Differential Equations with Bifocal Homoclinic Orbits

1997 ◽  
Vol 07 (01) ◽  
pp. 27-37 ◽  
Author(s):  
Paul Glendinning
Keyword(s):  
Differential Equations ◽  
Homoclinic Orbit ◽  
Stationary Point ◽  
Piecewise Linear ◽  
Global Bifurcation ◽  
Homoclinic Orbits ◽  
Bifurcation Phenomena ◽  
Global Bifurcation Theory ◽  
Linear Differential ◽  
Theoretical Results

Global bifurcation theory can be used to understand complicated bifurcation phenomena in families of differential equations. There are many theoretical results relating to systems having a homoclinic orbit biasymptotic to a stationary point at some value of the parameters, and these results depend upon the eigenvalues of the Jacobian matrix of the flow evaluated at the stationary point. Three important cases arise in the theoretical analysis, and there are many examples of systems which illustrate two of these three cases. We describe a construction which can be used to produce examples of the third case (bifocal homoclinic orbits), and use this construction to prove the existence of a bifocal homoclinic orbit in a simple piecewise linear differential equation.

HORSESHOES NEAR HOMOCLINIC ORBITS FOR PIECEWISE LINEAR DIFFERENTIAL SYSTEMS IN ℝ3

2007 ◽  
Vol 17 (04) ◽  
pp. 1171-1184 ◽  
Author(s):  
JAUME LLIBRE ◽  
ENRIQUE PONCE ◽  
ANTONIO E. TERUEL
Keyword(s):  
Periodic Orbits ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Homoclinic Orbits ◽  
Parametric Family ◽  
Homoclinic Loop ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Analytical Proof ◽  
Linear Differential

For a three-parametric family of continuous piecewise linear differential systems introduced by Arneodo et al. [1981] and considering a situation which is reminiscent of the Hopf-Zero bifurcation, an analytical proof on the existence of a two-parametric family of homoclinic orbits is provided. These homoclinic orbits exist both under Shil'nikov (0 < δ < 1) and non-Shil'nikov assumptions (δ ≥ 1). As it is well known for the case of differentiable systems, under Shil'nikov assumptions there exist infinitely many periodic orbits accumulating to the homoclinic loop. We also prove that this behavior persists at δ = 1. Moreover, for δ > 1 and sufficiently close to 1 we show that these periodic orbits persist but then they do not accumulate to the homoclinic orbit.

scholarly journals Interaction of homoclinic solutions and Hopf points in functional differential equations of mixed type

2009 ◽  
Vol 139 (1) ◽  
pp. 123-155 ◽  
Author(s):  
Marc Georgi
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Homoclinic Orbit ◽  
Mixed Type ◽  
Functional Differential Equations ◽  
Homoclinic Orbits ◽  
Homoclinic Solution ◽  
Equations Of Mixed Type ◽  
Functional Differential ◽  
Equation Of Mixed Type

We study a homoclinic bifurcation in a general functional differential equation of mixed type. More precisely, we investigate the case when the asymptotic steady state of a homoclinic solution undergoes a Hopf bifurcation. Bifurcations of this kind are diffcult to analyse due to the lack of Fredholm properties. In particular, a straightforward application of a Lyapunov–Schmidt reduction is not possible.As one of the main results we prove the existence of centre-stable and centre-unstable manifolds of steady states near homoclinic orbits. With their help, we can analyse the bifurcation scenario similar to the case for ordinary differential equations and can show the existence of solutions which bifurcate near the homoclinic orbit, are decaying in one direction and oscillatory in the other direction. These solutions can be visualized as an interaction of the homoclinic orbit and small periodic solutions that exist on account of the Hopf bifurcation, for exactly one asymptotic direction t→8 or t→−∞.

scholarly journals Limit Cycles Bifurcating from the Periodic Orbits of a Discontinuous Piecewise Linear Differentiable Center with Two Zones

2015 ◽  
Vol 25 (11) ◽  
pp. 1550144 ◽  
Author(s):  
Jaume Llibre ◽  
Douglas D. Novaes ◽  
Marco A. Teixeira
Keyword(s):  
Equilibrium Point ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Straight Line ◽  
Linear Differential

