DOUBLE HOPF BIFURCATION IN DELAYED VAN DER POL–DUFFING EQUATION

2013 ◽  
Vol 23 (01) ◽  
pp. 1350014 ◽  
Author(s):  
YUTING DING ◽  
WEIHUA JIANG ◽  
PEI YU
Keyword(s):  
Hopf Bifurcation ◽  
Critical Point ◽  
Duffing Equation ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Van Der Pol ◽  
Dimensional Parameter ◽  
Center Manifold Reduction ◽  
Double Hopf Bifurcation ◽  
Reduction Methods

In this paper, we study dynamics in delayed van der Pol–Duffing equation, with particular attention focused on nonresonant double Hopf bifurcation. Both multiple time scales and center manifold reduction methods are applied to obtain the normal forms near a double Hopf critical point. A comparison between these two methods is given to show their equivalence. Bifurcations are classified in a two-dimensional parameter space near the critical point. Numerical simulations are presented to demonstrate the applicability of the theoretical results.

scholarly journals Double Hopf Bifurcation in Microbubble Oscillators with Delay Coupling

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Jinbin Wang ◽  
Rui Zhang ◽  
Lifenq Ma
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Two Dimensional ◽  
Main Attention ◽  
Dimensional Parameter ◽  
Center Manifold Reduction ◽  
Double Hopf Bifurcation ◽  
Physical Phenomena ◽  
Manifold Reduction ◽  
Theoretical Results

Using center manifold reduction methodswe investigate the double Hopf bifurcation in the dynamics of microbubble with delay couplingwith main attention focused on nonresonant double Hopf bifurcation. We obtain the normal form of the system in the vicinity of the double Hopf point and classify the bifurcations in a two-dimensional parameter space near the critical point. Some numerical simulations support the applicability of the theoretical results. In particularwe give the explanation for some physical phenomena of the system using the obtained mathematical results.

Double-Hopf Bifurcation in an Oscillator With External Forcing and Time-Delayed Feedback Control

2005 ◽  
Author(s):  
Zhen Chen ◽  
Pei Yu
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Approximate Solutions ◽  
Delayed Feedback Control ◽  
Critical Conditions ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Feedback Controls ◽  
Double Hopf Bifurcation ◽  
Manifold Theory

In this paper an oscillator with time delayed velocity feedback controls is studied in detail. The particular attention is focused on internal double-Hopf bifurcation with an external exciting force. Linear analysis is used to find the critical conditions under which a double-Hopf bifurcation occurs. Then center manifold theory is applied to obtain an ODE system described on a four-dimensional center manifold. Further, the technique of multiple-time scales is employed to find the approximate solutions of periodic and quasi-periodic motions. Finally, numerical simulation results are presented to verify the analytical predictions. Also, for some certain parameter values, numerical results show chaotic attractors.

DOUBLE-HOPF BIFURCATION IN AN OSCILLATOR WITH EXTERNAL FORCING AND TIME-DELAYED FEEDBACK CONTROL

2006 ◽  
Vol 16 (12) ◽  
pp. 3523-3537 ◽  
Author(s):  
ZHEN CHEN ◽  
PEI YU
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Approximate Solutions ◽  
Delayed Feedback Control ◽  
Critical Conditions ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Feedback Controls ◽  
Chaotic Motions ◽  
Double Hopf Bifurcation

In this paper, an oscillator with time delayed velocity feedback controls is studied in detail. Particular attention is given to internal double-Hopf bifurcation with an external exciting force. Linear analysis is used to find the critical conditions under which double-Hopf bifurcation occurs. Then center manifold theory is applied to obtain an ODE system described on a four-dimensional center manifold. Further, the technique of multiple-time scales is employed to find the approximate solutions of periodic and quasi-periodic motions. Finally, numerical simulation results are presented to not only validate the analytical predictions, but also show chaotic motions for some certain parameter values.

Bifurcation analysis in a predator–prey model for the effect of delay in prey

2016 ◽  
Vol 09 (04) ◽  
pp. 1650061 ◽  
Author(s):  
Qiubao Wang
Keyword(s):  
Critical Point ◽  
Center Manifold ◽  
Dimensional Parameter ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Reduction Methods ◽  
Manifold Reduction ◽  
Theoretical Results

In this paper, we study dynamics in a predator–prey model with delay, in which predator can be infected, with particular attention focused on nonresonant double Hopf bifurcation. By using center manifold reduction methods, we obtain the equivalent normal forms near a double Hopf critical point in this system. Moreover, bifurcations are classified in a two-dimensional parameter space near the critical point. Numerical simulations are presented to demonstrate the applicability of the theoretical results.

