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Birhythmicity occurs in many natural and artificial systems. In this paper, we propose a self-feedback scheme to control birhythmicity. To establish the efficacy and generality of the proposed control scheme, we apply it on three birhythmic oscillators from diverse fields of natural science, namely, an energy harvesting system, the p53-Mdm2 network for protein genesis (the OAK model), and a glycolysis model (modified Decroly-Goldbeter model). Using the harmonic decomposition technique and energy balance method, we derive the analytical conditions for the control of birhythmicity. A detailed numerical bifurcation analysis in the parameter space establishes that the control scheme is capable of eliminating birhythmicity and it can also induce transitions between different forms of bistability. As the proposed control scheme is quite general, it can be applied for control of several real systems, particularly in biochemical and engineering systems.
Multistability appears in diverse forms, and their study is an exciting topic of research in science and engineering. A particular form of multistability is bistability: it shows many variants, such as the coexistence of two stable steady states (SSS), one stable steady state and one stable limit cycle (LC), two stable limit cycles, or two chaotic attractors. Birhythmicity is the phenomenon of coexistence of two stable limit cycles separated by an unstable limit cycle with different amplitudes and frequencies. In many physical systems, birhythmicity is undesirable as in energy harvesting systems, but in most biological systems, e.g., enzymatic oscillations, it is desirable. Therefore, control of birhythmicity is of utmost importance. Although the control of multistability is a well studied topic, the control of birhythmicity has not been explored to that extent. In this paper, we propose a control scheme that can effectively control and, whenever required, can eliminate birhythmicity. We theoretically explore and numerically establish the technique of control of birhythmicity and transitions to any desired attractor. A number of engineering and biological systems are investigated with the proposed control scheme to establish the efficacy and generality of the scheme. The main essence of this control scheme lies in the fact that it is easily realizable and offers an efficient mean to control birhythmicity.
