Stabilizing a homoclinic stripe

Homoclinic Stripe Patterns

In this paper, we study homoclinic stripe patterns in the two-dimensional generalized Gierer– Meinhardt equation, where we interpret this equation as a prototypical representative of a class of singularly perturbed monostable reaction-diffusion equations. The structure of a stripe pattern is essentially one-dimensional; therefore, we can use results from the literature to establish the existenc...

A homoclinic hierarchy

Homoclinic bifurcations in autonomous ordinaty differential equations provide useful organizing centres for the analysis of examples. There are four generic types of homoclinic bifurcation, depending on the dominant eigenvalues of the Jacobian matrix of the flow near a stationary point. A family of differential equations is presented which, for suitable choices of parameters, can exhibit each o...

Tessellation of a stripe.

This paper describes enumeration of a class of topologically distinct periodic divisions of a stripe. Optimization of the geometry of these periodic tilings--optimization that yields minimum total perimeter of the tiles--gives a set of physically plausible periodic structures of monodisperse, two-dimensional foams bounded by two parallel walls. Evaluation of the minimum total perimeters of the ...

Homoclinic Bifurcations

1. Introduction. We say that a one-parameter family of diffeomorphisms ip^: M — • M, p G R, has a homoclinic bifurcation, or a homoclinic tangency, for p = 0 if ipo has an orbit of nontransverse intersection of a stable and an unstable manifold, both of the same hyperbolic fixed point (or periodic point), which splits, for p > 0, into two orbits of transverse intersection of these stable and un...

Homoclinic Points