Construction of Higher-Dimensional Hyperchaotic Systems with a Maximum Number of Positive Lyapunov Exponents under Average Eigenvalue Criteria

Abstract

Over the last 40 years, the design of [Formula: see text]-dimensional hyperchaotic systems with a maximum number ([Formula: see text]) of positive Lyapunov exponents has been an open problem for research. Nowadays it is not difficult to design [Formula: see text]-dimensional hyperchaotic systems with less than ([Formula: see text]) positive Lyapunov exponents, but it is still extremely difficult to design an [Formula: see text]-dimensional hyperchaotic system with the maximum number ([Formula: see text]) of positive Lyapunov exponents. This paper aims to resolve this challenging problem by developing a chaotification approach using average eigenvalue criteria. The approach consists of four steps: (i) a globally bounded controlled system is designed based on an asymptotically stable nominal system with a uniformly bounded controller; (ii) a closed-loop pole assignment technique is utilized to ensure that the numbers of eigenvalues with positive real parts of the controlled system be equal to ([Formula: see text]) and ([Formula: see text]), respectively, at two saddle-focus equilibrium points; (iii) the number of average eigenvalues with positive real parts is ensured to be equal to ([Formula: see text]) for the controlled system over a given control period; (iv) the smallest value of the positive real parts of the average eigenvalues is ensured to be greater than a given threshold value. Finally, the paper is closed with some typical examples which illustrate the feasibility and performance of the proposed design methodology.