We study a class of discontinuous piecewise linear differential systems with two zones separated by the straight line x = 0. In x > 0, we have a linear saddle with its equilibrium point living in x > 0, and in x < 0 we have a linear differential center. Let p be the equilibrium point of this linear center, when p lives in x < 0, we say that it is real, and when p lives in x > 0 we say that it is virtual. We assume that this discontinuous piecewise linear differential system formed by the center and the saddle has a center q surrounded by periodic orbits ending in a homoclinic orbit of the saddle, independent if p is real, virtual or p is in x = 0. Note that q = p if p is real or p is in x = 0. We perturb these three classes of systems, according to the position of p, inside the class of all discontinuous piecewise linear differential systems with two zones separated by x = 0. Let N be the maximum number of limit cycles which can bifurcate from the periodic solutions of the center q with these perturbations. Our main results show that N = 2 when p is on x = 0, and N ≥ 2 when p is a real or virtual center. Furthermore, when p is a real center we found an example satisfying N ≥ 3.

scholarly journals A NUMERICAL TOOLBOX FOR HOMOCLINIC BIFURCATION ANALYSIS

1996 ◽  
Vol 06 (05) ◽  
pp. 867-887 ◽  
Author(s):  
A.R. CHAMPNEYS ◽  
YU. A. KUZNETSOV ◽  
B. SANDSTEDE
Keyword(s):  
Numerical Analysis ◽  
Chemical Kinetics ◽  
Bifurcation Analysis ◽  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Homoclinic Bifurcation ◽  
Codimension Two ◽  
Codimension One ◽  
Homoclinic Bifurcations ◽  
Theoretical Results

This paper presents extensions and improvements of recently developed algorithms for the numerical analysis of orbits homoclinic to equilibria in ODEs and describes the implementation of these algorithms within the standard continuation package AUTO86. This leads to a kind of toolbox, called HOMCONT, for analysing homoclinic bifurcations either as an aid to producing new theoretical results, or to understand dynamics arising from applications. This toolbox allows the continuation of codimension-one homoclinic orbits to hyperbolic or non-hyperbolic equilibria as well as detection and continuation of higher-order homoclinic singularities in more parameters. All known codimension-two cases that involve a unique homoclinic orbit are supported. Two specific example systems from ecology and chemical kinetics are analysed in some detail, allowing the reader to understand how to use the the toolbox for themselves. In the process, new results are also derived on these two particular models.

Stability and Perturbations of Homoclinic Loops in a Class of Piecewise Smooth Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550114 ◽  
Author(s):  
Shuang Chen ◽  
Zhengdong Du
Keyword(s):  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Piecewise Linear ◽  
Small Neighborhood ◽  
Homoclinic Orbits ◽  
Piecewise Smooth ◽  
Piecewise Smooth Systems ◽  
Planar System ◽  
The Stability ◽  
First Time

Like for smooth systems, a typical method to produce multiple limit cycles for a given piecewise smooth planar system is via homoclinic bifurcation. Previous works only focused on limit cycles that bifurcate from homoclinic orbits of piecewise-linear systems. In this paper, we consider for the first time the same problem for a class of general nonlinear piecewise smooth systems. By introducing the Dulac map in a small neighborhood of the hyperbolic saddle, we obtain the approximation of the Poincaré map for the nonsmooth homoclinic orbit. Then, we give conditions for the stability of the homoclinic orbit and conditions under which one or two limit cycles bifurcate from it. As an example, we construct a nonlinear piecewise smooth system with two limit cycles that bifurcate from a homoclinic orbit.

scholarly journals Shilnikov chaos, Filippov sliding and boundary equilibrium bifurcations

2018 ◽  
Vol 29 (5) ◽  
pp. 757-777 ◽  
Author(s):  
P. A. GLENDINNING
Keyword(s):  
Differential Equations ◽  
Homoclinic Orbit ◽  
Homoclinic Orbits ◽  
Piecewise Smooth ◽  
Boundary Equilibrium ◽  
The 1960S

In the 1960s, L.P. Shilnikov showed that certain homoclinic orbits for smooth families of differential equations imply the existence of chaos, and there are complicated sequences of bifurcations near the parameter value at which the homoclinic orbit exists. We describe how this analysis is modified if the differential equations are piecewise smooth and the homoclinic orbit has a sliding segment. Moreover, we show that the Shilnikov mechanism appears naturally in the unfolding of boundary equilibrium bifurcations in $\mathbb{R}^3$.

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