Feedback control of a nonlinear aeroelastic system with non-semi-simple eigenvalues at the critical point of Hopf bifurcation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Licai Wang ◽  
Yudong Chen ◽  
Chunyan Pei ◽  
Lina Liu ◽  
Suhuan Chen
Keyword(s):  
Hopf Bifurcation ◽  
Feedback Control ◽  
Critical Point ◽  
Control Parameter ◽  
Eigenvalue Problems ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
State Matrix ◽  
Aeroelastic System ◽  
Center Manifold Reduction

Abstract The feedback control of Hopf bifurcation of nonlinear aeroelastic systems with asymmetric aerodynamic lift force and nonlinear elastic forces of the airfoil is discussed. For the Hopf bifurcation analysis, the eigenvalue problems of the state matrix and its adjoint matrix are defined. The Puiseux expansion is used to discuss the variations of the non-semi-simple eigenvalues, as the control parameter passes through the critical value to avoid the difficulty for computing the derivatives of the non-semi-simple eigenvalues with respect to the control parameter. The method of multiple scales and center-manifold reduction are used to deal with the feedback control design of a nonlinear system with non-semi-simple eigenvalues at the critical point of the Hopf bifurcation. The first order approximate solutions are developed, which include gain vector and input. The presented methods are based on the Jordan form which is the simplest one. Finally, an example of an airfoil model is given to show the feasibility and for verification of the present method.

scholarly journals Double Hopf Bifurcation with Strong Resonances in Delayed Systems with Nonlinerities

2009 ◽  
Vol 2009 ◽  
pp. 1-16 ◽  
Author(s):  
J. Xu ◽  
K. W. Chung
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Nonlinear Systems ◽  
Efficient Method ◽  
Cubic Nonlinearity ◽  
Van Der Pol ◽  
Delayed Systems ◽  
Double Hopf Bifurcation ◽  
Codimension Two ◽  
System With Time Delay

An efficient method is proposed to study delay-induced strong resonant double Hopf bifurcation for nonlinear systems with time delay. As an illustration, the proposed method is employed to investigate the 1 : 2 double Hopf bifurcation in the van der Pol system with time delay. Dynamics arising from the bifurcation are classified qualitatively and expressed approximately in a closed form for either square or cubic nonlinearity. The results show that 1 : 2 resonance can lead to codimension-three and codimension-two bifurcations. The validity of analytical predictions is shown by their consistency with numerical simulations.

Equivalence of the MTS Method and CMR Method for Differential Equations Associated with Semisimple Singularity

2014 ◽  
Vol 24 (01) ◽  
pp. 1450003 ◽  
Author(s):  
Pei Yu ◽  
Yuting Ding ◽  
Weihua Jiang
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Delay Differential Equations ◽  
Normal Forms ◽  
Functional Differential Equations ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Center Manifold Reduction ◽  
Functional Differential ◽  
Delay Differential

In this paper, the equivalence of the multiple time scales (MTS) method and the center manifold reduction (CMR) method is proved for computing the normal forms of ordinary differential equations and delay differential equations. The delay equations considered include general delay differential equations (DDE), neutral functional differential equations (NFDE) (or neutral delay differential equations (NDDE)), and partial functional differential equations (PFDE). The delays involved in these equations can be discrete or distributed. Particular attention is focused on dynamics associated with the semisimple singularity, and both the MTS and CMR methods are applied to compute the normal forms near the semisimple singular point. For the ordinary differential equations (ODE), we show that the two methods are equivalent up to any order in computing the normal forms; while for the differential equations with delays, we obtain the conditions under which the normal forms, derived by using the MTS and CMR methods, are identical up to third order. Different types of practical examples with delays are presented to demonstrate the application of the theoretical results, associated with Hopf, Hopf-zero and double-Hopf singularities.

DYNAMICS OF RELAXATION OSCILLATIONS

2001 ◽  
Vol 11 (05) ◽  
pp. 1471-1482 ◽  
Author(s):  
PAUL E. PHILLIPSON ◽  
PETER SCHUSTER
Keyword(s):  
Perturbation Theory ◽  
Singular Perturbation ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Time Span ◽  
Singular Perturbation Theory ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Relaxation Oscillations ◽  
Van Der Pol

Relaxation oscillations are characteristic of periodic processes consisting of segments which differ greatly in time: a long-time span when the system is moving slowly and a relatively short time span when the system is moving rapidly. The period of oscillation, the sum of these contributions, is usually treated by singular perturbation theory which starts from the premise that the long span is asymptotically extended and the short span shrinks asymptotically to a single instant. Application of the theory involves the analysis of adjacent dynamical regions and multiple time scales. The relaxation oscillations of the Stoker–Haag piecewise-linear discontinuous equation and the van der Pol equation are investigated using a simpler analytical method requiring only the connection at a point of the two dynamical fast and slow regions. Compared to the results of singular perturbation theory, the quantitative results of the present method are more accurate in the Stoker–Haag case and marginally less accurate in the van der Pol case. The relative simplicity of the formulation suggests extension to three-dimensional systems where relaxation oscillations can become unstable leading to bistability, multiple periodicity and chaos.

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.

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