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.

scholarly journals Parabolic bursting, spike-adding, dips and slices in a minimal model

2019 ◽  
Vol 14 (4) ◽  
pp. 406
Author(s):  
Mathieu Desroches ◽  
Jean-Pierre Francoise ◽  
Martin Krupa
Keyword(s):  
Qualitative Analysis ◽  
Numerical Simulations ◽  
Phase Portrait ◽  
Minimal Model ◽  
Bifurcation Theory ◽  
Minimal System ◽  
Dimensional System ◽  
Slow Flow ◽  
Van Der Pol ◽  
One Dimensional

A minimal system for parabolic bursting, whose associated slow flow is integrable, is presented and studied both from the viewpoint of bifurcation theory of slow-fast systems, of the qualitative analysis of its phase portrait and of numerical simulations. We focus the analysis on the spike-adding phenomenon. After a reduction to a periodically forced one-dimensional system, we uncover the link with the dips and slices first discussed by J.E. Littlewood in his famous articles on the periodically forced van der Pol system.

scholarly journals Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic Excitation

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Xindong Ma ◽  
Shuqian Cao
Keyword(s):  
Natural Frequency ◽  
Coupling Effect ◽  
Excitation Frequency ◽  
Time Interval ◽  
Periodic Excitation ◽  
Bursting Oscillations ◽  
Fold Bifurcation ◽  
Exciting Frequency ◽  
Bursting Oscillation ◽  
Parameter Condition

The coupling effect of two different frequency scales between the exciting frequency and the natural frequency of the Shimizu-Morioka system with slow-varying periodic excitation is investigated. First, based on the analysis of the equilibrium states, homoclinic bifurcation, fold bifurcation, and supercritical Hopf bifurcation are observed in the system under a certain parameter condition. When the exciting frequency is much smaller than the natural frequency, we can regard the periodic excitation as a slow-varying parameter. Second, complicated dynamic behaviors are analyzed when the slow-varying parameter passes through different bifurcation points, of which the mechanisms of four different bursting patterns, namely, symmetric “homoclinic/homoclinic” bursting oscillation, symmetric “fold/Hopf” bursting oscillation, symmetric “fold/fold” bursting oscillation, and symmetric “Hopf/Hopf” bursting oscillation via “fold/fold” hysteresis loop, are revealed with different values of the parameterbby means of the transformed phase portrait. Finally, we can find that the time interval between two symmetric adjacent spikes of bursting oscillations exhibits dependency on the periodic excitation frequency.

BIFURCATION STRUCTURE OF A VAN DER POL OSCILLATOR SUBJECTED TO NONSINUSOIDAL PERIODIC EXCITATION

2012 ◽  
Vol 22 (01) ◽  
pp. 1250003 ◽  
Author(s):  
H. SIMO ◽  
P. WOAFO
Keyword(s):  
Experimental Investigation ◽  
Numerical Simulations ◽  
Control Parameter ◽  
Van Der Pol Oscillator ◽  
Periodic Excitation ◽  
Electronic Components ◽  
Bifurcation Structure ◽  
Van Der Pol ◽  
Bifurcation Points ◽  
Bifurcation Structures

Bifurcation structures of a Van der Pol oscillator subjected to the effects of nonsinusoidal excitations are obtained both numerically and experimentally. It is found that the bifurcation sequences are similar, but the ranges of a particular behavior and the bifurcation points of the control parameter are different. The experimental investigation using electronic components shows that results are similar to those observed from numerical simulations.

