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 betweenthe coupled systems and finding some completely initial condition-dependent constants of motion. In our scheme, we synchronize $i$ number $(1 \leq i \leq m − 1)$ of state variables completely and keep constant difference between $j$ $(1 \leq j \leq m −1, i + j = m)$ 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.

Multistability or coexistence of different chaotic attractors for a given set of parameters depending on the initial condition only is one of the most exciting phenomenon in dynamical systems. The schemes to designmultistability systems via coupling two identical or non-identical but the same-dimensional systems have been proposed earlier. Coupled different-dimensional systems are very useful to describe the real-world physical and biological systems. In this paper, a scheme for designing a multistable system by coupling two different-dimensional dynamical systems has been proposed. Coupled Lorenz and Lorenz–Stenflo systems have been considered to illustrate the scheme. The efficiency of the scheme is shown numerically, by presenting phase diagrams, bifurcation diagrams and variation of maximum Lyapunov exponents.