Hopf and resonant codimension two bifurcation in van der Pol equation with two time delays

2005 ◽  
Vol 23 (3) ◽  
pp. 857-871 ◽  
Author(s):  
Xiaofeng Liao
Keyword(s):  
Time Delays ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Codimension Two ◽  
Codimension Two Bifurcation

Stability and Bifurcation Analysis of a Network of Four Neurons With Time Delays

2010 ◽  
Vol 5 (4) ◽  
Author(s):  
Xiaochen Mao ◽  
Haiyan Hu
Keyword(s):  
Local Stability ◽  
Lyapunov Functional ◽  
Time Delays ◽  
Nearest Neighbor ◽  
Sufficient Conditions ◽  
Codimension Two ◽  
Codimension One ◽  
Dynamical Behaviors ◽  
Trivial Equilibrium ◽  
Codimension Two Bifurcation

This paper reveals the dynamical behaviors of a bidirectional neural network consisting of four neurons with delayed nearest-neighbor and shortcut connections. The criterion of the global asymptotic stability of the trivial equilibrium of the network is derived by means of a suitable Lyapunov functional. The local stability of the trivial equilibrium is investigated by analyzing the distributions of roots of the associated characteristic equation. The sufficient conditions for the existence of nontrivial synchronous and asynchronous equilibria and periodic oscillations arising from codimension one bifurcations are obtained. Multistability near the codimension two bifurcation points is presented. Numerical simulations are given to validate the theoretical analysis.

Fuzzy Codimension Two Bifurcations

2011 ◽  
Author(s):  
Ling Hong ◽  
Jian-Qiao Sun
Keyword(s):  
Symmetry Breaking ◽  
Mapping Method ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Generalized Cell Mapping ◽  
Codimension Two ◽  
Deterministic Systems ◽  
Breaking Parameter ◽  
Fuzzy Noise ◽  
Codimension Two Bifurcation

By means of fuzzy generalized cell mapping method, a Duffing-Van der Pol oscillator in the presence of fuzzy noise is studied in a regime where two symmetrically related fuzzy period-one attractors grow and merge as the intensity of fuzzy noise is increased. By introducing a small symmetry-breaking parameter to break the symmetry, the merging explosion bifurcation unfolds to a pattern of two catastrophic and explosive bifurcations. Considering both the intensity of fuzzy noise and the symmetry-breaking parameter together as controls, a codimension two bifurcation of fuzzy attractors is defined, and two examples of additive and multiplicative fuzzy noise are given. Such a codimension two bifurcation is fuzzy noise-induced effects which cannot be seen in the deterministic systems.

LOCAL STABILITY, HOPF AND RESONANT CODIMENSION-TWO BIFURCATION IN A HARMONIC OSCILLATOR WITH TWO TIME DELAYS

2001 ◽  
Vol 11 (08) ◽  
pp. 2105-2121 ◽  
Author(s):  
XIAOFENG LIAO ◽  
GUANRONG CHEN
Keyword(s):  
Harmonic Oscillator ◽  
Characteristic Equation ◽  
Local Stability ◽  
Time Delays ◽  
Zero Solution ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Codimension Two ◽  
Form Theory ◽  
Codimension Two Bifurcation

A harmonic oscillator with two discrete time delays is considered. The local stability of the zero solution of this equation is investigated by analyzing the corresponding transcendental characteristic equation of its linearized equation and employing the Nyquist criterion. Some general stability criteria involving the delays and the system parameters are derived. By choosing one of the delays as a bifurcation parameter, the model is found to undergo a sequence of Hopf bifurcation. The direction and stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Resonant codimension-two bifurcation is also found to occur in this model. A complete description is given to the location of points in the parameter space at which the transcendental characteristic equation possesses two pairs of pure imaginary roots, ±iω1, ±iω2 with ω1:ω2 = m:n, where m and n are positive integers. Some numerical examples are finally given for justifying the theoretical results.

scholarly journals On Periodic Perturbations of Asymmetric Duffing–Van-der-Pol Equation

