Perturbation Solution for Secondary Bifurcation in the Quadratically-Damped Mathieu Equation

2001 ◽  
Author(s):  
Deepak V. Ramani ◽  
Richard H. Rand ◽  
William L. Keith
Keyword(s):  
Perturbation Method ◽  
Limit Cycles ◽  
Bifurcation Point ◽  
Mathieu Equation ◽  
Bifurcation Curve ◽  
Mathieu Functions ◽  
Bifurcation Of Limit Cycles ◽  
Saddle Node Bifurcation ◽  
Secondary Bifurcation ◽  
Transition Curves

Abstract This paper concerns the quadratically-damped Mathieu equation:x..+(δ+ϵcos⁡t)x+x.|x.|=0. Numerical integration shows the existence of a secondary-bifurcation in which a pair of limit cycles come together and disappear (a saddle-node bifurcation of limit cycles). In δ–ϵ parameter space, this secondary bifurcation appears as a curve which emanates from one of the transition curves of the linear Mathieu equation for ϵ ≈ 1.5. The bifurcation point along with an approximation for the bifurcation curve is obtained by a perturbation method which uses Mathieu functions rather than the usual sines and cosines.

scholarly journals DYNAMICS OF DELAY-COUPLED EXCITABLE NEURAL SYSTEMS

2009 ◽  
Vol 19 (02) ◽  
pp. 745-753 ◽  
Author(s):  
M. A. DAHLEM ◽  
G. HILLER ◽  
A. PANCHUK ◽  
E. SCHÖLL
Keyword(s):  
Nonlinear Dynamics ◽  
Fixed Point ◽  
Limit Cycles ◽  
Coupling Strength ◽  
Neural Systems ◽  
Stable Fixed Point ◽  
Bifurcation Of Limit Cycles ◽  
Large Delay ◽  
Saddle Node Bifurcation ◽  
Delay Times

We study the nonlinear dynamics of two delay-coupled neural systems each modeled by excitable dynamics of FitzHugh–Nagumo type and demonstrate that bistability between the stable fixed point and limit cycle oscillations occurs for sufficiently large delay times τ and coupling strength C. As the mechanism for these delay-induced oscillations, we identify a saddle-node bifurcation of limit cycles.

Bifurcations in a Mathieu Equation With Cubic Nonlinearities: Part II

2002 ◽  
Author(s):  
Leslie Ng ◽  
Richard Rand
Keyword(s):  
Periodic Orbits ◽  
Bifurcation Point ◽  
Mathieu Equation ◽  
Analytic Approximation ◽  
Slow Flow ◽  
Method Of Averaging ◽  
Bifurcation Curve ◽  
Heteroclinic Bifurcation ◽  
Degenerate Bifurcation

In a previous paper [6], the authors investigated the dynamics of the equation: d2xdt2+(δ+εcost)x+εAx3+Bx2dxdt+Cxdxdt2+Ddxdt3=0. We used the method of averaging in the neighborhood of the 2:1 resonance in the limit of small forcing and small nonlinearity. We found that a degenerate bifurcation point occurs in the resulting slow flow and some of the bifurcations near this point were looked at. In this work we present additional results concerning the bifurcations around this point using analytic techniques and AUTO. An analytic approximation for a heteroclinic bifurcation curve is obtained. Additional results on the bifurcations of periodic orbits in the slow flow are also presented.

scholarly journals Fast-slow Variable Dissection with Two Slow Variables Related to Calcium Concentrations: A Case Study to Bursting in a Neural Pacemaker Model

2021 ◽  
Author(s):  
Yuye Li ◽  
Huaguang Gu ◽  
Yanbing Jia ◽  
Kaihua Ma
Keyword(s):  
Bifurcation Point ◽  
Slow Variable ◽  
Quiescent State ◽  
Bifurcation Curve ◽  
Fast Subsystem ◽  
Saddle Node Bifurcation ◽  
Initiation Point ◽  
Saddle Node ◽  
Pathological Pain ◽  
Slow Variables

Abstract Neuronal bursting is an electrophysiological behavior participating in physiological or pathological functions and a complex nonlinear alternating between burst and quiescent state modulated by slow variables. Identification of dynamics of bursting modulated by two slow variables is still an open problem. In the present paper, a novel fast-slow variable dissection method with two slow variables is proposed to analyze the complex bursting in a 4-dimensional neuronal model to describe bursting associated with pathological pain. The lumenal (Clum) and intracellular (Cin) calcium concentrations are the slowest variables respectively in the quiescent state and burst duration. Questions encountered when the traditional method with one low variable is used. When Clum is taken as slow variable, the burst is successfully identified to terminate near the saddle-homoclinic bifurcation point of the fast subsystem and begin not from the saddle-node bifurcation. With Cin chosen as slow variable, Clum value of initiation point is far from the saddle-node bifurcation point, due to Clum not contained in the equation of membrane potential. To overcome this problem, both Cin and Clum are regarded as slow variables, the two-dimensional fast subsystem exhibits a saddle-node bifurcation point, which is extended to a saddle-node bifurcation curve by introducing Clum dimension. Then, the initial point of burst is successfully identified to be near the saddle-node bifurcation curve. The results present a feasible method for fast-slow variable dissection and deep understanding to the complex bursting behavior with two slow variables, which is helpful for the modulation to pathological pain.

scholarly journals Bifurcation of Limit Cycles and Center Conditions for Two Families of Kukles-Like Systems with Nilpotent Singularities

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Peiluan Li ◽  
Yusen Wu ◽  
Xiaoquan Ding
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Singular Points ◽  
Center Problem ◽  
Bifurcation Of Limit Cycles ◽  
Center Conditions

We solve theoretically the center problem and the cyclicity of the Hopf bifurcation for two families of Kukles-like systems with their origins being nilpotent and monodromic isolated singular points.

BIFURCATION OF LIMIT CYCLES AND ISOCHRONOUS CENTERS FOR A QUARTIC SYSTEM

2013 ◽  
Vol 23 (10) ◽  
pp. 1350172 ◽  
Author(s):  
WENTAO HUANG ◽  
AIYONG CHEN ◽  
QIUJIN XU
Keyword(s):  
Limit Cycles ◽  
Necessary Conditions ◽  
Polynomial System ◽  
Isochronous Center ◽  
Eighth Order ◽  
Bifurcation Of Limit Cycles ◽  
Quartic Polynomial ◽  
Fine Focus ◽  
Sufficient And Necessary Conditions ◽  
Isochronous Centers

For a quartic polynomial system we investigate bifurcations of limit cycles and obtain conditions for the origin to be a center. Computing the singular point values we find also the conditions for the origin to be the eighth order fine focus. It is proven that the system can have eight small amplitude limit cycles in a neighborhood of the origin. To the best of our knowledge, this is the first example of a quartic system with eight limit cycles bifurcated from a fine focus. We also give the sufficient and necessary conditions for the origin to be an isochronous center.

scholarly journals Hopf Bifurcation of Limit Cycles in Discontinuous Liénard Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Special Form ◽  
Bifurcation Of Limit Cycles ◽  
Liénard Systems

We consider a class of discontinuous Liénard systems and study the number of limit cycles bifurcated from the origin when parameters vary. We establish a method of studying cyclicity of the system at the origin. As an application, we discuss some discontinuous Liénard systems of special form and study the cyclicity near the origin.

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