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

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Fixed points and Hartman theorem[edit]

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[edit]

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[edit]

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.