Hopf Bifurcations and Limit Cycles in Evolutionary Network Dynamics

2012 ◽  
Vol 11 (4) ◽  
pp. 1754-1784 ◽  
Author(s):  
Darren Pais ◽  
Carlos H. Caicedo-Nún͂ez ◽  
Naomi E. Leonard
Keyword(s):  
Limit Cycles ◽  
Network Dynamics ◽  
Hopf Bifurcations ◽  
Evolutionary Network

Hopf Bifurcations in Two-Strategy Delayed Replicator Dynamics

2016 ◽  
Vol 26 (01) ◽  
pp. 1650006 ◽  
Author(s):  
Elizabeth Wesson ◽  
Richard Rand ◽  
David Rand
Keyword(s):  
Limit Cycles ◽  
Replicator Dynamics ◽  
Hopf Bifurcations ◽  
Time Interval ◽  
Bifurcation Parameter ◽  
Replicator Equations

We investigate the dynamics of two-strategy replicator equations in which the fitness of each strategy is a function of the population frequencies delayed by a time interval [Formula: see text]. We analyze two models: in the first, all terms in the fitness are delayed, while in the second, only opposite-strategy terms are delayed. We compare the two models via a linear homotopy. Taking the delay [Formula: see text] as a bifurcation parameter, we demonstrate the existence of (nondegenerate) Hopf bifurcations in both models, and present an analysis of the resulting limit cycles using Lindstedt’s method.

Global Dynamics of a Parasite-Host Model with Nonlinear Incidence Rate

2015 ◽  
Vol 25 (08) ◽  
pp. 1550102 ◽  
Author(s):  
Yilei Tang
Keyword(s):  
Incidence Rate ◽  
Global Attractor ◽  
Exponential Function ◽  
Limit Cycles ◽  
Hopf Bifurcations ◽  
Global Dynamics ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Dynamical Behaviors ◽  
Fractional Function

The paper is concerned with the effect of a nonlinear incidence rate Sp Iq on dynamical behaviors of a parasite-host model. It is shown that the global attractor of the parasite-host model is an equilibrium if q = 1, which is similar to that of the parasite-host model with a nonlinear incidence rate of the fractional function [Formula: see text]. However, when q is greater than one, more positive equilibria appear and limit cycles arise from Hopf bifurcations at the positive equilibria for the model with the incidence rate Sp Iq. It reveals that the nonlinear incidence rate of the exponential function Sp Iq for generic p and q can lead to more complicated and richer dynamics than the bilinear incidence rate or the fractional incidence rate for this model.

ON THE BIRTH OF MULTIPLE LIMIT CYCLES IN NONLINEAR SYSTEMS

1996 ◽  
Vol 06 (12b) ◽  
pp. 2587-2603 ◽  
Author(s):  
JORGE L. MOIOLA ◽  
GUANRONG CHEN
Keyword(s):  
Limit Cycles ◽  
Nonlinear Dynamical Systems ◽  
Approximation Technique ◽  
Hopf Bifurcations ◽  
Systems Methodology ◽  
Nonlinear Dynamical ◽  
Parameter Perturbations ◽  
Feedback Control Systems ◽  
Stability Indexes ◽  
The Stability

Degenerate (or singular) Hopf bifurcations of a certain type determine the appearance of multiple limit cycles under system parameter perturbations. In the study of these degenerate Hopf bifurcations, computational formulas for the stability indexes (i.e., curvature coefficients) are essential. However, such formulas are very difficult to derive, and so are usually computed by different approximation methods. Inspired by the feedback control systems methodology and the harmonic balance approximation technique, higher-order approximate formulas for such curvature coefficients are derived in this paper in the frequency domain setting. The results obtained are then applied to a study of nonlinear dynamical systems within the region of one periodic solution, bypassing a direct investigation of the multiple limit cycles and some tedious discussion of the complex multiplicity issue. Finally, we will show that several types of stability bifurcations can be controlled based on the results obtained in this paper.

HOPF BIFURCATIONS FOR NEAR-HAMILTONIAN SYSTEMS

2009 ◽  
Vol 19 (12) ◽  
pp. 4117-4130 ◽  
Author(s):  
MAOAN HAN ◽  
JUNMIN YANG ◽  
PEI YU
Keyword(s):  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Efficient Algorithm ◽  
Hamiltonian Function ◽  
Melnikov Function ◽  
Quadratic System ◽  
New Method ◽  
Hopf Bifurcations ◽  
Computationally Efficient ◽  
Bifurcation Of Limit Cycles

In this paper, we consider bifurcation of limit cycles in near-Hamiltonian systems. A new method is developed to study the analytical property of the Melnikov function near the origin for such systems. Based on the new method, a computationally efficient algorithm is established to systematically compute the coefficients of Melnikov function. Moreover, we consider the case that the Hamiltonian function of the system depends on parameters, in addition to the coefficients involved in perturbations, which generates more limit cycles in the neighborhood of the origin. The results are applied to a quadratic system with cubic perturbations to show that the system can have five limit cycles in the vicinity of the origin.

QUALITATIVE ANALYSIS FOR THE MERKIN ENZYME REACTION SYSTEM

2002 ◽  
Vol 10 (02) ◽  
pp. 167-182
Author(s):  
YUQUAN WANG ◽  
ZUORUI SHEN
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Local Stability ◽  
Bifurcation Theory ◽  
Sufficient Conditions ◽  
Reaction System ◽  
Enzyme Reaction ◽  
Qualitative Theory ◽  
Positive Equilibrium ◽  
Hopf Bifurcations

Applying qualitative theory and Hopf bifurcation theory, we detailedly discuss the Merkin enzyme reaction system, and the sufficient conditions derived for the global stability of the unique positive equilibrium, the local stability of three equilibria and the existence of limit cycles. Meanwhile, we show that the Hopf bifurcations may occur. Using MATLAB software, we present three examples to simulate these conclusions in this paper.

CONDITIONS FOR APPEARANCE AND DISAPPEARANCE OF LIMIT CYCLES IN THE SHIMIZU–MORIOKA SYSTEM

2011 ◽  
Vol 21 (09) ◽  
pp. 2489-2503
Author(s):  
LINGLING LIU ◽  
BO GAO
Keyword(s):  
Numerical Simulations ◽  
Limit Cycles ◽  
Center Manifold ◽  
Normal Forms ◽  
Hopf Bifurcations ◽  
Algebraic Varieties ◽  
System A ◽  
Parameter Dependent ◽  
Generalized Lorenz Canonical Form ◽  
Two Equilibria

This paper deals with the Shimizu–Morioka system, a special generalized Lorenz canonical form. Using techniques of elimination in the computation of algebraic varieties we obtain parameter-dependent normal forms on a center manifold. Our computation shows that the maximal number of limit cycles produced from Hopf bifurcations is four and only even number of limit cycles can be bifurcated near the two equilibria because of [Formula: see text]-symmetry. Our parameter-dependent normal forms enable us to give parameter conditions for the cases of none, two and four limit cycles separately. Furthermore, considering exterior perturbations, we give conditions under which one or three limit cycles can be produced from Hopf bifurcations. Moreover, we also give conditions for fold bifurcations, under which limit cycles coincide or disappear. Finally, our results are illustrated by numerical simulations.

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