Complex Bursting Patterns in a van der Pol–Mathieu–Duffing Oscillator

2021 ◽  
Vol 31 (06) ◽  
pp. 2150082
Author(s):  
Xindong Ma ◽  
Jin Song ◽  
Mengke Wei ◽  
Xiujing Han ◽  
Qinsheng Bi
Keyword(s):  
Duffing Oscillator ◽  
Relaxation Oscillation ◽  
Multiple Frequency ◽  
Van Der Pol ◽  
Bursting Oscillations ◽  
Duffing System ◽  
Dynamical Behaviors ◽  
Parameter Interval ◽  
Bursting Oscillation

The pulse-shaped explosion (PSE), characterized by the pulse-shaped quantitative of system solutions varying dramatically, is a special route to bursting oscillations reported recently. This paper reports interesting dynamical behaviors related to the PSE of equilibria, and based on that, the complex bursting dynamics is investigated in a van der Pol–Mathieu–Duffing system with multiple-frequency slow-varying excitations. We find that bifurcations can be observed in a narrow parameter interval within PSE. We also show that two groups of bifurcations are symmetrically arranged on both sides of PSE, and each of which determines a different bursting part. Based on this, two compound bursting patterns, i.e. compound Hopf/Hopf bursting oscillation and compound subHopf/fold cycle bursting oscillation, and a novel type of relaxation oscillation (bursting oscillation of point/point) independent of bifurcations, are revealed. Our results enrich the knowledge of dynamical behaviors related to PSE as well as the possible routes to complex bursting dynamics.

scholarly journals On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol-Duffing system with slow-varying periodic excitation

2021 ◽  
Author(s):  
Xiaofang Zhang ◽  
Bin Zhang ◽  
Xiujing Han ◽  
Qinsheng Bi
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Multiple Equilibrium ◽  
Sudden Increase ◽  
Van Der Pol ◽  
Full System ◽  
Bursting Oscillations ◽  
Duffing System ◽  
Exciting Frequency ◽  
Different Types

Abstract The main purpose of the paper is to reveal the mechanism of certain special phenomena in bursting oscillations such as the sudden increase of the spiking amplitude. When multiple equilibrium points coexist in a dynamical system, several types of stable attractors via different bifurcations from these points may be observed with the variation of parameters, which may interact with each other to form other types of bifurcations. Here we take the modified van der Pol-Duffing system as an example, in which periodic parametric excitation is introduced. When the exciting frequency is far less than the natural frequency, bursting oscillations may appear. By regarding the exciting term as a slow-varying parameter, the number of the equilibrium branches in the fast generalized autonomous subsystem varies from one to five with the variation of the slow-varying parameter, on which different types of bifurcations, such as Hopf and pitch fork bifurcations, can be observed. The limit cycles, including the cycles via Hopf bifurcations and the cycles near the homo-clinic orbit may interact with each other to form the fold limit cycle bifurcations. With the increase of the exciting amplitude, different stable attractors and bifurcations of the generalized autonomous fast subsystem involve the full system, leading to different types of bursting oscillations. Fold limit cycle bifurcations may cause the sudden change of the spiking amplitude, since at the bifurcation points, the trajectory may oscillate according to different stable limit cycles with obviously different amplitudes. At the pitch fork bifurcation point, two possible jumping ways may result in two coexisted asymmetric bursting attractors, which may expand in the phase space to interact with each other to form an enlarged symmetric bursting attractor with doubled period. The inertia of the movement along the stable equilibrium may cause the trajectory to pass across the related bifurcations, leading to the delay effect of the bifurcations. Not only the large exciting amplitude, but also the large value of the exciting frequency may increase inertia of the movement, since in both the two cases, the change rate of the slow-varying parameter may increase. Therefore, a relative small exciting frequency may be taken in order to show the possible influence of all the equilibrium branches and their bifurcations on the dynamics of the full system.

scholarly journals Bursting Oscillation and Its Mechanism of a Generalized Duffing–Van der Pol System with Periodic Excitation

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Youhua Qian ◽  
Danjin Zhang ◽  
Bingwen Lin
Keyword(s):  
Numerical Simulations ◽  
Phase Portrait ◽  
Time Scales ◽  
Bifurcation Analysis ◽  
Hot Spot ◽  
Periodic Excitation ◽  
Fast Subsystem ◽  
Van Der Pol ◽  
Bursting Oscillations ◽  
Bursting Oscillation

The complex bursting oscillation and bifurcation mechanisms in coupling systems of different scales have been a hot spot domestically and overseas. In this paper, we analyze the bursting oscillation of a generalized Duffing–Van der Pol system with periodic excitation. Regarding this periodic excitation as a slow-varying parameter, the system can possess two time scales and the equilibrium curves and bifurcation analysis of the fast subsystem with slow-varying parameters are given. Through numerical simulations, we obtain four kinds of typical bursting oscillations, namely, symmetric fold/fold bursting, symmetric fold/supHopf bursting, symmetric subHopf/fold cycle bursting, and symmetric subHopf/subHopf bursting. It is found that these four kinds of bursting oscillations are symmetric. Combining the transformed phase portrait with bifurcation analysis, we can observe bursting oscillations obviously and further reveal bifurcation mechanisms of these four kinds of bursting oscillations.

Prediction of Solutions of the Duffing System with Tuned Mass Damper

2020 ◽  
Vol 22 (4) ◽  
pp. 983-990
Author(s):  
Konrad Mnich
Keyword(s):  
Dynamical System ◽  
Duffing Oscillator ◽  
Probabilistic Approach ◽  
Nonlinear Dynamical System ◽  
Tuned Mass Damper ◽  
Nonlinear Dynamical ◽  
Duffing System ◽  
Mass Damper ◽  
Basin Stability ◽  
Stability Concept

AbstractIn this work we analyze the behavior of a nonlinear dynamical system using a probabilistic approach. We focus on the coexistence of solutions and we check how the changes in the parameters of excitation influence the dynamics of the system. For the demonstration we use the Duffing oscillator with the tuned mass absorber. We mention the numerous attractors present in such a system and describe how they were found with the method based on the basin stability concept.

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