Chaotic behavior and modified function projective synchronization of a simple system with one stable equilibrium

By introducing a feedback control to a proposed Sprott E system, an extremely complex chaotic attractor with only one stable equilibrium is derived. The system evolves into periodic and chaotic behaviors by detailed numerical as well as theoretical analysis. Analysis results show that chaos also can be generated via a period-doubling bifurcation when the system has one and only one stable equilibrium. Based on Lyapunov stability theory, the adaptive control law and the parameter update law are derived to achieve modified function projective synchronized between the extended Sprott E system and original Sprott E system. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.