Design of multistable systems via partial synchronization

Many researchers introduce schemes for designing multistable systems by coupling two identical systems. In this paper, we introduce a generalized scheme for designing multistable systems by coupling two different dynamical systems. The basic idea of the scheme is to design partial synchronization of states between the coupled systems and finding some completely initial condition-dependent constants of motion. In our scheme, we synchronize i number (\(1\le i \le m-1\)) of state variables completely and keep constant difference between j (\(1\le j\le m-1\), \(i+j=m\)) number of state variables of two coupled m-dimensional different dynamical systems to obtain multistable behaviour. We illustrate our scheme for coupled Lorenz and Lu systems. Numerical simulation results consisting of phase diagram, bifurcation diagram and maximum Lyapunov exponents are presented to show the effectiveness of our scheme.