x)\in U{\mbox{ for all }}t\geq 0\}}$

${\displaystyle W_{loc}^{u}=\{x\in U\|\phi _{t}(x)\rightarrow x^{*}{\mbox{ as }}t\rightarrow -\infty ,{\mbox{ and }}\phi _{t}(x)\in U{\mbox{ for all }}t\leq 0\}}$

where ${\displaystyle U}$ is a neighborhood of the fixed point ${\displaystyle X^{*}}$.

Theorem:

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

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

eqalphchoose

${\displaystyle x_{\alpha }=x^{*}+\alpha e_{u}.}$

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

The number ${\displaystyle \alpha }$ in equation eqalphchoose has to be small enough for the linear approximation to be accurate. Typically, to choose ${\displaystyle \alpha }$ one compares the distant between the images of ${\displaystyle x_{\alpha }}$ given by the linearized dynamics and the exact dynamics. If it is too large, then ${\displaystyle \alpha }$ 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 ${\displaystyle U=0}$ as one sided Poincare section. (The 'side' of the section is here defined by ${\displaystyle U}$ 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 ${\displaystyle H(X)=P(X)-X}$ where ${\displaystyle P(X)}$ is the Poincare map associated to our system which can be written as :

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

${\displaystyle X(0)=X_{0}}$

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

eqnewton

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

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

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

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

${\displaystyle X(0)=X_{0}}$

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

${\displaystyle \Phi ^{0}=Id}$

where ${\displaystyle DF_{X,\epsilon }}$ is the Jacobian of ${\displaystyle F_{\epsilon }}$ in ${\displaystyle X}$, and ${\displaystyle X_{0}}$ 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 ${\displaystyle 0.003}$.

We have the relation:

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

where ${\displaystyle T}$ 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 ${\displaystyle \Phi ^{T}}$ which is one.