Designing Hyperchaotic Systems With Any Desired Number of Positive Lyapunov Exponents via A Simple Model

This paper introduces a new and unified approach for designing desirable dissipative hyperchaotic systems. Based on the anti-control principle of continuous-time systems, a nominal system of
$n ~~(n \geq 5)$
independent first-order linear differential equations are coupled through all state variables, making the controlled system be in a closed-loop cascade-coupling form, where each equation contains only two state variables therefore the system is quite simple. Based on this setting, a simple model for dissipative hyperchaotic systems is constructed, with an adjustable parameter which can ensure the dissipation of the system. In the closed-loop cascade-coupling form, it is shown that all the eigenvalues are symmetrically distributed in a circumferential manner. Consequently, a universal law is derived on the relationship of the number of positive Lyapunov exponents and the number of positive real parts of its Jacobian eigenvalues. For the above-mentioned simple model, the number of positive Lyapunov exponents for any
$n$
-dimensional dissipative hyperchaotic system is given by
$N= {\rm round}((n-1)/2)$
,
$n \geq 5$
. Therefore, in theory, the system can generate any desired number of positive Lyapunov exponents as long as the dimension of the system is sufficiently high. Thus, the proposed method provides a new approach for purposefully constructing desirable dissipative hyperchaotic systems. Finally, two examples are given to demonstrate the feasibility of the proposed design method.