Multiple Bifurcations and Complex Responses of Nonlinear Time-Delay Oscillators

2021 ◽  
Author(s):  
Xiaochen Mao ◽  
Fuchen Lei ◽  
Xingyong Li ◽  
Weijie Ding ◽  
Tiantian Shi
Keyword(s):  
Time Delay ◽  
Time Delays ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Electronic Circuit ◽  
Dynamic Features ◽  
Nonlinear Circuit ◽  
Amplitude Death ◽  
Van Der Pol ◽  
Dynamical Behaviors

Abstract In this paper, the dynamical properties of multiple van der Pol-Duffing oscillators with time delays are studied. The amplitude death and bifurcation curves in the parameter plane are determined by using the space decomposition method. Different patterns of bifurcated solutions are given on the basis of the symmetric bifurcation theory. The properties of bifurcated solutions are shown by using the norm forms on the center manifold. The interactions of bifurcations are discussed and their dynamical behaviors are shown. An electronic circuit platform is implemented by means of nonlinear circuit and time delay circuit. The revealed behaviors of the circuit reach an agreement with the obtained results. It is shown that the nonlinearity and time delays have great effects on the system performance and can induce interesting and abundant dynamic features.

scholarly journals Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease with Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yuanyuan Chen ◽  
Ya-Qing Bi
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Local Analysis ◽  
Normal Form Theory ◽  
Form Theory ◽  
Vector Borne ◽  
The Stability ◽  
Delay Differential

A delay-differential modelling of vector-borne is investigated. Its dynamics are studied in terms of local analysis and Hopf bifurcation theory, and its linear stability and Hopf bifurcation are demonstrated by studying the characteristic equation. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold argument.

scholarly journals Bifurcations and Dynamics of the Rb-E2F Pathway Involving miR449

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-20
Author(s):  
Lingling Li ◽  
Jianwei Shen
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Numerical Simulations ◽  
Bifurcation Theory ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Delay Model ◽  
Dynamical Behaviors ◽  
Theoretical Results

We focused on the gene regulative network involving Rb-E2F pathway and microRNAs (miR449) and studied the influence of time delay on the dynamical behaviors of Rb-E2F pathway by using Hopf bifurcation theory. It is shown that under certain assumptions the steady state of the delay model is asymptotically stable for all delay values; there is a critical value under another set of conditions; the steady state is stable when the time delay is less than the critical value, while the steady state is changed to be unstable when the time delay is greater than the critical value. Thus, Hopf bifurcation appears at the steady state when the delay passes through the critical value. Numerical simulations were presented to illustrate the theoretical results.

scholarly journals Dynamic complexity and bifurcation analysis of a host–parasitoid model with Allee effect and Holling type III functional response

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hua Liu ◽  
Kai Zhang ◽  
Yong Ye ◽  
Yumei Wei ◽  
Ming Ma
Keyword(s):  
Local Stability ◽  
Bifurcation Theory ◽  
Allee Effect ◽  
Center Manifold ◽  
Discrete Time System ◽  
Time System ◽  
Dynamic Complexity ◽  
Center Manifold Theorem ◽  
Dynamical Behaviors ◽  
Local Stability Analysis

AbstractIn this paper, we focus on dynamics in a basic discrete-time system of host–parasitoid interaction. We perform local stability analysis of this system. Furthermore, both flip and Neimark–Sacker bifurcations are also analyzed in the interior of $R_{ +}^{2}$R+2 by using center manifold theorem and bifurcation theory. Finally, numerical simulations are deployed to validate our results with theoretical analysis and to exhibit the dynamical behaviors.

scholarly journals Marine Autopilots’ Multipurpose Control Laws Synthesis for Actuators Time Delay

2020 ◽  
Vol 8 (7) ◽  
pp. 477 ◽  
Author(s):  
Evgeny I. Veremey ◽  
Sergei V. Pogozhev ◽  
Margarita V. Sotnikova
Keyword(s):  
Time Delay ◽  
Time Delays ◽  
Closed Loop ◽  
Controller Design ◽  
Dynamic Features ◽  
Closed Loop System ◽  
Control Laws ◽  
New Approach ◽  
Feedback Synthesis ◽  
Autopilot Systems

One analytical design problem involves constructing control laws for marine autopilot systems. Despite numerous known solutions, this problem can still be further developed by taking into account the actual conditions of the control system operation. An important issue for discussion is the feedback synthesis for marine ships with time delays in their rudders’ actuators. In this work, a new approach is proposed for providing all the desirable dynamic features of a closed-loop system with autopilot while taking into account the presence of a time delay. This approach is based on the predictive compensation of time delays via the specific transformation of an initially given reference controller with a special multipurpose structure. The applicability and effectiveness of the proposed method is further illustrated by a practical example of a controller design.

