scholarly journals The Focus Case of a Nonsmooth Rayleigh-Duffing Oscillator

2021 ◽  
Author(s):  
Zhaoxia Wang ◽  
Hebai Chen ◽  
Yilei Tang
Keyword(s):  
Limit Cycle ◽  
Duffing Oscillator ◽  
Pitchfork Bifurcation ◽  
Homoclinic Bifurcation ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Bifurcation Curve ◽  
Limit Cycle Bifurcation ◽  
Theoretical Results ◽  
Double Limit

Abstract In this paper, we study the global dynamics of a nonsmooth Rayleigh-Duffing equation x¨ + ax˙ + bx˙|x˙| + cx + dx3 = 0 for the case d > 0, i.e., the focus case. The global dynamics of this nonsmooth Rayleigh-Duffing oscillator for the case d < 0, i.e., the saddle case, has been studied completely in the companion volume [Int. J. Non-Linear Mech., 129 (2021) 103657]. The research for the focus case is more complex than the saddle case, such as the appearance of five limit cycles and the gluing bifurcation which means that two double limit cycle bifurcation curves and one homoclinic bifurcation curve are very adjacent occurs. We present bifurcation diagram, including one pitchfork bifurcation curve, two Hopf bifurcation curves, two double limit cycle bifurcation curves and one homoclinic bifurcation curve. Finally, numerical phase portraits illustrate our theoretical results.

Global Dynamics of Memristor Oscillator

2016 ◽  
Vol 26 (12) ◽  
pp. 1650198 ◽  
Author(s):  
Hebai Chen
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Pitchfork Bifurcation ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Generalized Hopf Bifurcation ◽  
Limit Cycle Bifurcation ◽  
Sector Method ◽  
Double Limit ◽  
The Continuum

In this paper, we investigate the global dynamics of a memristor oscillator [Formula: see text] which comes from [Corinto et al., 2011], where [Formula: see text], and [Formula: see text]. Clearly, the case [Formula: see text] is trivial. So far, all results of this oscillator were given only for the case [Formula: see text], where the set of equilibria may change among a singleton, three points and a singular continuum and at most one limit cycle can arise and no limit cycles arise from the continuum. Compared with the case [Formula: see text], this oscillator displays more complicated dynamics for the case when [Formula: see text]. More clearly, one limit cycle may arise from the continuum and at most three limit cycles appear in the case of three equilibria, where generalized pitchfork bifurcation, saddle-node bifurcation, generalized Hopf bifurcation, double limit cycle bifurcation and homoclinic bifurcation may occur. Finally all global phase portraits are given for [Formula: see text] cases on the Poincaré disc, where a generalized normal sector method is applied. Moreover, our partial analytical results are demonstrated by numerical examples.

scholarly journals Complex Dynamics of an Impulsive Chemostat Model

2019 ◽  
Vol 29 (08) ◽  
pp. 1950101 ◽  
Author(s):  
Jin Yang ◽  
Yuanshun Tan ◽  
Robert A. Cheke
Keyword(s):  
Periodic Solution ◽  
Global Stability ◽  
Substrate Concentration ◽  
Complex Dynamics ◽  
Control Strategies ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Chemostat Model ◽  
Existence And Stability ◽  
Theoretical Results

We propose a novel impulsive chemostat model with the substrate concentration as the basis for the implementation of control strategies, and then investigate the model’s global dynamics. The exact domains of the impulsive and phase sets are discussed in the light of phase portraits of the model, and then we define the Poincaré map and study its complex properties. Furthermore, the existence and stability of the microorganism eradication periodic solution are addressed, and the analysis of a transcritical bifurcation reveals that an order-1 periodic solution is generated. We also provide the conditions for the global stability of an order-1 periodic solution and show the existence of order-[Formula: see text] [Formula: see text] periodic solutions. Moreover, the PRCC results and bifurcation analyses not only substantiate our results, but also indicate that the proposed system exists with complex dynamics. Finally, biological implications related to the theoretical results are discussed.

