Coexistence of stable states in a parametric family of bimodal maps

We demonstrate the emergence of monostability and bistability in Lyapunov sense in discrete-time nonlinear dynamics and discuss the properties associated with this behavior. Specifically, we introduce the necessary conditions to ensure the occurrence of the bistability phenomenon within a parametric family of bimodal maps, based on the difference map. The bimodal map is defined within a regular partition consisting of two subintervals, and we present three case studies: the first case corresponds to keeping the value of the first modal fixed, while the second modal changes its value according to a parameter. In the second case, the value assigned to the first modal changes according to another parameter while the value of the second modal remains fixed. In the third case, both values assigned to the modals change according to a bifurcation parameter. Bifurcation diagrams are shown for the three case studies and let us identify invariant sets or trapping regions in each subinterval. Subsequently, two invariant sets are defined to enable the bistability phenomenon. Therefore, monostability and bistability phenomena appear in parametric families according to the control of invariant sets and trapping regions by varying a bifurcation parameter. The numerical results align with the developed theory. We show an application of the family of bimodal maps to generate a reconfigurable multivibrator-classified as astable, bistable and monostable.