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We study the dynamic control of birhythmicity under an impulsive feedback control scheme where the feedback is made ON for a certain rather small period of time and for the rest of the time, it is kept OFF. We show that, depending on the height and width of the feedback pulse, the system can be brought to any of the desired limit cycles of the original birhythmic oscillation. We derive a rigorous analytical condition of controlling birhythmicity using the harmonic decomposition and energy balance methods. The efficacy of the control scheme is investigated through numerical analysis in the parameter space. We demonstrate the robustness of the control scheme in a birhythmic electronic circuit where the presence of noise and parameter fluctuations are inevitable. Finally, we demonstrate the applicability of the control scheme in controlling birhythmicity in diverse engineering and biochemical systems and processes, such as an energy harvesting system, a glycolysis process, and a p53-mdm2 network.
Birhythmicity is a potential variant of multistability exhibiting the coexistence of two stable limit cycles with different amplitudes and frequencies separated by an unstable limit cycle. The birhythmicity is perilous for many physical systems, whereas it is obvious and desirable in most biological systems. Hence, the control of birhythmicity deserves much attention. The control of multistability is a well studied topic, but surprisingly the control of birhythmicity is not. In this paper, we study and establish the efficacy of our impulsive feedback control scheme on controlling birhythmicity. The main advantage of the proposed scheme is that the control is required for a minimal period of time and then it may be switched off. We theoretically explore the control scheme and numerically establish the generality of the scheme. We demonstrate that the scheme is robust enough to work in a practical electronic circuit experiment in the presence of noise and fluctuations.
