Coupled Oscillatory Systems with đť”»4 Symmetry and Application to van der Pol Oscillators

2016 ◽  
Vol 26 (08) ◽  
pp. 1650141 ◽  
Author(s):  
Adrian C. Murza ◽  
Pei Yu
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Coupled Oscillators ◽  
Invariant Manifolds ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
Oscillatory Systems ◽  
Weakly Coupled ◽  
Form Theory ◽  
Van Der Pol Oscillators

In this paper, we study the dynamics of autonomous ODE systems with [Formula: see text] symmetry. First, we consider eight weakly-coupled oscillators and establish the condition for the existence of stable heteroclinic cycles in most generic [Formula: see text]-equivariant systems. Then, we analyze the action of [Formula: see text] on [Formula: see text] and study the pattern of periodic solutions arising from Hopf bifurcation. We identify the type of periodic solutions associated with the pairs [Formula: see text] of spatiotemporal or spatial symmetries, and prove their existence by using the [Formula: see text] Theorem due to Hopf bifurcation and the [Formula: see text] symmetry. In particular, we give a rigorous proof for the existence of a fourth branch of periodic solutions in [Formula: see text]-equivariant systems. Further, we apply our theory to study a concrete case: two coupled van der Pol oscillators with [Formula: see text] symmetry. We use normal form theory to analyze the periodic solutions arising from Hopf bifurcation. Among the families of the periodic solutions, we pay particular attention to the phase-locked oscillations, each of them being embedded in one of the invariant manifolds, and identify the in-phase, completely synchronized motions. We derive their explicit expressions and analyze their stability in terms of the parameters.

scholarly journals Stability and Hopf-Bifurcation Analysis of Delayed BAM Neural Network under Dynamic Thresholds

2009 ◽  
Vol 14 (4) ◽  
pp. 435-461 ◽  
Author(s):  
P. D. Gupta ◽  
N. C. Majee ◽  
A. B. Roy
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Global Bifurcation ◽  
Distributed Delay ◽  
Degree Theory ◽  
Homotopy Invariance ◽  
Normal Form Theory ◽  
Form Theory ◽  
Bam Neural Network ◽  
Uniqueness Of Equilibrium

In this paper the dynamics of a three neuron model with self-connection and distributed delay under dynamical threshold is investigated. With the help of topological degree theory and Homotopy invariance principle existence and uniqueness of equilibrium point are established. The conditions for which the Hopf-bifurcation occurs at the equilibrium are obtained for the weak kernel of the distributed delay. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and central manifold theorem. Lastly global bifurcation aspect of such periodic solutions is studied. Some numerical simulations for justifying the theoretical analysis are also presented.

scholarly journals Dynamical Analysis in a Delayed Predator-Prey System with Stage-Structure for Both the Predator and the Prey

10.1155/2014/909238 ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-19
Author(s):  
Zizhen Zhang ◽  
Huizhong Yang
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
The Stability

A predator-prey system with two delays and stage-structure for both the predator and the prey is considered. Sufficient conditions for the local stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays are obtained by analyzing the distribution of the roots of the associated characteristic equation. Specially, the direction of the Hopf bifurcation and the stability of the periodic solutions bifurcating from the Hopf bifurcation are determined by applying the normal form theory and center manifold argument. Some numerical simulations for justifying the theoretical analysis are also provided.

scholarly journals Direction and Stability of Bifurcating Periodic Solutions in a Delay-Induced Ecoepidemiological System

10.1155/2011/978387 ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-25 ◽  
Author(s):  
N. Bairagi
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Critical Value ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Reproduction Period ◽  
The Stability ◽  
Parameter Values ◽  
Bifurcating Periodic Solution

A SI-type ecoepidemiological model that incorporates reproduction delay of predator is studied. Considering delay as parameter, we investigate the effect of delay on the stability of the coexisting equilibrium. It is observed that there is stability switches, and Hopf bifurcation occurs when the delay crosses some critical value. By applying the normal form theory and the center manifold theorem, the explicit formulae which determine the stability and direction of the bifurcating periodic solutions are determined. Computer simulations have been carried out to illustrate different analytical findings. Results indicate that the Hopf bifurcation is supercritical and the bifurcating periodic solution is stable for the considered parameter values. It is also observed that the quantitative level of abundance of system populations depends crucially on the delay parameter if the reproduction period of predator exceeds the critical value.

scholarly journals Hopf bifurcation in a delayed predator-prey system with asymmetric functional response and additional food

AIMS Mathematics ◽  
10.3934/math.2021708 ◽  
2021 ◽  
Vol 6 (11) ◽  
pp. 12225-12244
Author(s):  
Luoyi Wu ◽  
◽  
Hang Zheng ◽  
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Functional Response ◽  
Additional Food ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Local Hopf Bifurcation ◽  
Prey System ◽  
Form Theory

<abstract><p>In this paper, a delayed predator-prey system with additional food and asymmetric functional response is investigated. We discuss the local stability of equilibria and the existence of local Hopf bifurcation under the influence of the time delay. By using the normal form theory and center manifold theorem, the explicit formulas which determine the properties of bifurcating periodic solutions are obtained. Further, we prove that global periodic solutions exist after the second critical value of delay via Wu's theory. Finally, the correctness of the previous theoretical analysis is demonstrated by some numerical cases.</p></abstract>

scholarly journals Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays

10.1155/2011/569141 ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Xiaobing Zhou ◽  
Murong Jiang ◽  
Xiaomei Cai
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Distributed Delays ◽  
Hopf Bifurcations ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Discrete Time Delay ◽  
Form Theory

We consider the van der Pol equation with discrete and distributed delays. Linear stability of this equation is investigated by analyzing the transcendental characteristic equation of its linearized equation. It is found that this equation undergoes a sequence of Hopf bifurcations by choosing the discrete time delay as a bifurcation parameter. In addition, the properties of Hopf bifurcation were analyzed in detail by applying the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate and verify the theoretical analysis.

