Chaos Suppression via Integrative Time Delay Control

2020 ◽  
Vol 30 (14) ◽  
pp. 2050208
Author(s):  
Ayman A. Arafa ◽  
Yong Xu ◽  
Gamal M. Mahmoud
Keyword(s):  
Time Delay ◽  
Chaotic System ◽  
Chaotic Systems ◽  
Time Interval ◽  
General Strategy ◽  
Normal Form Theory ◽  
Chaos Suppression ◽  
Form Theory ◽  
The Stability ◽  
Time Delayed Feedback

A general strategy for suppressing chaos in chaotic Burke–Shaw system using integrative time delay (ITD) control is proposed, as an example. The idea of ITD is that the feedback is integrated over a time interval. Physically, the chaotic system responds to the average information it receives from the feedback. The main feature of integrative is that the stability of the chaotic system occurs over a wider range of the space parameters. Controlling chaotic systems with ITD has not been discussed before as far as we know. Stability and the existence of Hopf bifurcation are studied which demonstrate that the switch stability occurs at critical values of the time delay. Employing the normal form theory and center manifold argument, an explicit formula is derived to determine the stability and the direction of the bifurcating periodic solutions. Numerically, the bifurcation diagram and the eigenvalues of the corresponding characteristic equations are computed to supply a clear interpretation for suppressing chaos via ITD. Furthermore, ITD method is compared with the time delayed feedback (TDF) control numerically. This comparison shows that the stability area with ITD is larger than TDF which demonstrates the feasibility and effectiveness of the ITD. Other examples of chaotic systems can be similarly investigated.

scholarly journals Dynamics Analysis of a Mathematical Model for New Product Innovation Diffusion

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Chunru Li ◽  
Zujun Ma
Keyword(s):  
Mathematical Model ◽  
Time Delay ◽  
Media Coverage ◽  
New Product ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Significant Difference ◽  
Form Theory ◽  
The Difference ◽  
The Stability

In this paper, a mathematical model with time-delay-related parameters and media coverage to describe the diffusion process of new products is proposed, in which the time-delay-related parameters denote the stage in which potential customers decide whether to adopt a new product. Then, the stability and the Hopf bifurcation of the proposed model are analyzed in detail. The center manifold theorem and the normal form theory are used to investigate the stability of the bifurcating periodic solution. Moreover, a numerical simulation is conducted to investigate the difference between the model with delay-dependent parameters and that with delay-independent parameters. The results show that there is significant difference between the two models.

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 Control Scheme for a Fractional-Order Chaotic Genesio-Tesi Model

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Changjin Xu ◽  
Peiluan Li ◽  
Maoxin Liao ◽  
Zixin Liu ◽  
Qimei Xiao ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Fractional Order ◽  
Characteristic Equation ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Feedback Controllers ◽  
Control Scheme ◽  
The Stability ◽  
Time Delayed Feedback

In this paper, based on the earlier research, a new fractional-order chaotic Genesio-Tesi model is established. The chaotic phenomenon of the fractional-order chaotic Genesio-Tesi model is controlled by designing two suitable time-delayed feedback controllers. With the aid of Laplace transform, we obtain the characteristic equation of the controlled chaotic Genesio-Tesi model. Then by regarding the time delay as the bifurcation parameter and analyzing the characteristic equation, some new sufficient criteria to guarantee the stability and the existence of Hopf bifurcation for the controlled fractional-order chaotic Genesio-Tesi model are derived. The research shows that when time delay remains in some interval, the equilibrium point of the controlled chaotic Genesio-Tesi model is stable and a Hopf bifurcation will happen when the time delay crosses a critical value. The effect of the time delay on the stability and the existence of Hopf bifurcation for the controlled fractional-order chaotic Genesio-Tesi model is shown. At last, computer simulations check the rationalization of the obtained theoretical prediction. The derived key results in this paper play an important role in controlling the chaotic behavior of many other differential chaotic systems.

Bifurcations of Time-Delay-Induced Multiple Transitions Between In-Phase and Anti-Phase Synchronizations in Neurons with Excitatory or Inhibitory Synapses

2019 ◽  
Vol 29 (11) ◽  
pp. 1950147 ◽  
Author(s):  
Li Li ◽  
Zhiguo Zhao ◽  
Huaguang Gu
Keyword(s):  
Time Delay ◽  
Inhibitory Synapses ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Subthreshold Oscillations ◽  
Dynamical Behaviors ◽  
Stable Periodic ◽  
Form Theory ◽  
The Stability ◽  
Manifold Reduction

Time-delay-induced synchronous behaviors and synchronization transitions have been widely investigated for coupled neurons, and they play important roles for physiological functions. In the present study, time-delay-induced synchronized subthreshold oscillations were simulated, and the bifurcations underlying the synchronized behaviors were identified for a pair of coupled FitzHugh–Nagumo neurons. Multiple transitions between in-phase and anti-phase synchronizations induced by the time delay were simulated for the inhibitory and excitatory couplings. Subcritical or supercritical Hopf bifurcations and the stability of the Hopf-bifurcating periodic subthreshold oscillations were acquired using center manifold reduction and normal form theory. The in-phase or anti-phase synchronizations of the stable periodic subthreshold oscillations, which appear for multiple values of the time delay, were interpreted with the related eigenspace. The distributions of the different dynamical behaviors, including the synchronizations and bifurcations in the two-parameter plane of the time delay and coupling strength, were acquired for both types of synapses, and the different roles of the inhibitory and excitatory couplings on the synchronization transitions were compared.

