Bifurcation analysis in the system with the existence of two stable limit cycles and a stable steady state
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ORIGINAL PAPER
Bifurcation analysis in the system with the existence of two stable limit cycles and a stable steady state Lijuan Ning
Received: 1 April 2020 / Accepted: 5 August 2020 © Springer Nature B.V. 2020
Abstract Van der Pol–Duffing oscillator, which can be used a model for many dynamical system, has been widely concerned. However, most of the systems by scholars are either stable steady states or limit cycles. Here, the self-sustained oscillator with the coexistence of steady state and limit cycles, which is famous for describing the flutter of airfoils with large span ratio in low-speed wind tunnels, is treated in this paper. Using the energy balance method, the deterministic bifurcation of the tristable system with time-delay feedback is investigated. The presence of time-delay feedback expands the bifurcation range of the parameters, making the bifurcation phenomenon more abundant. In addition, according to the stationary probability density function obtained by the stochastic averaging method, stochastic bifurcation of the system with timedelay feedback and noise is explored theoretically. The numerical results confirm the correctness of the theoretical analysis. Transition between the unimodal structure, the bimodal structure and the trimodal structure is found. Many rich bifurcations are available by adjusting the time-delay and noise intensity, which may be conductive to achieve the desired phenomenon in the real-world application.
L. Ning (B) School of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710119, People’s Republic of China e-mail: [email protected]
Keywords Deterministic bifurcation · Stochastic bifurcation · Time-delay feedback · Self-sustained oscillator
1 Introduction Nonlinear oscillators, which are adopted to model many natural systems ranging from biology, physics, chemistry to engineering, have aroused widespread attention. As one of the most widely used nonlinear oscillators, self-sustained oscillators first presented by Andronov [1] are especially appreciated due to possessing abundant oscillation phenomena, particularly limit cycles (LCs), which are ubiquitous and can be traced in the heart rhythm of second, glycolysis of minute and circadian rhythm over 24 h, and even epidemiology over years. A LC oscillator has a fascinating magic that damps itself when it grows too much and provides energy when it grows too small. Self-sustained oscillators with LC oscillations are becoming more and more popular, and there has been an explosive development in the study of the dynamic behavior of self-sustained oscillators with LCs, involving the fields of biology, biochemistry, physics and engineering [2–7]. Birhythmicity characterized by two stable LCs with various sizes and frequencies, it has displayed a seductive charm in describing enzymes-substrates reactions [8,9], biology glycolysis oscillator [10,11], circadian oscillators [12], and some biochemical reactions [13,14]. Birhythmic system, by nature, is self-sustained
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