Hyperchaos, adaptive control and synchronization of a novel 5-D hyperchaotic system with three positive Lyapunov exponents and its SPICE implementation

In this research work, a twelve-term novel 5-D hyperchaotic Lorenz system with three quadratic nonlinearities has been derived by adding a feedback control to a ten-term 4-D hyperchaotic Lorenz system (Jia, 2007) with three quadratic nonlinearities. The 4-D hyperchaotic Lorenz system (Jia, 2007) has the Lyapunov exponents L 1 = 0 . 3684 , L 2 = 0 . 2174 , L 3 = 0 and L 4 = − 12 . 9513, and the Kaplan-Yorke dimension of this 4-D system is found as D KY = 3 . 0452. The 5-D novel hyperchaotic Lorenz system proposed in this work has the Lyapunov exponents L 1 = 0 . 4195 , L 2 = 0 . 2430 , L 3 = 0 . 0145 , L 4 = 0 and L 5 = − 13 . 0405, and the Kaplan-Yorke dimension of this 5-D system is found as D KY = 4 . 0159. Thus, the novel 5-D hyperchaotic Lorenz system has a maximal Lyapunov exponent (MLE), which is greater than the maximal Lyapunov exponent (MLE) of the 4-D hyperchaotic Lorenz system. The 5-D novel hyperchaotic Lorenz system has a unique equilibrium point at the origin, which is a saddle-point and hence unstable. Next, an adaptive controller is designed to stabilize the novel 5-D hyperchaotic Lorenz system with unknown system parameters. Moreover, an adaptive controller is designed to achieve global hyperchaos synchronization of the identical novel 5-D hyperchaotic Lorenz systems with unknown system parameters. Finally, an electronic circuit realization of the novel 5-D hyperchaotic Lorenz system using SPICE is described in detail to confirm the feasibility of the theoretical model.