Odd and Even Functions in the Design Problem of New Chaotic Attractors

Abstract

Let [Formula: see text] be a chaotic attractor generated by a quadratic system of ordinary differential equations [Formula: see text]. A method for constructing new chaotic attractors based on the attractor [Formula: see text] is proposed. The idea of the method is to replace the state vector [Formula: see text] located on the right side of the original system with new vector [Formula: see text]; where [Formula: see text], [Formula: see text], and [Formula: see text] are odd power functions; [Formula: see text]. (In other words, a state feedback [Formula: see text] is introduced into the right side of the system under study: [Formula: see text].) As a result, the newly obtained system generates new chaotic attractors, which are topologically not equivalent (generally speaking) to the attractor [Formula: see text]. In addition, for an antisymmetric neural ODE system with a homoclinic orbit connected at a saddle point, the conditions for the occurrence of chaotic dynamics are found.