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.

scholarly journals "Study of effect of the inductor on dynamical behavior for electronic circuit by using nonlinear part from type memductance

2021 ◽  
Vol 8 (1) ◽  
pp. 98-108
Author(s):  
Abo-Talib .Y. Abbas
Keyword(s):  
Hopf Bifurcation ◽  
Phase Portrait ◽  
Bifurcation Diagram ◽  
Electronic Circuit ◽  
Dynamical Behavior ◽  
Stability Zone ◽  
Nonlinear Part ◽  
Bifurcation Line ◽  
Stability Maps ◽  
The Stability

"The effect of the coil on the dynamical behavior of the electronic circuit with nonlinear part memductance was studied, where we studied four coil values: L = 90mH, L = 80mH, L = 70mH and L = 100mH with changing the value of the first capacitance c1. We obtained stability maps consisting of two regions (the dynamical behavior zone and the stability zone), where the dividing line is the Hopf bifurcation line, as well as the bifurcation diagram for each value from the coil and the first capacitance was obtained, as well as the phase portrait diagram in which we obtained the chaotic attractions And the period of the first, second, third, fourth, fifth, sixth and eighth.

Dynamics in a Van Der Pol Model with Delay

2012 ◽  
Vol 204-208 ◽  
pp. 4586-4589
Author(s):  
Chang Jin Xu ◽  
Pei Luan Li ◽  
Ling Yun Yao
Keyword(s):  
Asymptotic Behavior ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Parameter ◽  
Bifurcation Problem ◽  
Van Der Pol ◽  
Van Der Pol Model ◽  
The Stability

In this paper, the dynamics of a van der pol model with delay are considered. It is shown that the asymptotic behavior depends crucially on the time delay parameter. By regarded the delay as a bifurcation parameter, we are particularly interested in the study of the Hopf bifurcation problem. The length of delay which preserves the stability of the equilibrium is calculated. Some numerical simulations for justifying the analytical findings are included.

Nonlinear Analysis of Hysteretic Modulation-Based Sliding Mode Controlled Quadratic Buck–Boost Converter

2018 ◽  
Vol 28 (02) ◽  
pp. 1950025 ◽  
Author(s):  
Muppala Kumar Kavitha ◽  
Anbukumar Kavitha
Keyword(s):  
Hopf Bifurcation ◽  
Phase Portrait ◽  
Sliding Mode ◽  
Stability Boundary ◽  
Low Frequency ◽  
Nonlinear Behavior ◽  
Input Voltage ◽  
Boost Converter ◽  
Complex Conjugate ◽  
The Stability

In this paper, the dynamics of hysteresis current-controlled quadratic buck-boost converter is investigated in detail. The system model is derived based on the sliding mode approach and also in its dimensionless form for algebraic brevity. The stability of the system is disclosed with the aid of the movement of eigenvalues. Onset of Hopf bifurcation is identified when the complex conjugate eigenvalue pair crosses the imaginary axis of the complex plane. The stability boundary is drawn to benefit the power electronics engineer for a stable and reliable design. The computer simulation of the switched model is performed using MATLAB/Simulink software to uncover the sequential occurrence of nonlinear behavior exhibited due to Hopf bifurcation for variation in control parameters and the input voltage. The phase portrait disclosing the subtle periodicity is plotted at different operating points to elicit that the stable period-1 attractor bifurcates to the quasi-periodic orbit and finally to a limit cycle. The precise dynamic of the phase portrait is also captured using the Poincare section. Experimental outputs are presented for confirming the low-frequency bifurcation scenario witnessed in the simulated and analytical results.

scholarly journals Another New Chaotic System: Bifurcation and Chaos Control

2020 ◽  
Vol 30 (11) ◽  
pp. 2050161
Author(s):  
Arnob Ray ◽  
Dibakar Ghosh
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Period Doubling ◽  
Slow Manifold ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Dimensional Parameter ◽  
Lyapunov Coefficient ◽  
The Stability

We propose a new simple three-dimensional continuous autonomous model with two nonlinear terms and observe the dynamical behavior with respect to system parameters. This system changes the stability of fixed point via Hopf bifurcation and then undergoes a cascade of period-doubling route to chaos. We analytically derive the first Lyapunov coefficient to investigate the nature of Hopf bifurcation. We investigate well-separated regions for different kinds of attractors in the two-dimensional parameter space. Next, we introduce a timescale ratio parameter and calculate the slow manifold using geometric singular perturbation theory. Finally, the chaotic state annihilates by decreasing the value of the timescale ratio parameter.

