Chapter 1 - Dynamics, Circuit Design, Synchronization, and Fractional-Order Form of a No-Equilibrium Chaotic System

Abstract

Systems without equilibrium such as electromechanical models with rotation and electrical circuits with cylindrical phase space were studied a long time ago. However, chaotic systems without equilibrium have received significant attention recently after the introduction of hidden attractors. Interestingly, an attractor of a no-equilibrium system is hidden because its basin of attraction does not intersect with any neighborhood of an unstable fixed point. This chapter presents a 3D no-equilibrium system with hidden chaotic attractors. The fundamental qualitative properties of the proposed no-equilibrium system are discovered by using phase portraits, Poincaré map, bifurcation diagram, and Lyapunov exponents. We have designed an electronic circuit to confirm the physical implementation of the theoretical no-equilibrium system. In addition, global chaos antisynchronization of the proposed system is investigated and confirmed by numerical simulations. Finally the fractional-order form of the proposed no-equilibrium chaotic system is studied in detail.