STABILITY AND HOPF BIFURCATION OF A PREDATOR–PREY SYSTEM WITH TIME DELAY AND HOLLING TYPE-II FUNCTIONAL RESPONSE

2009 ◽  
Vol 02 (02) ◽  
pp. 139-149 ◽  
Author(s):  
LINGSHU WANG ◽  
RUI XU ◽  
GUANGHUI FENG
Keyword(s):  
Time Delay ◽  
Functional Response ◽  
Type Ii ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Form Theory ◽  
The Stability ◽  
Type Ii Functional Response

A predator–prey model with time delay and Holling type-II functional response is investigated. By choosing time delay as the bifurcation parameter and analyzing the associated characteristic equation of the linearized system, the local stability of the system is investigated and Hopf bifurcations are established. The formulae determining the direction of bifurcations and the stability of bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. Numerical simulations are carried out to illustrate the theoretical results.

scholarly journals Hopf Bifurcation Analysis and Chaos Control of a Chaotic System withoutilnikov Orbits

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Na Li ◽  
Wei Tan ◽  
Huitao Zhao
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic System ◽  
Chaos Control ◽  
Chaotic Oscillation ◽  
Delayed Feedback Control ◽  
Normal Form Theory ◽  
Local Hopf Bifurcation ◽  
Dynamical Behaviors ◽  
Form Theory ◽  
The Stability

This paper mainly investigates the dynamical behaviors of a chaotic system withoutilnikov orbits by the normal form theory. Both the stability of the equilibria and the existence of local Hopf bifurcation are proved in view of analyzing the associated characteristic equation. Meanwhile, the direction and the period of bifurcating periodic solutions are determined. Regarding the delay as a parameter, we discuss the effect of time delay on the dynamics of chaotic system with delayed feedback control. Finally, numerical simulations indicate that chaotic oscillation is converted into a steady state when the delay passes through a certain critical value.

scholarly journals Bifurcation Analysis for a Kind of Nonlinear Finance System with Delayed Feedback and Its Application to Control of Chaos

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Rongyan Zhang
Keyword(s):  
Center Manifold ◽  
Chaotic Oscillation ◽  
Delayed Feedback ◽  
Stable Steady State ◽  
Normal Form Theory ◽  
Finance System ◽  
Stable Periodic ◽  
Form Theory ◽  
The Stability ◽  
Time Delayed Feedback

A kind of nonlinear finance system with time-delayed feedback is considered. Firstly, by employing the polynomial theorem to analyze the distribution of the roots to the associate characteristic equation, the conditions of ensuring the existence of Hopf bifurcation are given. Secondly, by using the normal form theory and center manifold argument, we derive the explicit formulas determining the stability, direction, and other properties of bifurcating periodic solutions. Finally, we give several numerical simulations, which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable steady state or a stable periodic orbit.

scholarly journals Bifurcation Analysis for a Predator-Prey Model with Time Delay and Delay-Dependent Parameters

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Changjin Xu
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Form Theory ◽  
The Stability ◽  
Manifold Theory ◽  
Delay Dependent

A class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. Its linear stability is investigated and Hopf bifurcation is demonstrated. Using normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained. Finally, numerical simulations are performed to verify the analytical results.

scholarly journals Hopf Bifurcation Analysis for the Model of the Chemostat with One Species of Organism

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Haiyun Bai ◽  
Yanhui Zhai
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Center Manifold ◽  
Stability Switches ◽  
Chemostat Model ◽  
Explicit Formulas ◽  
Normal Form Theory ◽  
Form Theory ◽  
The Stability

We research the dynamics of the chemostat model with time delay. The conclusion confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. By using the normal form theory and center manifold method, we derive the explicit formulas determining the stability and direction of bifurcating periodic solutions. Finally, some numerical simulations are given to illustrate the effectiveness of our results.

scholarly journals Modelling the Drugs Therapy for HIV Infection with Discrete-Time Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Xueyong Zhou ◽  
Xiangyun Shi
Keyword(s):  
Time Delay ◽  
Hiv Infection ◽  
Discrete Time ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Discrete Time Delay ◽  
Form Theory ◽  
The Stability ◽  
Delay Differential ◽  
Two Equilibria

A discrete-time-delay differential mathematical model that described HIV infection of CD4+T cells with drugs therapy is analyzed. The stability of the two equilibria and the existence of Hopf bifurcation at the positive equilibrium are investigated. Using the normal form theory and center manifold argument, the explicit formulas which determine the stability, the direction, and the period of bifurcating periodic solutions are derived. Numerical simulations are carried out to explain the mathematical conclusions.

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