Delay-Induced Double Hopf Bifurcations in a System of Two Delay-Coupled van der Pol–Duffing Oscillators

2015 ◽  
Vol 25 (04) ◽  
pp. 1550058 ◽  
Author(s):  
Heping Jiang ◽  
Tonghua Zhang ◽  
Yongli Song
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Hopf Bifurcations ◽  
Van Der Pol ◽  
Normal Form Theory ◽  
Double Hopf Bifurcation ◽  
Codimension Two ◽  
Form Theory ◽  
Weak Resonance ◽  
Delay Differential

In this paper, we investigate the codimension-two double Hopf bifurcation in delay-coupled van der Pol–Duffing oscillators. By using normal form theory of delay differential equations, the normal form associated with the codimension-two double Hopf bifurcation is calculated. Choosing appropriate values of the coupling strength and the delay can result in nonresonance and weak resonance double Hopf bifurcations. The dynamical classification near these bifurcation points can be explicitly determined by the corresponding normal form. Periodic, quasi-periodic solutions and torus are found near the bifurcation point. The numerical simulations are employed to support the theoretical results.

scholarly journals Modulation of contractions in the small intestine indicate desynchronization via supercritical Andronov–Hopf bifurcation

Scientific Reports ◽  
2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Sean P. Parsons ◽  
Jan D. Huizinga
Keyword(s):  
Small Intestine ◽  
Hopf Bifurcation ◽  
Coupled Oscillators ◽  
Interstitial Cells Of Cajal ◽  
Van Der Pol ◽  
Physical Systems ◽  
Fourier Frequency ◽  
Van Der Pol Oscillators ◽  
A Chain ◽  
Cells Of Cajal

Abstract The small intestine is covered by a network of coupled oscillators, the interstitial cells of Cajal (ICC). These oscillators synchronize to generate rhythmic phase waves of contraction. At points of low coupling, oscillations desynchronise, frequency steps occur and every few waves terminates as a dislocation. The amplitude of contractions is modulated at frequency steps. The phase difference between contractions at a frequency step and a proximal reference point increased slowly at first and then, just at the dislocation, increased rapidly. Simultaneous frequency and amplitude modulation (AM/FM) results in a Fourier frequency spectrum with a lower sideband, a so called Lashinsky spectrum, and this was also seen in the small intestine. A model of the small intestine consisting of a chain of coupled Van der Pol oscillators, also demonstrated simultaneous AM/FM at frequency steps along with a Lashinsky spectrum. Simultaneous AM/FM, together with a Lashinsky spectrum, are predicted to occur when periodically-forced or mutually-coupled oscillators desynchronise via a supercritical Andronov–Hopf bifurcation and have been observed before in other physical systems of forced or coupled oscillators in plasma physics and electrical engineering. Thus motility patterns in the intestine can be understood from the viewpoint of very general dynamical principles.

scholarly journals Hopf bifurcation of a delayed worm model with two latent periods

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Juan Liu ◽  
Zizhen Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Sensor Network ◽  
Center Manifold ◽  
Sufficient Conditions ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory ◽  
Worm Model ◽  
Distribution Of Roots ◽  
Influential Parameters

Abstract We investigate a delayed epidemic model for the propagation of worm in wireless sensor network with two latent periods. We derive sufficient conditions for local stability of the worm-induced equilibrium of the system and the existence of Hopf bifurcation by regarding different combination of two latent time delays as the bifurcation parameter and analyzing the distribution of roots of the associated characteristic equation. In particular, we investigate the direction and stability of the Hopf bifurcation by means of the normal form theory and center manifold theorem. To verify analytical results, we present numerical simulations. Also, the effect of some influential parameters of sensor network is properly executed so that the oscillations can be reduced and removed from the network.

scholarly journals Dynamical analysis of a giving up smoking model with time delay

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zizhen Zhang ◽  
Ruibin Wei ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

scholarly journals Dynamical Behavior in a Four-Dimensional Neural Network Model with Delay

10.1155/2012/397146 ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Changjin Xu ◽  
Peiluan Li
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Network Model ◽  
Neural Network Model ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Normal Form Theory ◽  
Form Theory ◽  
Explicit Formulae ◽  
Delay Differential

A four-dimensional neural network model with delay is investigated. With the help of the theory of delay differential equation and Hopf bifurcation, the conditions of the equilibrium undergoing Hopf bifurcation are worked out by choosing the delay as parameter. Applying the normal form theory and the center manifold argument, we derive the explicit formulae for determining the properties of the bifurcating periodic solutions. Numerical simulations are performed to illustrate the analytical results.

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