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:

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

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 $W_{loc}^{s}$ and $W_{loc}^{u}$ of a fixed point $x^{*}$ are

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

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

where $U$ 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 $W_{loc}^{s}$ and $W_{loc}^{u}$ 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 $x^{*}$. $W_{loc}^{s}$ and $W_{loc}^{u}$ 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.