Universalities in the chaotic generalized Moore & Spiegel equations

Abstract

By investigating the topology of chaotic solutions to the generalized Moore and Spiegel equations, we address the question how the solutions of nonlinear dynamical systems are dependent on the nature of the nonlinearity. In these generalized jerk equations, the single nonlinear term has a parity which depends on unu̇. The system has an inversion symmetry when n is even and no symmetry property when n is odd. It is shown that the topology of chaotic solutions only depends on the parity of n, that is, on the symmetry properties and not on the degree of nonlinearity. The value of n only affects the possibility to develop the chaotic solution, that is, to increase the number of unstable periodic orbits within the attractor.