Role of Fear in a Predator–Prey Model with Beddington–DeAngelis Functional Response

2019 ◽  
Vol 74 (7) ◽  
pp. 581-595 ◽  
Author(s):  
Saheb Pal ◽  
Subrata Majhi ◽  
Sutapa Mandal ◽  
Nikhil Pal
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Response ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Prey Interaction ◽  
Lyapunov Coefficient ◽  
The Stability ◽  
The Impact

AbstractIn the present article, we investigate the impact of fear effect in a predator–prey model, where predator–prey interaction follows Beddington–DeAngelis functional response. We consider that due to fear of predator the birth rate of prey population reduces. Mathematical properties, such as persistence, equilibria analysis, local and global stability analysis, and bifurcation analysis, have been investigated. We observe that an increase in the cost of fear destabilizes the system and produces periodic solutions via supercritical Hopf bifurcation. However, with further increase in the strength of fear, system undergoes another Hopf bifurcation and becomes stable. The stability of the Hopf-bifurcating periodic solutions is obtained by computing the first Lyapunov coefficient. Our results suggest that fear of predation risk can have both stabilizing and destabilizing effects.

scholarly journals Stability Switches and Double Hopf Bifurcation Analysis on Two-Degree-of-Freedom Coupled Delay van der Pol Oscillator

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2444
Author(s):  
Yani Chen ◽  
Youhua Qian
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Degree Of Freedom ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Double Hopf Bifurcation ◽  
Central Manifold ◽  
The Stability ◽  
Two Degree Of Freedom

In this paper, the normal form and central manifold theories are used to discuss the influence of two-degree-of-freedom coupled van der Pol oscillators with time delay feedback. Compared with the single-degree-of-freedom time delay van der Pol oscillator, the system studied in this paper has richer dynamical behavior. The results obtained include: the change of time delay causing the stability switching of the system, and the greater the time delay, the more complicated the stability switching. Near the double Hopf bifurcation point, the system is simplified by using the normal form and central manifold theories. The system is divided into six regions with different dynamical properties. With the above results, for practical engineering problems, we can perform time delay feedback adjustment to make the system show amplitude death, limit loop, and so on. It is worth noting that because of the existence of unstable limit cycles in the system, the limit cycle cannot be obtained by numerical solution. Therefore, we derive the approximate analytical solution of the system and simulate the time history of the interaction between two frequencies in Region IV.

Bursting Types and Bifurcation Analysis in the Pre-Bötzinger Complex Respiratory Rhythm Neuron

2017 ◽  
Vol 27 (01) ◽  
pp. 1750010 ◽  
Author(s):  
Jing Wang ◽  
Bo Lu ◽  
Shenquan Liu ◽  
Xiaofang Jiang
Keyword(s):  
Hopf Bifurcation ◽  
Respiratory Rhythm ◽  
Vital Role ◽  
Dynamical Analysis ◽  
Bifurcation Curve ◽  
Excitable Cells ◽  
Theoretical Derivation ◽  
Fold Bifurcation ◽  
Bötzinger Complex ◽  
Lyapunov Coefficient

Many types of neurons and excitable cells could intrinsically generate bursting activity, even in an isolated case, which plays a vital role in neuronal signaling and synaptic plasticity. In this paper, we have mainly investigated bursting types and corresponding bifurcations in the pre-Bötzinger complex respiratory rhythm neuron by using fast–slow dynamical analysis. The numerical simulation results have showed that for some appropriate parameters, the neuron model could exhibit four distinct types of fast–slow bursters. We also explored the bifurcation mechanisms related to these four types of bursters through the analysis of phase plane. Moreover, the first Lyapunov coefficient of the Hopf bifurcation, which can decide whether it is supercritical or subcritical, was calculated with the aid of MAPLE software. In addition, we analyzed the codimension-two bifurcation for equilibria of the whole system and gave a detailed theoretical derivation of the Bogdanov–Takens bifurcation. Finally, we obtained expressions for a fold bifurcation curve, a nondegenerate Hopf bifurcation curve, and a saddle homoclinic bifurcation curve near the Bogdanov–Takens bifurcation point.

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 Response of the Duffing-Van der Pol Oscillator under Position Feedback Control with Two Time Delays

2011 ◽  
Vol 18 (1-2) ◽  
pp. 377-386
Author(s):  
Xinye Li ◽  
Huabiao Zhang ◽  
Lijuan Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Trivial Solution ◽  
Time Delays ◽  
Linear Feedback ◽  
Nonlinear Feedback ◽  
Nontrivial Solutions ◽  
Van Der Pol ◽  
Position Feedback ◽  
The Stability

In this paper, the dynamics of Duffing-van der Pol oscillators under linear-plus-nonlinear position feedback control with two time delays is studied analytically and numerically. By the averaging method, together with truncation of Taylor expansions for those terms with time delay, the slow-flow equations are obtained from which the trivial and nontrivial solutions can be found. It is shown that the trivial solution can be stabilized by appropriate gain and time delay in linear feedback although it loses its stability via Hopf bifurcation and results in periodic solution for uncontrolled systems. And the stability of the trivial solution is independent of nonlinear feedback. Different from the case of the trivial solution, the stability of nontrivial solutions is also associated with nonlinear feedback besides linear feedback. Non-trivial solutions may lose their stability via saddle-node or Hopf bifurcation and the resulting response of the system may be quasi-periodic or chaotic. The feedback gains and time delays have great effects on the amplitude of the periodic solutions and their bifurcation control. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.

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