Constructing Higher-Dimensional Nondegenerate Hyperchaotic Systems with Multiple Controllers

Abstract

This paper proposes a new approach for constructing higher-dimensional nondegenerate hyperchaotic system with multiple controllers. Here, the so-called higher-dimensional nondegenerate hyperchaotic system means that it can be provided with a maximum number of positive Lyapunov exponents, which has been an open problem for research in recent years. The details of design are given by three steps as follows: (i) Design an [Formula: see text]-dimensional nominal matrix and similarity transformation matrix, and get an asymptotic stable nominal system; (ii) Add a master controller for the nominal matrix and get the controlled system. Then, find suitable control positions such that the controlled system satisfies the average eigenvalue criterion, i.e. the number of average eigenvalues with positive real parts of all Jacobi matrices over a given period of time is equal to ([Formula: see text]), and the maximum value of average eigenvalues with positive real parts is greater than a given threshold [Formula: see text]; (iii) Add nonmaster controllers, and the control positions are fixed and parameters are given in advance. So it can generate nondegenerate hyperchaotic systems with ([Formula: see text]) positive Lyapunov exponents. Finally, two typical examples are given to show the feasibility and effectiveness of the proposed method.