DEPARTMENT OF MATHEMATICAL SCIENCES

APPLIED AND COMPUTATIONAL MATHEMATICS SEMINAR

**Speaker:** Stanislaus Maier-Paape, RWTH, Aachen, Germany

**Title: ***
Computer assisted proofs of heteroclinic connections
*

**Abstract:**
As the title suggests, we propose a method which allows to prove
with computer aid connecting orbits of dynamical systems generated
by an ODE.
We want to prove the existence of connecting orbits between two
hyperbolic equilibria $x^R$ (repeller) and
$x^A$ (attractor) of this ODE, whose Morse indices are $k$ and
$k-1$, respectively.
The main idea is to analytically investigate neighborhoods of the
equilibria (to obtain for instance good estimates on the location of
their invariant manifolds in form of box enclosures), and combine
this with a fast rigorous transport algorithm for the ODE (like e.g.
the CAPD software, http://capd.wsb-nlu.edu.pl).
We transport small neighborhoods of a part of the unstable manifold
of $x^R$ by means of an enclosing algorithm from CAPD into a
neighborhood of $x^A$. Comparing this transported enclosure with a
box enclosure of the stable manifold of $x^A$, we can state
conditions whose verification with the computer results in a proof
that both manifolds intersect, giving the desired connecting orbit.
Our method applies for the cases $k=1$, $k=2$ and by time reversal
for the cases $k = n$ and $k = n-1$ ($n=$ space dimension).
Although we clearly yet are not even close, we think this method
does have the potential to be applicable for all $k \in \{1, \ldots,
n\}$, if not for evolution equations in infinite dimensions.

**Time:** Friday, Sept. 19, 2008, 1:30-2:30 p.m.

**Place:** Science and Tech I, Room 242

Department of Mathematical Sciences

George Mason University

4400 University Drive, MS 3F2

Fairfax, VA 22030-4444

http://math.gmu.edu/

Tel. 703-993-1460, Fax. 703-993-1491