scholarly journals On Stability Switches and Bifurcation of the Modified Autonomous Van der Pol–Duffing Equations via Delayed State Feedback Control

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2336
Author(s):  
Tiao-Yang Cai ◽  
Hui-Long Jin ◽  
Hong Yu ◽  
Xiang-Peng Xie
Keyword(s):  
Hopf Bifurcation ◽  
State Feedback ◽  
Nonlinear Dynamical Systems ◽  
State Feedback Control ◽  
Dynamic State ◽  
Stability Switches ◽  
Positive Real ◽  
Van Der Pol ◽  
Cubic Equation ◽  
Manifold Theory

This paper considers the Modified Autonomous Van der Pol–Duffing equation subjected to dynamic state feedback, which can well characterize the dynamic behaviors of the nonlinear dynamical systems. Both the issues of local stability switches and the Hopf bifurcation versus time delay are investigated. Associating with the τ decomposition strategy and the center manifold theory, the delay stable intervals and the direction and stability of the Hopf bifurcation are all determined. Specifically, the computation of purely imaginary roots (symmetry to the real axis), the positive real root formula for cubic equation and the sophisticated bilinear form of adjoint operators are proposed, which make the calculations mentioned in our discussion unified and simple. Finally, the typical numerical examples are shown to illustrate the correctness and effectiveness of the practical technique.

scholarly journals Double Hopf Bifurcation of a Hematopoietic System with State Feedback Control

2018 ◽  
Vol 10 (4) ◽  
pp. 116
Author(s):  
Suqi Ma
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
State Feedback ◽  
Threshold Value ◽  
State Feedback Control ◽  
Hematopoietic Stem ◽  
Double Hopf Bifurcation ◽  
Periodical Solution ◽  
Varying Time Delay ◽  
Manifold Theory

The dynamics of a system composed of hematopoietic stem cells and its relationship with neutrophils is ubiquitous due to periodic oscillating behavior induce cyclical neutropenia. Underlying the methodology of state feedback control with two time delays, double Hopf bifurcation occurs as varying time delay to reach its threshold value. By applying center manifold theory, the analytical analysis of system exposed the different dynamical feature in the classified regimes near double Hopf point. The novel dynamics as periodical solution and quasi-periodical attractor coexistence phenomena are explored and verified  by numerical simulation.

scholarly journals An Approach to Robust Control of the Hopf Bifurcation

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Giacomo Innocenti ◽  
Roberto Genesio ◽  
Alberto Tesi
Keyword(s):  
Hopf Bifurcation ◽  
State Feedback ◽  
Center Manifold ◽  
State Feedback Control ◽  
Control Input ◽  
Nonlinear Transformations ◽  
Third Order ◽  
Novel Approach ◽  
Linear And Nonlinear ◽  
Nonlinear State

The paper illustrates a novel approach to modify the Hopf bifurcation nature via a nonlinear state feedback control, which leaves the equilibrium properties unchanged. This result is achieved by recurring to linear and nonlinear transformations, which lead the system to locally assume the ordinary differential equation representation. Third-order models are considered, since they can be seen as proper representatives of a larger class of systems. The explicit relationship between the control input and the Hopf bifurcation nature is obtained via a frequency approach, that does not need the computation of the center manifold.

scholarly journals Hopf Bifurcation Control in a Delayed Predator-Prey System with Prey Infection and Modified Leslie-Gower Scheme

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
State Feedback ◽  
Center Manifold ◽  
Controlled System ◽  
Predator Prey ◽  
Hybrid Controller ◽  
Prey System ◽  
The Stability ◽  
Manifold Theory ◽  
Bifurcation And Stability

Hopf bifurcation of a delayed predator-prey system with prey infection and the modified Leslie-Gower scheme is investigated. The conditions for the stability and existence of Hopf bifurcation of the system are obtained. The state feedback and parameter perturbation are used for controlling Hopf bifurcation in the system. In addition, direction of Hopf bifurcation and stability of the bifurcated periodic solutions of the controlled system are obtained by using normal form and center manifold theory. Finally, numerical simulation results are presented to show that the hybrid controller is efficient in controlling Hopf bifurcation.

scholarly journals Hopf Bifurcation Analysis for the Modified Rayleigh Price Model with Time Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Yanhui Zhai ◽  
Haiyun Bai ◽  
Ying Xiong ◽  
Xiaona Ma
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Normal Form ◽  
Center Manifold ◽  
Center Manifold Theory ◽  
Price Model ◽  
Stability Switches ◽  
Normal Form Theory ◽  
Form Theory ◽  
Manifold Theory

This paper mainly modifies and further develops the Reyleigh price model. By modifying the basic Reyleigh model, we can more accurately illustrate the economic phenomena with price varying. First, we research the dynamics of the modified Reyleigh model with time delay. By employing the normal form theory and center manifold theory, we obtain some testable results on these issues. The conclusion confirms that a Hopf bifurcation occurs due to the existence of stability switches when the delay varies. Finally, some numerical simulations are given to illustrate the effectiveness of our results.

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