Multistable behaviour of coupled Lorenz-Stenflo systems

In this paper, we propose three different schemes for designing multistable systems coupling Lorenz-Stenflo (LS) systems. In all of these three schemes the coupled LS-systems have been reduced to a single modified LS-system. Theoretically, pitchfork bifurcation and Hopf bifurcation conditions of the modified LS-system are derived. Phase diagrams are presented to show the multistable nature of the coupled LS systems for different initial conditions. One parameter bifurcation analysis is done with respect to difference in initial conditions of the two systems. Two parameter bifurcation analysis results are also presented. Our most important observation is that in coupling two m-dimensional dynamical systems multistable nature can be obtained if i number of variables of the two systems are completely synchronized and j number of variables keep a constant difference between them, where i + j = m and 1 <= i, j <= m - 1. Our observation may be applicable for designing physically or biologically useful multistable systems.