A New Chaotic System with Coexisting Attractors

Abstract

The investigation of new chaotic systems in spherical coordinates has been one of the present exciting research directions in exploring new chaotic systems. In this paper, a new system in spherical coordinates is presented. The appealing feature of the proposed system is that the dynamics of the system cannot pass through a sphere of a specific radius and stop as soon as the solution crosses the sphere in Cartesian coordinates. So, the system’s attractors are limited to be located on one side of the sphere and cannot touch it. Moreover, the reason for this phenomenon is that the velocity of a system’s variable becomes zero for a specific value of that variable. The proposed system has three unstable equilibrium points and four hidden attractors, including a limit cycle and a strange attractor inside and a limit cycle and a strange attractor outside the sphere. The system’s dynamical properties are investigated with the help of bifurcation diagrams and the calculation of Lyapunov exponents. The basin of attraction for the system’s attractors is also studied. Finally, the system is controlled or stabilized using the impulsive control theory.