2014 ◽  
Vol 24 (05) ◽  
pp. 1450061 ◽  
Author(s):  
Albert D. Morozov ◽  
Olga S. Kostromina
Keyword(s):  
Saddle Point ◽  
Limit Cycles ◽  
Small Neighborhood ◽  
Integrable Equation ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Periodic Perturbations ◽  
Autonomous Equation ◽  
Nonautonomous Equation ◽  
Time Periodic

Time-periodic perturbations of an asymmetric Duffing–Van-der-Pol equation close to an integrable equation with a homoclinic "figure-eight" of a saddle are considered. The behavior of solutions outside the neighborhood of "figure-eight" is studied analytically. The problem of limit cycles for an autonomous equation is solved and resonance zones for a nonautonomous equation are analyzed. The behavior of the separatrices of a fixed saddle point of the Poincaré map in the small neighborhood of the unperturbed "figure-eight" is ascertained. The results obtained are illustrated by numerical computations.

NEW PHENOMENA IN THE NEIGHBORHOOD OF THE CODIMENSION-TWO BIFURCATION IN THE TWIN-WELL DUFFING OSCILLATOR

2000 ◽  
Vol 10 (06) ◽  
pp. 1367-1381 ◽  
Author(s):  
W. SZEMPLIŃSKA-STUPNICKA ◽  
A. ZUBRZYCKI ◽  
E. TYRKIEL
Keyword(s):  
Bifurcation Point ◽  
Chaotic Attractor ◽  
Duffing Oscillator ◽  
Codimension Two ◽  
Periodic Attractors ◽  
The Cross ◽  
Complex Phenomena ◽  
New Phenomena ◽  
Secondary Bifurcations ◽  
Codimension Two Bifurcation

In this paper, we study effects of the secondary bifurcations in the neighborhood of the primary codimension-two bifurcation point. The twin-well potential Duffing oscillator is considered and the investigations are focused on the new scenario of destruction of the cross-well chaotic attractor. The phenomenon belongs to the category of the subduction scenario and relies on the replacement of the cross-well chaotic attractor by a pair of unsymmetric 2T-periodic attractors. The exploration of a sequence of accompanying bifurcations throws more light on the complex phenomena that may occur in the neighborhood of the primary codimension-two bifurcation point. It shows that in the close vicinity of the point there appears a transition zone in the system parameter plane, the zone which separates the two so-far investigated scenarios of annihilation of the cross-well chaotic attractor.

On Integrating Factors and a Perturbation Method

1995 ◽  
Author(s):  
W. T. van Horssen
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
First Integrals ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
Integrating Factors ◽  
Complete Solutions

Abstract In this paper the fundamental concept (due to Euler, 1734) of how to make a first order ordinary differential equation exact by means of integrating factors, is extended to n-th order (n ≥ 2) ordinary differential equations and to systems of first order ordinary differential equations. For new classes of differential equations first integrals or complete solutions can be constructed. Also a perturbation method based on integrating factors can be developed. To show how this perturbation method works the method is applied to the well-known Van der Pol equation.

Bogdanov–Takens and Triple Zero Bifurcations of Coupled van der Pol–Duffing Oscillators with Multiple Delays

2017 ◽  
Vol 27 (09) ◽  
pp. 1750133 ◽  
Author(s):  
Xia Liu ◽  
Tonghua Zhang
Keyword(s):  
Delay Differential Equations ◽  
Time Delays ◽  
Normal Forms ◽  
Multiple Delays ◽  
Third Order ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
Form Theory ◽  
Delay Differential ◽  
Theoretical Results

In this paper, the Bogdanov–Takens (B–T) and triple zero bifurcations are investigated for coupled van der Pol–Duffing oscillators with [Formula: see text] symmetry, in the presence of time delays due to the intrinsic response and coupling. Different from previous works, third order unfolding normal forms associated with B–T and triple zero bifurcations are needed, which are obtained by using the normal form theory of delay differential equations. Numerical simulations are also presented to illustrate the theoretical results.

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