Limit cycle oscillation and multiple entrainment phenomena in a duffing oscillator under time-delayed displacement feedback

2015 ◽  
Vol 23 (17) ◽  
pp. 2742-2756 ◽  
Author(s):  
RK Mitra ◽  
S Chatterjee ◽  
AK Banik
Keyword(s):  
Frequency Response ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Duffing Oscillator ◽  
A Priori ◽  
High Amplitude ◽  
Limit Cycle Oscillation ◽  
Global Dynamics ◽  
Response Curves ◽  
Damped System

The Duffing oscillator under time-delayed displacement feedback is investigated to study the effect of intentional time-delay on the global dynamics of the oscillator. From the free vibration study performed by employing the describing function method it is observed that for the undamped oscillator, an infinite number of limit cycles is present for all possible values of gain and delay. The number of stable and unstable limit cycles in the gain versus delay plane is studied region wise with the help of limit cycle stability lines. Secondly, in a damped system, the number of limit cycles is finite and depends upon the values of gain, delay and damping coefficient from which the maximum number of limit cycles, their frequencies and amplitudes are obtained. When the system is excited by harmonic forcing, these limit cycles exhibit the phenomena of multiple entrainments and their frequency response curves become very complex and most often results in the very high amplitude oscillations. The study of the forced damped oscillator is therefore carried out by applying the method of slowly varying parameter and the frequency response curves for period-1 responses are analyzed. Further, with the a priori knowledge of possible stable and unstable limit cycles obtained by the application of semi-analytical methods, the various instability phenomena due to subharmonic and quasiperiodic responses have also been investigated by numerical simulation using Simulink in the different parametric ranges.

scholarly journals On resonances under quasi-periodic perturbations of systems with a double limit cycle, close to two-dimensional nonlinear Hamiltonian systems

2021 ◽  
Vol 23 (1) ◽  
pp. 11-27
Author(s):  
Olga S. Kostromina
Keyword(s):  
Numerical Calculation ◽  
Limit Cycle ◽  
Autonomous System ◽  
Theoretical Study ◽  
Type Equation ◽  
Phase Portraits ◽  
Two Dimensional ◽  
Resonant Structures ◽  
Periodic Perturbations ◽  
Double Limit

The effect of multi-frequency quasi-periodic perturbations on systems close to twodimensional nonlinear Hamiltonian ones is studied. It is assumed that the corresponding perturbed autonomous system has a double limit cycle. Analysis of the Poincar´e–Pontryagin function constructed for the autonomous system makes it possible to establish the presence of such a cycle. When the condition of commensurability of the natural frequency of the corresponding unperturbed Hamiltonian system with the frequencies of the quasi-periodic perturbation is fulfilled, the unperturbed level becomes resonant. Resonant structures essentially depend on whether the selected resonance levels coincide with the levels that generate limit cycles in the autonomous system. An averaged system is obtained that describes the topology of the neighborhoods of resonance levels. Possible phase portraits of the averaged system are established near the bifurcation case, when the resonance level coincides with the level in whose neighborhood the corresponding autonomous system has a double limit cycle. To illustrate the results obtained, the results of a theoretical study and of a numerical calculation are presented for a specific pendulum-type equation under two-frequency quasi-periodic perturbations.

scholarly journals On two-frequency quasi-periodic perturbations of systems close to two-dimensional Hamiltonian ones with a double limit cycle

2021 ◽  
Vol 31 (1) ◽  
pp. 35-49
Author(s):  
O.S. Kostromina
Keyword(s):  
Limit Cycle ◽  
Nonlinear Oscillations ◽  
Autonomous Systems ◽  
Type Equation ◽  
Unperturbed System ◽  
Two Dimensional ◽  
Periodic Perturbations ◽  
Averaged Systems ◽  
Theoretical Results ◽  
Double Limit

The problem of the effect of two-frequency quasi-periodic perturbations on systems close to arbitrary nonlinear two-dimensional Hamiltonian ones is studied in the case when the corresponding perturbed autonomous systems have a double limit cycle. Its solution is important both for the theory of synchronization of nonlinear oscillations and for the theory of bifurcations of dynamical systems. In the case of commensurability of the natural frequency of the unperturbed system with frequencies of quasi-periodic perturbation, resonance occurs. Averaged systems are derived that make it possible to ascertain the structure of the resonance zone, that is, to describe the behavior of solutions in the neighborhood of individual resonance levels. The study of these systems allows determining possible bifurcations arising when the resonance level deviates from the level of the unperturbed system, which generates a double limit cycle in a perturbed autonomous system. The theoretical results obtained are applied in the study of a two-frequency quasi-periodic perturbed pendulum-type equation and are illustrated by numerical computations.

