Control of birhythmicity: A self-feedback approach

Abstract

Birhythmicity occurs in many natural and artificial systems. In this paper, we propose a selffeedback 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.

Multi-rhythmicity generated by coupling two cellular rhythms

“…Also less common are examples of birhythmicity, which involves the coexistence between two stable oscillatory regimes. Birhythmicity has been observed in a variety of models for cellular oscillatory processes [18–26], as well as in physical systems [3] for which some theoretical studies aim at controlling the phenomenon to transform it into monorhythmicity [3,27]. Trirhythmicity, i.e.…”

Time‐Delayed Feedback Control in the Multiple Attractors Wind‐Induced Vibration Energy Harvesting System

“…It should be noted that the theoretical method in our paper is more accurate than the theoretical method proposed by Debabrata Biswas and his collaborators [37]. For the same parameters as in Figure 3 in [37], one can obtain that the Hopf bifurcation of the control parameter is approximately 0.0953 by using the continuation package XPPAUT. The theoretical value of the Hopf bifurcation of the control parameter in [2] is = = 0.1.…”

“…The physical model consists of a flexible beam with distributed piezoelectric patches and an electrical circuit having a load resistance . The dimensionless form of the original model is dominated by the following set of equations [2,37]:…”

Network synchronization, stability and rhythmic processes in a diffusive mean-field coupled SEIR model

“…Birhythmic oscillations which appear in living systems are the intracellular Ca2+ oscillations [7] , glycolytic oscillations and enzymatic reactions [8] , [9] , [10] , [11] , the circadian oscillation in Period (PER) and Timeless (TIM) proteins in Drosophila [12] , the cyclic AMP signaling system of the slime mold Dictyostelium discoideum [13] and rhythm that arises in population dynamics [14] . Aside from living systems, many artificial systems also exhibit birhythmic oscillations (e.g., the wind-induced mechanical energy harvesting system [15] , [16] ). Another important concept of rhythmic oscillation is rhythmogenesis which is the inverse of the amplitude death.…”