SERIES-MODE PITCHFORK-HYSTERESIS BURSTING OSCILLATIONS AND THEIR DYNAMICAL MECHANISMS

doi:10.60520459-1879-18-223

Abstract

Abstract:

Bursting oscillations is a spontaneous physical phenomenon existing in natural science, which has various patterns according to their dynamical regimes. For instance, bursting of point-point type means bursting patterns related to transition behaviors among different equilibrium attractors. Pitchfork-hysteresis bursting, induced by delayed pitchfork bifurcation, is a kind of point-point type bursting pattern showing simple dynamical characteristics. The present paper takes the Duffing system with multiple-frequency parametric excitations as an example in order to reveal bursting patterns, related to delayed pitchfork bifurcation, showing complex characteristics, i.e., the series-mode pitchfork-hysteresis bursting oscillations. We considered the case when one excitation frequency is an integer multiple of the other, obtained the fast subsystem and the slow variable of the Duffing system by frequency-transformation fast-slow analysis, and analyzed bifurcation behaviors of the fast subsystem. Our study shows that two or multiple pitchfork bifurcation points can be observed in the fast subsystem, and thus two or multiple pitchfork-hysteresis bursting patterns are created when the slow variable passes through these points. In particular, the pitchfork-hysteresis bursting patterns are connected in series, and as a result, the so-called series-mode pitchfork-hysteresis bursting oscillations are generated. Besides, the effects of parameters on the series-mode pitchfork-hysteresis bursting oscillations are analyzed. It is found that the damping of the system and the maximum excitation amplitude show no qualitative impact on corresponding dynamical mechanisms, while the smaller one may lead to vanish of busting oscillations. Our findings reveal the road from simple dynamical characteristics of point-point type bursting oscillation related to complex one, thereout, a complement and expansion for nowadays bursting dynamics is obtained.

Key words: Duffing system, slowly varying periodic parameters, delayed pitchfork bifurcation, series-mode bursting oscillations, frequency-transformation fast-slow analysis

CLC Number:

O322

Zhang Yi, Han Xiujing, Bi Qinsheng. SERIES-MODE PITCHFORK-HYSTERESIS BURSTING OSCILLATIONS AND THEIR DYNAMICAL MECHANISMS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(1): 228-236.

Figures/Tables 8

Fig. 1 The pitchfork-hysteresis bursting oscillations. The system parameters are fixed at $\delta = 0.6$,$\beta _1 = 2.0$,$\beta _2 = 0$,\\ $\omega _1 = 0.01$ and $\omega _2 = 0.01$...

Fig.2 Transition from "pitchfork-hysteresis" bursting oscillations (a) to "series-mode pitchfork-hysteresis" bursting oscillations (b-d) ...

Fig.3 Fast-slow analysis of the bursting oscillations in Fig. 2...

Fig.4 Bifurcation sets of the fast subsystem (3) on parameter plane $(\beta _2 ,\gamma )$ ...

Fig.5 "Series-mode pitchfork-hysteresis" bursting oscillations of system (1) for $\beta _2 = 0.9$...

Fig.6 The existence of asymptotic line leads to the disappearance of bursting oscillations...

Fig.7 Bifurcation sets on parameter plane $(\beta _1 ,\gamma )$ ...

Fig.8 Bursting oscillations related to different values of$\beta _1 $, where the other parameters are the same as in Fig. 2(b) ...

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