Infinite coexisting attractors in an autonomous hyperchaotic megastable oscillator and linear quadratic regulator-based control and synchronization

The coexistence of different attractors with fixed parameters affords versatility in the performance of the dynamical system. As a result, recent research has focused on defining nonlinear oscillators with infinite coexisting attractors, the vast majority of which are non-autonomous systems. In this study, we present an infinite number of equilibriums achieved with the simplest autonomous megastable oscillator. To begin, we explore the symmetry property of distinct coexisting attractors, such as periodic, quasi-periodic, and chaotic attractors. The dynamical properties of the proposed system are further investigated using phase portraits, Lyapunov exponents spectrum, and bifurcation diagram using the local maxima of the state variables. The detailed investigation reveals that the proposed system has a wide range of dynamical properties ranging from periodic oscillations to hyperchaos. Linear quadratic regulator (LQR) control-based controllers are designed to control the chaos in the proposed system and also achieve synchronization between proposed systems that have different initial conditions. The effectiveness of the controller designed based on LQR is presented by simulation results. It is shown that the proposed system converges asymptotically to the desired equilibrium point, and the synchronization between the slave and the master systems is achieved by converging the error to zero after controllers are activated.