Analysis and Control of a 4-D Novel Hyperchaotic System

Abstract

Hyperchaotic systems are defined as chaotic systems with more than one positive Lyapunov exponent. Combined with one null Lyapunov exponent along the flow and one negative Lyapunov exponent to ensure boundedness of the solution, the minimal dimension for a continuous hyperchaotic system is four. The hyperchaotic systems are known to have important applications in secure communications and cryptosystems. First, this work describes an eleven-term 4-D novel hyperchaotic system with four quadratic nonlinearities. The qualitative properties of the novel hyperchaotic system are described in detail. The Lyapunov exponents of the system are obtained as \( L_{1} = 0.7781,L_{2} = 0.2299,L_{3} = 0 \) and \( L_{4} = - 12.5062 \). The maximal Lyapunov exponent of the system (MLE) is \( L_{1} = 0.7781 \). The Lyapunov dimension of the novel hyperchaotic system is obtained as \( D_{L} = 3.0806 \). Next, the work describes an adaptive controller design for the global chaos control of the novel hyperchaotic system. The main result for the adaptive controller design has been proved using Lyapunov stability theory. MATLAB simulations are described in detail for all the main results derived in this work for the eleven-term 4-D novel hyperchaotic system with four quadratic nonlinearities.