Connecting orbits in highdimensional dynamical systems
Fronts and pulses (traveling waves) in parabolic systems are an important class of localized structures which transport information in diffusive media. They have numerous applications in transport models of chemical and biological systems (see e.g. the reaction-diffusion equations, the models of nerve conduction and of population dynamics in [Murray 89]).
Nonlinear reaction functions are important for the form and velocity of the wave.
For their numerical computation one has to solve boundary value problems on the whole real line or in cylindric domains. Such a traveling wave solution can be viewed as an orbit that connects two steady states of an appropriate first order differential equation.
The spectra of the boundary value problem linearized at the wave profile determine it's stability.
Apart from the application to traveling waves, connecting orbits can be found as solutions of parabolic systems,
e.g. homoclinic orbits as bifurcation points of branches of periodic orbits
or heteroclinic orbits as parts of the global attractor.
For these orbits we investigate the effects of semi-discretizations (Galerkin or finite elements in space),
continous in time.
In this way one obtains high-dimensional dynamical systems for which
connecting orbits are computed using appropriate projection boundary conditions.