Saddle-node bifurcation of limit cycles in a feedback system with rate limiter

2001 ◽  
Author(s):  
I. Alcala ◽  
F. Gordillo ◽  
J. Aracil
Keyword(s):  
Limit Cycles ◽  
Feedback System ◽  
Bifurcation Of Limit Cycles ◽  
Saddle Node Bifurcation ◽  
Saddle Node

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.

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 Algebraic Limit Cycles in Piecewise Linear Differential Systems

2018 ◽  
Vol 28 (03) ◽  
pp. 1850039 ◽  
Author(s):  
Claudio A. Buzzi ◽  
Armengol Gasull ◽  
Joan Torregrosa
Keyword(s):  
Limit Cycles ◽  
Piecewise Linear ◽  
Saddle Node Bifurcation ◽  
Differential Systems ◽  
Saddle Node ◽  
Linear Differential Systems ◽  
Linear Differential ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

This paper is devoted to study the algebraic limit cycles of planar piecewise linear differential systems. In particular, we present examples exhibiting two explicit hyperbolic algebraic limit cycles, as well as some one-parameter families with a saddle-node bifurcation of algebraic limit cycles. We also show that all degrees for algebraic limit cycles are allowed.

SEMISTABLE LIMIT CYCLES THAT BIFURCATE FROM CENTERS

2003 ◽  
Vol 13 (11) ◽  
pp. 3489-3498 ◽  
Author(s):  
HECTOR GIACOMINI ◽  
MIREILLE VIANO ◽  
JAUME LLIBRE
Keyword(s):  
Parameter Space ◽  
Limit Cycles ◽  
Analytic Functions ◽  
Arbitrary Order ◽  
Perturbation Parameter ◽  
Analytic Method ◽  
Analytic Center ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Bifurcation Surface

Suppose that the differential system [Formula: see text] has a center at the origin for ε=0, where P0, Q0, aij and bij are analytic functions in their variables, such that aij(0)=bij(0)=0. We present an analytic method to compute the semistable limit cycles which bifurcate from the periodic orbits of the analytic center, up to an arbitrary order in the perturbation parameter ε. We also provide an algorithm for the computation of the saddle–node bifurcation hypersurface of limit cycles in the parameter space {aij,bij}1≤i,j≤m. As an example, we apply the method to compute, first, the anal ytic expression of the unique semistable limit cycle of the Liénard system [Formula: see text] and second, an approximation of the saddle-node bifurcation surface of limit cycles in the parameter space (a1, a3, a5). Both computations are valid for ε sufficiently small.

GLUING BIFURCATIONS IN CHUA OSCILLATOR

2006 ◽  
Vol 16 (12) ◽  
pp. 3497-3508 ◽  
Author(s):  
PRODYOT KUMAR ROY ◽  
SYAMAL KUMAR DANA
Keyword(s):  
Limit Cycles ◽  
Control Parameter ◽  
Homoclinic Orbits ◽  
Oscillator Model ◽  
Saddle Node Bifurcation ◽  
Chua’S Oscillator ◽  
Saddle Node ◽  
Chua Oscillator

Gluing bifurcation in a modified Chua's oscillator is reported. Keeping other parameters fixed when a control parameter is varied in the modified oscillator model, two symmetric homoclinic orbits to saddle focus at origin, which are mirror images of each other, are glued together for a particular value of the control parameter. In experiments, two asymmetric limit cycles are homoclinic to the saddle focus origin for different values of the control parameter. However, imperfect gluing bifurcation has been observed, in experiments, when one stable and unstable limit cycles merge to the saddle focus origin via saddle-node bifurcation.

Attractor-preserving control to avoid saddle-node bifurcation

2014 ◽  
Vol 2 ◽  
pp. 150-153
Author(s):  
Daisuke Ito ◽  
Tetsushi Ueta ◽  
Shigeki Tsuji ◽  
Kazuyuki Aihara
Keyword(s):  
Saddle Node Bifurcation ◽  
Saddle Node

BASIN ORGANIZATION PRIOR TO A TANGLED SADDLE-NODE BIFURCATION

1991 ◽  
Vol 01 (01) ◽  
pp. 107-118 ◽  
Author(s):  
MOHAMED S. SOLIMAN ◽  
J. M. T. THOMPSON
Keyword(s):  
Fractal Structure ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Homoclinic Connections

Heteroclinic and homoclinic connections of saddle cycles play an important role in basin organization. In this study, we outline how these events can lead to an indeterminate jump to resonance from a saddle-node bifurcation. Here, due to the fractal structure of the basins in the vicinity of the saddle-node, we cannot predict to which available attractor the system will jump in the presence of even infinitesimal noise.

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