On Generalized Hopf Bifurcations

1984 ◽  
Vol 106 (4) ◽  
pp. 327-334 ◽  
Author(s):  
K. Huseyin ◽  
A. S. Atadan
Keyword(s):  
Hopf Bifurcation ◽  
Additional Condition ◽  
System Parameter ◽  
Transversality Condition ◽  
Existence Condition ◽  
The Other ◽  
Hopf Bifurcations ◽  
Bifurcation Phenomenon ◽  
Bifurcation Phenomena ◽  
Lumped Parameter

Two distinct degenerate Hopf bifurcation phenomena associated with autonomous lumped-parameter systems are explored in great detail via the intrinsic harmonic balancing method. It is assumed that the Hopf’s transversality condition is violated and certain other conditions prevail. In one of the cases, the system exhibits a cusp shape bifurcation path which exists for either positive or negative values of the system parameter. On the other hand, the second case is concerned with a tangential bifurcation phenomenon which may not be exhibited unless an additional condition is satisfied. This existence condition is obtained in the course of analysis. The distinctive feature of the paper is that the results concerning the bifurcating paths and limit cycles are given in general, explicit forms which are expected to be very useful in a variety of applications.

Heteroclinic Trajectory and Hopf Bifurcation in an Extended Lorenz System

2018 ◽  
Vol 28 (09) ◽  
pp. 1850111
Author(s):  
Xianyi Li ◽  
Haijun Wang
Keyword(s):  
Hopf Bifurcation ◽  
Lyapunov Function ◽  
Numerical Simulations ◽  
Lorenz System ◽  
The Other ◽  
Hopf Bifurcations ◽  
Limit Sets ◽  
Initial Values ◽  
Heteroclinic Trajectory ◽  
The One

This note revisits an extended Lorenz system, which was presented in the paper entitled “Hopf bifurcations in an extended Lorenz system” by Zhou et al. [2017]. On the one hand, one points out and corrects some wrong results in that paper on the Hopf bifurcation at the symmetric equilibria [Formula: see text] and [Formula: see text]. On the other hand, combining Lyapunov function and the concepts of [Formula: see text]- and [Formula: see text]-limit sets, it is rigorously proved that there exists two and only two heteroclinic trajectories but no homoclinic trajectories under some certain conditions of its parameters and initial values. In addition, numerical simulations illustrate the consistence with the theoretical conclusions. The results together not only improve and complement the known ones, but also provide support in some future applications.

Analysis of the interaction among weed, inorganic fertilizer and herbivore in paddy ecosystem in fallow season

2017 ◽  
Vol 10 (08) ◽  
pp. 1750120 ◽  
Author(s):  
Meihong Xiang ◽  
Zhaohua Wu ◽  
Tiejun Zhou
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Point ◽  
Equation Model ◽  
Inorganic Fertilizer ◽  
Stable Region ◽  
Positive Equilibrium ◽  
Bifurcation Phenomenon ◽  
Bifurcation Phenomena ◽  
Paddy Ecosystem ◽  
Positive Equilibrium Point

Paddy growth is influenced by the amount of inorganic fertilizer in paddy ecosystem in fallow season. To discover the interaction among weed, inorganic fertilizer and herbivore in the system, we put forward a differential equation model and investigate its properties. Results show that the system has a weed and herbivore extinct equilibrium and a herbivore extinct equilibrium. The two equilibria are proven to be unstable using the center manifold method. Under certain conditions, the system also has a positive equilibrium point. We give the stable region and the unstable region of the positive equilibrium point, which are determined by some parameters. We find that the system has the Hopf bifurcation phenomenon, and give the critical value of Hopf bifurcation by taking a system parameter as the bifurcation parameter. By comparing the equilibrium states between a paddy ecosystem with herbivore and one without herbivore, we find that the content of inorganic fertilizer can be improved by putting herbivore into a paddy field. An example is given to illustrate the feasibility of the results. Numerical simulation shows that Hopf bifurcation phenomena exist in the system.