Bifurcation Control for a Kind of Non-Autonomous System with Time Delay

2010 ◽  
Vol 34-35 ◽  
pp. 1752-1756
Author(s):  
Chang Zhao Qian ◽  
Zhi Wen Wang ◽  
Chuang Wen Dong ◽  
Yang Liu
Keyword(s):  
Time Delay ◽  
Perturbation Method ◽  
Time Delays ◽  
Autonomous System ◽  
Primary Resonance ◽  
Stable Region ◽  
Bifurcation Equation ◽  
Van Der Pol ◽  
Average Equation ◽  
System With Time Delay

A forced van der Pol system with two time-delays is studied. The central aim is analyzing primary resonance of this system. Perturbation method is used to obtain the average equation and bifurcation equation with time-delays. Based on the average equation, the stable region of this system is discussed. Based on the bifurcation equation, the multivalued property of response amplitude is studied. The result indicates that this system can be well controlled with time delays.

scholarly journals Hopf Bifurcation of an Improved SLBS Model under the Influence of Latent Period

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Chunming Zhang ◽  
Wanping Liu ◽  
Jing Xiao ◽  
Yun Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Dynamical Behaviors ◽  
Latent Time ◽  
Normal Form Method ◽  
The Stability ◽  
Theoretical Results

A model applicable to describe the propagation of computer virus is developed and studied, along with the latent time incorporated. We regard time delay as a bifurcating parameter to study the dynamical behaviors including local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when the time delay passes through a sequence of critical values. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions is given by using the normal form method and center manifold theorem. Finally, illustrative examples are given to support the theoretical results.

Dynamical Behaviors in Complex-Valued Love Model With or Without Time Delays

2017 ◽  
Vol 27 (13) ◽  
pp. 1750200 ◽  
Author(s):  
Wei Deng ◽  
Xiaofeng Liao ◽  
Tao Dong
Keyword(s):  
Time Delays ◽  
Romantic Relationship ◽  
Center Manifold ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Dynamical Behaviors ◽  
Real Domain ◽  
Form Theory ◽  
Complex Valued ◽  
Love Dynamics

In this paper, a novel version of nonlinear model, i.e. a complex-valued love model with two time delays between two individuals in a love affair, has been proposed. A notable feature in this model is that we separate the emotion of one individual into real and imaginary parts to represent the variation and complexity of psychophysiological emotion in romantic relationship instead of just real domain, and make our model much closer to reality. This is because love is a complicated cognitive and social phenomenon, full of complexity, diversity and unpredictability, which refers to the coexistence of different aspects of feelings, states and attitudes ranging from joy and trust to sadness and disgust. By analyzing associated characteristic equation of linearized equations for our model, it is found that the Hopf bifurcation occurs when the sum of time delays passes through a sequence of critical value. Stability of bifurcating cyclic love dynamics is also derived by applying the normal form theory and the center manifold theorem. In addition, it is also shown that, for some appropriate chosen parameters, chaotic behaviors can appear even without time delay.

scholarly journals The Dynamical Behaviors for a Class of Immunogenic Tumor Model with Delay

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Ping Bi ◽  
Zijian Liu ◽  
Mutei Damaris Muthoni ◽  
Jianhua Pang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Tumor Model ◽  
Numerical Examples ◽  
Dynamical Behaviors ◽  
Existence And Stability ◽  
Discrete Delays

This paper aims at studying the model proposed by Kuznetsov and Taylor in 1994. Inspired by Mayer et al., time delay is introduced in the general model. The dynamic behaviors of this model are studied, which include the existence and stability of the equilibria and Hopf bifurcation of the model with discrete delays. The properties of the bifurcated periodic solutions are studied by using the normal form on the center manifold. Numerical examples and simulations are given to illustrate the bifurcation analysis and the obtained results.

scholarly journals Stability and Hopf Bifurcation for a Delayed SLBRS Computer Virus Model

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Computer Virus ◽  
Center Manifold Theory ◽  
Dynamical Behaviors ◽  
Virus Model ◽  
Normal Form Method ◽  
Manifold Theory

By incorporating the time delay due to the period that computers use antivirus software to clean the virus into the SLBRS model a delayed SLBRS computer virus model is proposed in this paper. The dynamical behaviors which include local stability and Hopf bifurcation are investigated by regarding the delay as bifurcating parameter. Specially, direction and stability of the Hopf bifurcation are derived by applying the normal form method and center manifold theory. Finally, an illustrative example is also presented to testify our analytical results.

scholarly journals The Bifurcation of Two Invariant Closed Curves in a Discrete Model

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Yingying Zhang ◽  
Yicang Zhou
Keyword(s):  
Discrete Model ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Euler Method ◽  
Flip Bifurcation ◽  
Closed Curves ◽  
Dynamical Behaviors ◽  
Discrete Population ◽  
Forward Euler ◽  
Forward Euler Method

A discrete population model integrated using the forward Euler method is investigated. The qualitative bifurcation analysis indicates that the model exhibits rich dynamical behaviors including the existence of the equilibrium state, the flip bifurcation, the Neimark-Sacker bifurcation, and two invariant closed curves. The conditions for existence of these bifurcations are derived by using the center manifold and bifurcation theory. Numerical simulations and bifurcation diagrams exhibit the complex dynamical behaviors, especially the occurrence of two invariant closed curves.

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