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.

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 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.

DYNAMICS AND DOUBLE HOPF BIFURCATIONS OF THE ROSE–HINDMARSH MODEL WITH TIME DELAY

2009 ◽  
Vol 19 (11) ◽  
pp. 3733-3751 ◽  
Author(s):  
SUQI MA ◽  
ZHAOSHENG FENG ◽  
QISHAI LU
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Value Of Time ◽  
Asymptotically Stable ◽  
Universal Unfolding ◽  
Double Hopf Bifurcation ◽  
Bifurcation Points ◽  
The Rose

In this paper, we are concerned with the Rose–Hindmarsh model with time delay. By applying the generalized Sturm criterion, a number of imaginary roots of the characteristic equation are classified. The absolutely stable regions for any value of time delay are detected. By the continuous software DDE-Biftool, both the Hopf bifurcation curves and double Hopf bifurcation points are illustrated in parametric spaces. The normal form and universal unfolding at double Hopf bifurcation points are considered by the center manifold method. Some examples also indicate that the corresponding unique attractor near each double Hopf point is asymptotically stable.

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 FOR STUART–LANDAU SYSTEM WITH NONLINEAR DELAY FEEDBACK AND DELAY-DEPENDENT PARAMETERS

2007 ◽  
Vol 10 (04) ◽  
pp. 423-448 ◽  
Author(s):  
SUQI MA ◽  
QISHAO LU ◽  
S. JOHN HOGAN
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Resonant Oscillation ◽  
Double Hopf Bifurcation ◽  
Dependent Parameter ◽  
Stable Periodic ◽  
Delay Feedback Control ◽  
Delay Dependent ◽  
Nonlinear Delay

A Stuart–Landau system under delay feedback control with the nonlinear delay-dependent parameter e-pτ is investigated. A geometrical demonstration method combined with theoretical analysis is developed so as to effectively solve the characteristic equation. Multi-stable regions are separated from unstable regions by allocations of Hopf bifurcation curves in (p,τ) plane. Some weak resonant and non-resonant oscillation phenomena induced by double Hopf bifurcation are discovered. The normal form for double Hopf bifurcation is deduced. The local dynamical behavior near double Hopf bifurcation points are also clarified in detail by using the center manifold method. Some states of two coexisting stable periodic solutions are verified, and some torus-broken procedures are also traced.

Stability and Hopf bifurcation analysis of a new four-dimensional hyper-chaotic system

2020 ◽  
Vol 34 (29) ◽  
pp. 2050327
Author(s):  
Liangqiang Zhou ◽  
Ziman Zhao ◽  
Fangqi Chen
Keyword(s):  
Hopf Bifurcation ◽  
Chaotic System ◽  
Bifurcation Theory ◽  
Center Manifold ◽  
Equilibrium Points ◽  
Chaotic Attractors ◽  
Local Dynamic ◽  
Chaotic Motions ◽  
System Parameters ◽  
Lyapunov Coefficient

With both analytical and numerical methods, local dynamic behaviors including stability and Hopf bifurcation of a new four-dimensional hyper-chaotic system are studied in this paper. All the equilibrium points and their stability conditions are obtained with the Routh–Hurwitz criterion. It is shown that there may exist one, two, or three equilibrium points for different system parameters. Via Hopf bifurcation theory, parameter conditions leading to Hopf bifurcation is presented. With the aid of center manifold and the first Lyapunov coefficient, it is also presented that the Hopf bifurcation is supercritical for some certain parameters. Finally, numerical simulations are given to confirm the analytical results and demonstrate the chaotic attractors of this system. It is also shown that the system may evolve chaotic motions through periodic bifurcations or intermittence chaos while the system parameters vary.

scholarly journals Valuation of Credit Derivatives with Multiple Time Scales in the Intensity Model

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Beom Jin Kim ◽  
Chan Yeol Park ◽  
Yong-Ki Ma
Keyword(s):  
Time Scales ◽  
Yield Curve ◽  
Credit Derivatives ◽  
Credit Default Swap ◽  
Approximate Solutions ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Bond Price ◽  
Defaultable Bond ◽  
Default Intensity

We propose approximate solutions for pricing zero-coupon defaultable bonds, credit default swap rates, and bond options based on the averaging principle of stochastic differential equations. We consider the intensity-based defaultable bond, where the volatility of the default intensity is driven by multiple time scales. Small corrections are computed using regular and singular perturbations to the intensity of default. The effectiveness of these corrections is tested on the bond price and yield curve by investigating the behavior of the time scales with respect to the relevant parameters.

Size-dependent vibrations of a micro beam conveying fluid and resting on an elastic foundation

2016 ◽  
Vol 23 (7) ◽  
pp. 1106-1114 ◽  
Author(s):  
Saim Kural ◽  
Erdoğan Özkaya
Keyword(s):  
Elastic Foundation ◽  
Couple Stress Theory ◽  
Fluid Velocity ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Approximate Solutions ◽  
Modified Couple Stress Theory ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Classical Beam Theory

In this study, fluid conveying continuous media was considered as micro beam. Unlike the classical beam theory, the effects of shear stress on micro-structure's dynamic behavior not negligible. Therefore, modified couple stress theory (MCST) were used to see the effects of being micro-sized. By using Hamilton's principle, the nonlinear equations of motion for the fluid conveying micro beam were obtained. Micro beam was considered as resting on an elastic foundation. The obtained equations of motion were became independence from material and geometric structure by nondimensionalization. Approximate solutions of the system were achieved with using the multiple time scales method (a perturbation method). The effects of micro-structure, spring constant, the occupancy rate of micro beam, the fluid velocity on natural frequency and solutions were researched. MCST compared with classical beam theory and showed that beam models that based on classical beam theory are not capable of describing the size effects. Comparisons of classical beam theory and MCST were showed in graphics and these graphics also proved that obtained mathematical model suitable for describe the behavior of normal sized beams.

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