Slow–Fast Behaviors and Their Mechanism in a Periodically Excited Dynamical System with Double Hopf Bifurcations

2021 ◽  
Vol 31 (08) ◽  
pp. 2130022
Author(s):  
Miaorong Zhang ◽  
Xiaofang Zhang ◽  
Qinsheng Bi
Keyword(s):  
Dynamical System ◽  
Hopf Bifurcation ◽  
Frequency Domain ◽  
Autonomous System ◽  
Single Mode ◽  
Phase Portraits ◽  
Hopf Bifurcations ◽  
Double Hopf Bifurcation ◽  
Exciting Frequency ◽  
Slow Passage

This paper focuses on the influence of two scales in the frequency domain on the behaviors of a typical dynamical system with a double Hopf bifurcation. By introducing an external periodic excitation to the normal form of the vector field with double Hopf bifurcation at the origin and taking the exciting frequency far less than the natural frequency, a theoretical model with two scales in the frequency domain is established. Regarding the whole exciting term as a slow-varying parameter leads to a generalized autonomous system, in which the equilibrium branches and their bifurcations with the variation of the slow-varying parameter can be derived. With the increase of the exciting amplitude, different types of bifurcations may be involved in the generalized autonomous system, resulting in several qualitatively different forms of bursting attractors, the mechanism of which is presented by overlapping the transformed phase portraits and the bifurcations of the equilibrium branches. It is found that the single mode 2D torus may evolve to the bursting attractors with mixed modes, in which the trajectory alternates between the single mode oscillations and the mixed mode oscillations. Furthermore, the transitions between the quiescent states and the spiking states may not occur exactly at the bifurcation points because of the slow passage effect, while Hopf bifurcations may cause different forms of repetitive spiking oscillations.

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.

Application of the Hopf bifurcation theory to limit cycle prediction of short journal bearings with isotropic roughness effects

2007 ◽  
Vol 221 (8) ◽  
pp. 869-879 ◽  
Author(s):  
J-R Lin
Keyword(s):  
Surface Roughness ◽  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Critical Region ◽  
System Parameter ◽  
Journal Bearings ◽  
Roughness Effects ◽  
Critical Limit ◽  
Bifurcation Behaviour ◽  
Bearing System

On the basis of the Christensen stochastic theory, the effects of isotropic surface roughness upon the bifurcation behaviour of a short journal-bearing system are investigated. By applying the Hopf bifurcation theorem to the non-linear equations of motion of rough journal bearings, the steady-state performance, the linear characteristics, and the weakly non-linear bifurcationphenomenaarepresented. For the short bearing with length-to-diameter ratio l = 0. 5, the onset of oil-whirl rough bearing system can manifest a bifurcation behaviour exhibiting subcritical limit cycles or super-critical limit cycles for running speeds near the bifurcation point. For a particular system parameter, the effects of isotropic surface roughness are found to enlarge the size of sub-critical limit cycles and super-critical limit cycles when the Hopf bifurcation occurs. On the whole, the roughness effects of isotropic surface patterns upon the Hopf bifurcation phenomena of the short-bearing system are more pronounced for a smaller system parameter ( p = 0. 4 in the sub-critical region and Sp = 0. 05 in the super-critical region) and a higher roughness parameter (Λ = 0. 4).

Influence of Hopf Bifurcation Dynamics on Conduction Failure of Action Potentials Along Nerve C-Fiber

2019 ◽  
Vol 29 (07) ◽  
pp. 1950093 ◽  
Author(s):  
Xinjing Zhang ◽  
Huaguang Gu
Keyword(s):  
Hopf Bifurcation ◽  
Coupling Strength ◽  
The Other ◽  
Dimensional Parameter ◽  
External Stimulation ◽  
C Fiber ◽  
Current Threshold ◽  
Stimulation Period ◽  
Conduction Failure ◽  
A Chain

