Designing Hyperchaotic Systems With <newline/>Any Desired Number of Positive Lyapunov <newline/>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 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n ~~(n \geq 5)$</tex></formula> 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 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex></formula> -dimensional dissipative hyperchaotic system is given by <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$N= {\rm round}((n-1)/2)$</tex></formula> , <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$n \geq 5$</tex></formula> . 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.