Periodic Motions in a Coupled Van Der Pol-Duffing Oscillator

2017 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C. J. Luo
Keyword(s):  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Eigenvalue Analysis ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

In this paper, periodic motions of a periodically forced, coupled van der Pol-Duffing oscillator are predicted analytically. The coupled van der Pol-Duffing oscillator is discretized for the discrete mapping. The periodic motions in such a coupled van der Pol-Duffing oscillator are obtained from specified mapping structures, and the corresponding stability and bifurcation analysis are carried out by eigenvalue analysis. Based on the analytical prediction, the initial conditions of periodic motions are used for numerical simulations.

scholarly journals Fast-Slow Coupling Dynamics Behavior of the van der Pol-Rayleigh System

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3004
Author(s):  
Danjin Zhang ◽  
Youhua Qian
Keyword(s):  
Hopf Bifurcation ◽  
Phase Portrait ◽  
External Excitation ◽  
Fast Subsystem ◽  
Van Der Pol ◽  
Fold Bifurcation ◽  
Modes Of Vibration ◽  
Lyapunov Coefficient ◽  
The Stability ◽  
Rayleigh System

In this paper, the dynamic behavior of the van der Pol-Rayleigh system is studied by using the fast–slow analysis method and the transformation phase portrait method. Firstly, the stability and bifurcation behavior of the equilibrium point of the system are analyzed. We find that the system has no fold bifurcation, but has Hopf bifurcation. By calculating the first Lyapunov coefficient, the bifurcation direction and stability of the Hopf bifurcation are obtained. Moreover, the bifurcation diagram of the system with respect to the external excitation is drawn. Then, the fast subsystem is simulated numerically and analyzed with or without external excitation. Finally, the vibration behavior and its generation mechanism of the system in different modes are analyzed. The vibration mode of the system is affected by both the fast and slow varying processes. The mechanisms of different modes of vibration of the system are revealed by the transformation phase portrait method, because the system trajectory will encounter different types of attractors in the fast subsystem.

Bursting oscillations in a slow-varying periodically excited vector field with Bogdanov–Takens bifurcation

2021 ◽  
pp. 107754632199358
Author(s):  
Shuqian Wu ◽  
Qinsheng Bi
Keyword(s):  
Vector Field ◽  
Normal Form ◽  
Phase Portrait ◽  
Bifurcation Analysis ◽  
Bifurcation Point ◽  
Topological Structure ◽  
Dynamical Evolution ◽  
Delay Effect ◽  
Universal Unfolding ◽  
Bursting Oscillations

The main purpose of the article is to classify all the possible bursting oscillations in a vector field with Bogdanov–Takens bifurcation at the origin. Based on the universal unfolding of the normal form of the vector field, the topological structure in the neighborhood of the bifurcation point on the unfolding parameters is presented. Replacing one of the unfolding parameters by a slow-varying periodic exciting term, the coupling of two scales in frequency domain involves the vector field, which may lead to the bursting oscillations. According to the bifurcation analysis, we focus on three typical cases to investigate the dynamical evolution with the increase of the exciting amplitude. By introducing the transformed phase portrait, the mechanism of bursting oscillations can be presented. Three types of bifurcations, that is, fold, Hopf, and saddle on the limit cycle bifurcations may cause the alternations of the trajectory between the quiescent states and the spiking states, different combinations of which may result in different bursting attractors. Furthermore, the inertia of the movement may result in the delay effect of the bifurcation, which may lead to the disappearance of the bifurcation influence and the corresponding spiking oscillations.

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.

Bifurcation Analysis of Piezoelectric Bi-Stable Plates

2014 ◽  
Author(s):  
Lihua Chen ◽  
Ma Yepeng ◽  
Wei Zhang
Keyword(s):  
Numerical Simulations ◽  
Phase Portrait ◽  
Bifurcation Analysis ◽  
Nonlinear Dynamic ◽  
Piezoelectric Effect ◽  
Kutta Method ◽  
Bifurcation Diagrams ◽  
Nonlinear Dynamic Model ◽  
Piezoelectric Patch ◽  
Runge Kutta Method

The complex nonlinear dynamic behaviors of the composite bi-stable plates with piezoelectric patch are analyzed. Based on the Vo n Karman hypothesis and Hamilton’s principle, the nonlinear dynamic model is derived. Temperature and piezoelectric effect are also considered in the model. Numerical simulations are performed to study the nonlinear vibration response of the composite bi-stable plate using the Runge-Kutta method. The analysis of the phase portrait, waveforms and bifurcation diagrams of numerical simulations shows that the period, multi-period and chaotic responses can be observed with the variation of the excitation in frequency and amplitude.

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