Local Bifurcation Analysis and Global Dynamics Estimation of a Novel 4-Dimensional Hyperchaotic System

2017 ◽  
Vol 27 (02) ◽  
pp. 1750021 ◽  
Author(s):  
Leilei Zhou ◽  
Zengqiang Chen ◽  
Jiezhi Wang ◽  
Qing Zhang
Keyword(s):  
Complex Dynamics ◽  
Optimization Method ◽  
Pitchfork Bifurcation ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Hyperchaotic System ◽  
Center Manifold Theorem ◽  
Local Bifurcation Analysis ◽  
Lyapunov Function Method ◽  
The Stability

In this paper, we present a novel 4-dimensional (4D) smooth quadratic autonomous hyperchaotic system with complex dynamics. In order to investigate the dynamics evolution of the system, the Lyapunov exponent spectrum, bifurcation diagram and various phase portraits are provided. The local dynamics of this hyperchaotic system, such as the stability, pitchfork bifurcation, and Hopf bifurcation of equilibrium point, are analyzed by using the center manifold theorem and bifurcation theory. About the global dynamics, the ultimate bound sets of the system are found by combining the Lyapunov function method and appropriate optimization method. Numerical simulations are given to demonstrate the emergence of the two bifurcations and show the ultimate boundary regions.

scholarly journals Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Yanju Xiao ◽  
Weipeng Zhang ◽  
Guifeng Deng ◽  
Zhehua Liu
Keyword(s):  
Sufficient Conditions ◽  
Global Dynamics ◽  
Sis Model ◽  
Delayed Treatment ◽  
Sis Epidemic Model ◽  
Treatment Function ◽  
Lyapunov Coefficient ◽  
Saturated Treatment ◽  
Theoretical Results ◽  
Disease Free

This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.

Global Dynamics of a Holling-II Amensalism System with Nonlinear Growth Rate and Allee Effect on the First Species

2021 ◽  
Vol 31 (03) ◽  
pp. 2150050
Author(s):  
Demou Luo ◽  
Qiru Wang
Keyword(s):  
Growth Rate ◽  
Allee Effect ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Nonlinear Growth ◽  
Trivial Equilibrium ◽  
Existence And Stability ◽  
Stable Equilibria ◽  
Theoretical Results

Of concern is the global dynamics of a two-species Holling-II amensalism system with nonlinear growth rate. The existence and stability of trivial equilibrium, semi-trivial equilibria, interior equilibria and infinite singularity are studied. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, the global dynamics of the model is performed. Next, we incorporate Allee effect on the first species and offer a new analysis of equilibria and bifurcation discussion of the model. Finally, several numerical examples are performed to verify our theoretical results.

Bursting Oscillations and Experimental Verification of a Rucklidge System

2021 ◽  
Vol 31 (08) ◽  
pp. 2130023
Author(s):  
Zhijun Li ◽  
Siyuan Fang ◽  
Minglin Ma ◽  
Mengjiao Wang
Keyword(s):  
Electronic Devices ◽  
Pitchfork Bifurcation ◽  
Phase Portraits ◽  
Formation Mechanisms ◽  
Analysis Method ◽  
Good Qualitative Agreement ◽  
Bursting Oscillations ◽  
Real Time Measurement ◽  
Time Varying Parameter ◽  
Measurement Results

Bursting oscillations are ubiquitous in multi-time scale systems and have attracted widespread attention in recent years. However, research on experimental demonstration of the bursting oscillations induced by delayed bifurcation is very rarely reported. In this paper, a parametrically driven Rucklidge system is introduced and a distinct delayed behavior is observed when the time-varying parameter passes through the pitchfork bifurcation point. Different bursting patterns induced by such a delayed behavior are numerically investigated under different excitation amplitudes based on the fast–slow analysis method. Furthermore, in order to reproduce the bursting electronic signals and explore the underlying formation mechanisms experimentally, a real physical circuit of the parametrically driven Rucklidge system is developed by using off-the-shelf electronic devices. The real-time measurement results such as time series, phase portraits and transformed phase portraits are in good qualitative agreement with those obtained from the numerical computations. The experimental evidence to verify bursting oscillations induced by delayed pitchfork bifurcation is thus provided in this study.

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