It is natural to ask the following question:
What can we say about the solutions of eqnl based on our
knowledge of eql?
has no zero or purely imaginary eigenvalues, then there is a homeomorphism
defined on some neighborhood
locally taking orbits of the nonlinear flow
to those of the linear flow
The homeomorphism preserves the sense of orbits and can be chosen to preserve parametrization by time.
has no eigen values with zero real part,
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
Consider for example:
Eigenvalues of the linear part are .
If : a spiral sink, if : a repelling source,
if a center (hamiltonian system).
The local stable and unstable manifolds and of
a fixed point are
where is a neighborhood of the fixed point .
(Stable manifold theorem for a fixed point). Let be a hyperbolic
fixed point. There exist local stable and unstable manifold
and of the same dimesnion and as
those of the eigenspaces , and of the linearized system,
and tangent to and at . and
are as smooth as the function .
An algorithm to get unstable and stable manifolds is given in
It basically consists in finding an point sufficiently
close to the fixed
point , 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
For discrete time system, one has to integrate forward in time the
dynamics for points in the segment
where is the application.
in equation eqalphchoose has to be small
enough for the linear approximation to be accurate. Typically, to choose
one compares the distant between the images of
given by the linearized dynamics and the exact dynamics. If
it is too large, then
is divided by 2. The process is iterated
untill an acceptable accuracy is reached.
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 as one sided Poincare section.
(The 'side' of the section is here defined by becoming
Let us recall the main steps in locating periodic orbits
by using the Poincare map method :
the Newton-Raphson algorithm to the application
is the Poincare map
associated to our system which can be written as :
where denotes the set of the control
Namely, the Newton-Raphson algorithm is here:
is the Jacobian of the Poincare map
The jacobian of poincare map
needed in the scheme
of equation eqnewton is computed via
the integration of
the dynamical system:
is the Jacobian of
is a Point of the Poincare section.
We chose a Runge--Kutta scheme, fourth order
for the time integration of the whole previous system.
The time step was .
We have the relation:
is the time needed at which the trajectory
crosses le Poincare section again.
Note that a good test for the accuracy of the integration
is to check that on a periodic orbit, there is one eigenvalue of
which is one.