Homoclinic hyperchaos — an explicit example

Abstract

A four-variable, 2-domain piecewise-linear C 1, ordinary differential equation is indicated explicitly. Two parameter sets (and initial conditions) implying a homoclinic return to a generalized saddle focus are given with finite precision. The numerically larger eigenvalue is negative, the two positive eigenvalues — one for the complex-conjugate pair, the other real — are equal. Only under this singular condition does the extended neighborhood in parameter space in which topological hyperchaos survives include the homoclinic case itself. This does not diminish the power of the homoclinicity criterion. The second set of parameters and initial conditions is valid for arbitrarily high dimension numbers. It implies existence of maximum hyperchaos in an extended neighborhood. Analytical and numerical studies on both the C 1 and the C ∞ version of the equation are possible.