Contrary to faithful conduction of every action potential or spike along the axon, some spikes induced by the external stimulation with a high frequency at one end of the unmyelinated nerve fiber (C-fiber) disappear during the conduction process to the other end, which leads to conduction failure. Many physiological functions such as information coding or pathological pain are involved. In the present paper, the dynamic mechanism of the conduction failure is well interpreted by two characteristics of the focus near Hopf bifurcation of the Hodgkin–Huxley (HH) model. One is that the current threshold to evoke a spike from the after-potential corresponding to the focus exhibits damping oscillations, and the other is that the damping oscillations exhibit an internal period. A chain network model composed of HH neurons and stimulated by the external periodic stimulation is used to stimulate C-fiber. In the two-dimensional parameter space of the stimulation period and coupling strength, the conduction failure appears for the coupling strength lower than that of the faithful conduction, which is due to some maximal values of the coupling current for low coupling strength not being strong enough to evoke spikes, and the coupling strength threshold between the faithful conduction and conduction failure exhibiting damping oscillations with respect to the stimulation period, due to the damping oscillations of the current threshold. The damping oscillations of the coupling strength exhibit close correlations to those of the current threshold. The coupling strength for the conduction failure exhibits maximal values as the stimulation period is approximated to 1-, 2-, 3- or 4-times of the internal period and the maximal values decrease with increasing stimulation period. In addition, the correspondence between the simulation results and the previous experimental observations is discussed. The results present deep insights into the dynamics of the conduction failure with Hopf bifurcation and are helpful to investigate the influence of other modulation factors on the conduction failure.

TWO-TO-ONE RESONANT HOPF BIFURCATIONS IN A QUADRATICALLY NONLINEAR OSCILLATOR INVOLVING TIME DELAY

2012 ◽  
Vol 22 (03) ◽  
pp. 1250060 ◽  
Author(s):  
J. C. JI ◽  
X. Y. LI ◽  
Z. LUO ◽  
N. ZHANG
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Periodic Solutions ◽  
Nonlinear Oscillator ◽  
Critical Time ◽  
Zero Solution ◽  
Delayed Feedback Control ◽  
Hopf Bifurcations ◽  
Centre Manifold ◽  
Normal Form Theory

The trivial equilibrium of a weakly nonlinear oscillator having quadratic nonlinearities under a delayed feedback control can change its stability via a single Hopf bifurcation as the time delay increases. Double Hopf bifurcation occurs when the characteristic equation has two pairs of purely imaginary solutions. An interaction of resonant Hopf–Hopf bifurcations may be possible when the two critical time delays corresponding to the two Hopf bifurcations have the same value. With the aid of normal form theory and centre manifold theorem as well as the method of multiple scales, the present paper studies the dynamics of a quadratically nonlinear oscillator involving time delay in the vicinity of the point of two-to-one resonances of Hopf–Hopf bifurcations. The ratio of the frequencies of two Hopf bifurcations is numerically found to be nearly equal to two. The two resonant Hopf bifurcations can generate two respective periodic solutions. Consequently, the centre manifold corresponding to these two solutions is determined by a set of four first-order differential equations under two-to-one internal resonances. It is shown that the amplitudes of the two bifurcating periodic solutions admit the trivial solution and two-mode solutions for the averaged equations on the centre manifolds. Correspondingly, the cumulative behavior of the original nonlinear oscillator exhibits the initial equilibrium and a quasi-periodic motion having two frequencies. Illustrative examples are given to show the unstable zero solution, stable zero solution, and stable two-mode solution of the nonlinear oscillator under the two-to-one resonant Hopf–Hopf interactions.

scholarly journals Stability and Hopf Bifurcation of Delayed Predator-Prey System Incorporating Harvesting

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Fengying Wei ◽  
Lanqi Wu ◽  
Yuzhi Fang
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Prey System ◽  
Theoretical Predictions ◽  
Form Theory ◽  
The Stability

A kind of delayed predator-prey system with harvesting is considered in this paper. The influence of harvesting and delay is investigated. Our results show that Hopf bifurcations occur as the delayτpasses through critical values. By using of normal form theory and center manifold theorem, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are obtained. Finally, numerical simulations are given to support our theoretical predictions.

scholarly journals Stability and Hopf Bifurcation Analysis on a Bazykin Model with Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Jianming Zhang ◽  
Lijun Zhang ◽  
Chaudry Masood Khalique
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Center Manifold ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Finite Delay ◽  
The Stability ◽  
Predator System

The dynamics of a prey-predator system with a finite delay is investigated. We show that a sequence of Hopf bifurcations occurs at the positive equilibrium as the delay increases. By using the theory of normal form and center manifold, explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.

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