Dynamical Systems with a Codimension-One Invariant Manifold: The Unfoldings and Its Bifurcations

2015 ◽  
Vol 25 (06) ◽  
pp. 1550091 ◽  
Author(s):  
Kie Van Ivanky Saputra
Keyword(s):  
Dynamical System ◽  
Invariant Manifold ◽  
Normal Forms ◽  
Volterra Model ◽  
Infection Model ◽  
Global Bifurcations ◽  
Codimension Two ◽  
Codimension One ◽  
Variation Of Parameters ◽  
Local And Global Bifurcations

We investigate a dynamical system having a special structure namely a codimension-one invariant manifold that is preserved under the variation of parameters. We derive conditions such that bifurcations of codimension-one and of codimension-two occur in the system. The normal forms of these bifurcations are derived explicitly. Both local and global bifurcations are analyzed and yield the transcritical bifurcation as the codimension-one bifurcation while the saddle-node–transcritical interaction and the Hopf–transcritical interactions as the codimension-two bifurcations. The unfolding of this degeneracy is also analyzed and reveal global bifurcations such as homoclinic and heteroclinic bifurcations. We apply our results to a modified Lotka–Volterra model and to an infection model in HIV diseases.

ROBUST BURSTING TO THE ORIGIN: HETEROCLINIC CYCLES WITH MAXIMAL SYMMETRY EQUILIBRIA

2005 ◽  
Vol 15 (09) ◽  
pp. 2819-2832 ◽  
Author(s):  
DAVID HAWKER ◽  
PETER ASHWIN
Keyword(s):  
Limit Cycle ◽  
Vector Fields ◽  
Global Bifurcations ◽  
Heteroclinic Cycles ◽  
Maximal Symmetry ◽  
Codimension One ◽  
Local And Global Bifurcations ◽  
Resonance Bifurcation

Robust attracting heteroclinic cycles have been found in many models of dynamics with symmetries. In all previous examples, robust heteroclinic cycles appear between a number of symmetry broken equilibria. In this paper we examine the first example where there are robust attracting heteroclinic cycles that include the origin, i.e. a point with maximal symmetry. The example we study is for vector fields on ℝ3 with (ℤ2)3 symmetry. We list all possible generic (codimension one) local and global bifurcations by which this cycle can appear as an attractor; these include a resonance bifurcation from a limit cycle, direct bifurcation from a stable origin and direct bifurcation from other and more familiar robust heteroclinic cycles.

MULTIPARAMETRIC BIFURCATIONS IN AN ENZYME-CATALYZED REACTION MODEL

2005 ◽  
Vol 15 (03) ◽  
pp. 905-947 ◽  
Author(s):  
E. FREIRE ◽  
L. PIZARRO ◽  
A. J. RODRÍGUEZ-LUIS ◽  
F. FERNÁNDEZ-SÁNCHEZ
Keyword(s):  
Periodic Orbits ◽  
Dynamical Behavior ◽  
Homoclinic Orbits ◽  
Reaction Model ◽  
Numerical Continuation ◽  
Global Bifurcations ◽  
Codimension One ◽  
Local Bifurcations ◽  
Homoclinic Connections ◽  
Local And Global Bifurcations

An exhaustive analysis of local and global bifurcations in an enzyme-catalyzed reaction model is carried out. The model, given by a planar five-parameter system of autonomous ordinary differential equations, presents a great richness of bifurcations. This enzyme-catalyzed model has been considered previously by several authors, but they only detected a minimal part of the dynamical and bifurcation behavior exhibited by the system. First, we study local bifurcations of equilibria up to codimension-three (saddle-node, cusps, nondegenerate and degenerate Hopf bifurcations, and nondegenerate and degenerate Bogdanov–Takens bifurcations) by using analytical and numerical techniques. The numerical continuation of curves of global bifurcations allows to improve the results provided by the study of local bifurcations of equilibria and to detect new homoclinic connections of codimension-three. Our analysis shows that such a system exhibits up to sixteen different kinds of homoclinic orbits and thirty different configurations of equilibria and periodic orbits. The coexistence of up to five periodic orbits is also pointed out. Several bifurcation sets are sketched in order to show the dynamical behavior the system exhibits. The different codimension-one and -two bifurcations are organized around five codimension-three degeneracies.

LOCAL AND GLOBAL BIFURCATIONS TO LIMIT CYCLES IN A CLASS OF LIÉNARD EQUATION

2007 ◽  
Vol 17 (01) ◽  
pp. 183-198 ◽  
Author(s):  
PEI YU
Keyword(s):  
Numerical Simulation ◽  
Phase Space ◽  
Limit Cycles ◽  
Normal Forms ◽  
Polynomial Function ◽  
Local Limit ◽  
Liénard Equation ◽  
Global Bifurcations ◽  
Local And Global Bifurcations ◽  
Lienard Equation

In this paper, we study limit cycles in the Liénard equation: ẍ + f(x)ẋ + g(x) = 0 where f(x) is an even polynomial function with degree 2m, while g(x) is a third-degree, odd polynomial function. In phase space, the system has three fixed points, one saddle point at the origin and two linear centers which are symmetric about the origin. It is shown that the system can have 2m small (local) limit cycles in the vicinity of two focus points and several large (global) limit cycles enclosing all the small limit cycles. The method of normal forms is employed to prove the existence of the small limit cycles and numerical simulation is used to show the existence of large limit cycles.

A PIECEWISE LINEAR DYNAMICAL SYSTEM WITH TWO DROPPING SECTIONS

2009 ◽  
Vol 19 (04) ◽  
pp. 1367-1372 ◽  
Author(s):  
VALERY A. GAIKO ◽  
WIM T. VAN HORSSEN
Keyword(s):  
Dynamical System ◽  
Linear Function ◽  
Limit Cycles ◽  
Piecewise Linear ◽  
Nonlinear Function ◽  
Piecewise Linear Function ◽  
Singular Points ◽  
Linear Dynamical System ◽  
Global Bifurcations ◽  
Local And Global Bifurcations

In this paper, we consider a planar dynamical system with a piecewise linear function containing two dropping sections and approximating some continuous nonlinear function. Studying all possible local and global bifurcations of its limit cycles, we prove that such a piecewise linear dynamical system, with five singular points, can have at most four limit cycles, three of which surround the foci one by one and the fourth limit cycle surrounds all of the singular points of this system.

scholarly journals Bifurcations of Nonconstant Solutions of the Ginzburg-Landau Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-19
Author(s):  
Norimichi Hirano ◽  
Sławomir Rybicki
Keyword(s):  
Landau Equation ◽  
Conley Index ◽  
Global Bifurcations ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Local And Global Bifurcations

We study local and global bifurcations of nonconstant solutions of the Ginzburg-Landau equation from the families of constant ones. As the topological tools we use the equivariant Conley index and the degree for equivariant gradient maps.

Codimension ${\bi n}$ flips

1998 ◽  
Vol 18 (5) ◽  
pp. 1115-1137 ◽  
Author(s):  
JAQUES GHEINER
Keyword(s):  
Global Bifurcations ◽  
Local And Global Bifurcations

The generic unfolding of codimension $n$ flips (one eigenvalue $-1$ and the others with norm different from 1) embedded in a Morse–Smale diffeomorphism is analyzed. Local and global bifurcations are described.

Sign in / Sign up

Export Citation Format

Share Document