5-D Hyperchaotic and Chaotic Systems with Non-hyperbolic Equilibria and Many Equilibria

Abstract

In the present decade, chaotic systems are used and appeared in many fields like in information security, communication systems, economics, bioengineering, mathematics, etc. Thus, developing of chaotic dynamical systems is most interesting and desirable in comparison with dynamical systems with regular behaviour. The chaotic systems are categorised into two groups. These are (i) system with self-excited attractors and (ii) systems with hidden attractors. A self-excited attractor is generated depending on the location of its unstable equilibrium point and in such case, the basin of attraction touches the equilibria. But, in the case of hidden attractors, the basin of attraction does not touch the equilibria and also finding of such attractors is a difficult task. The systems with (i) no equilibrium point and (ii) stable equilibrium points belong to the category of hidden attractors. Recently chaotic systems with infinitely many equilibria/a line of equilibria are also considered under the cattegory of hidden attractors. Higher dimensional chaotic systems have more complexity and disorders compared with lower dimensional chaotic systems. Recently, more attention is given to the development of higher dimensional chaotic systems with hidden attractors. But, the development of higher dimensional chaotic systems having both hidden attractors and self-excited attractors is more demanding. This chapter reports three hyperchaotic and two chaotic, 5-D new systems having the nature of both the self-excited and hidden attractors. The systems have non-hyperbolic equilibria, hence, belong to the category of self-excited attractors. Also, the systems have many equilibria, and hence, may be considered under the category of a chaotic system with hidden attractors. A systematic procedure is used to develop the new systems from the well-known 3-D Lorenz chaotic system. All the five systems exhibit multistability with the change of initial conditions. Various theoretical and numerical tools like phase portrait, Lyapunov spectrum, bifurcation diagram, Poincare map, and frequency spectrum are used to confirm the chaotic nature of the new systems. The MATLAB simulation results of the new systems are validated by designing their circuits and realising the same.