scholarly journals Bifurcations of Orbit and Inclination Flips Heteroclinic Loop with Nonhyperbolic Equilibria

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Fengjie Geng ◽  
Junfang Zhao
Keyword(s):  
Periodic Orbits ◽  
Homoclinic Orbits ◽  
Transcritical Bifurcation ◽  
Heteroclinic Orbits ◽  
Heteroclinic Loop ◽  
Orbit Flip ◽  
Inclination Flip ◽  
Hyperbolic Saddle

The bifurcations of heteroclinic loop with one nonhyperbolic equilibrium and one hyperbolic saddle are considered, where the nonhyperbolic equilibrium is supposed to undergo a transcritical bifurcation; moreover, the heteroclinic loop has an orbit flip and an inclination flip. When the nonhyperbolic equilibrium does not undergo a transcritical bifurcation, we establish the coexistence and noncoexistence of the periodic orbits and homoclinic orbits. While the nonhyperbolic equilibrium undergoes the transcritical bifurcation, we obtain the noncoexistence of the periodic orbits and homoclinic orbits and the existence of two or three heteroclinic orbits.

Homoclinic and Heteroclinic Bifurcations Close to a Twisted Heteroclinic Cycle

1998 ◽  
Vol 08 (02) ◽  
pp. 359-375 ◽  
Author(s):  
Martín G. Zimmermann ◽  
Mario A. Natiello
Keyword(s):  
Reaction Diffusion ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Reaction Diffusion Equation ◽  
Heteroclinic Cycle ◽  
Flip Bifurcation ◽  
Double Loop ◽  
Orbit Flip ◽  
Inclination Flip ◽  
Two Parameters

We study the interaction of a transcritical (or saddle-node) bifurcation with a codimension-0/codimension-2 heteroclinic cycle close to (but away from) the local bifurcation point. The study is motivated by numerical observations on the traveling wave ODE of a reaction–diffusion equation. The manifold organization is such that two branches of homoclinic orbits to each fixed point are created when varying the two parameters controlling the codimension-2 loop. It is shown that the homoclinic orbits may become degenerate in an orbit-flip bifurcation. We establish the occurrence of multi-loop homoclinic and heteroclinic orbits in this system. The double-loop homoclinic orbits are shown to bifurcate in an inclination-flip bifurcation, where a Smale's horseshoe is found.

Bifurcations Near the Weak Type Heterodimensional Cycle

2014 ◽  
Vol 24 (09) ◽  
pp. 1450112 ◽  
Author(s):  
Xingbo Liu
Keyword(s):  
Coordinate System ◽  
Periodic Orbits ◽  
Weak Type ◽  
Local Coordinate System ◽  
Heteroclinic Orbit ◽  
Homoclinic Orbits ◽  
Small Perturbations ◽  
Orbit Flip ◽  
Bifurcation Phenomena ◽  
Inclination Flip

The aim of this paper is to show the bifurcation phenomena near the weak type heterodimensional cycle when the orbit flip and inclination flip occur simultaneously in its nontransversal heteroclinic orbit. With the aid of a suitable local coordinate system, the Poincaré map is constructed. By means of the bifurcation equations, the persistence of heterodimensional cycles, the coexistence of the heterodimensional cycle and periodic orbits or homoclinic orbits, and the existence of bifurcation surfaces of homoclinic orbits or the periodic orbits are discussed under small perturbations. Moreover, an example is given to show the existence of the system which has a heterodimensional cycle with orbit flip and inclination flip.

NONRESONANT BIFURCATIONS OF HETEROCLINIC LOOPS WITH ONE INCLINATION FLIP

2011 ◽  
Vol 21 (01) ◽  
pp. 255-273 ◽  
Author(s):  
SHULIANG SHUI ◽  
JINGJING LI ◽  
XUYANG ZHANG
Keyword(s):  
Periodic Orbit ◽  
Homoclinic Orbit ◽  
Vector Fields ◽  
Heteroclinic Orbits ◽  
Dimensional Vector ◽  
Heteroclinic Loop ◽  
Stable Foliation ◽  
Local Coordinates ◽  
Inclination Flip

Heteroclinic bifurcations in four-dimensional vector fields are investigated by setting up local coordinates near a heteroclinic loop. This heteroclinic loop consists of two principal heteroclinic orbits, but there is one stable foliation that involves an inclination flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit, and 1-periodic orbit are studied. Also, the nonexistence, existence of the 2-homoclinic and 2-periodic orbit are demonstrated.

Inclination-flip homoclinic orbits arising from orbit-flip

Nonlinearity ◽  
2001 ◽  
Vol 14 (2) ◽  
pp. 379-393 ◽  
Author(s):  
C A Morales ◽  
M J Pacifico
Keyword(s):  
Homoclinic Orbits ◽  
Orbit Flip ◽  
Inclination Flip

BIFURCATIONS OF GENERIC HETEROCLINIC LOOP ACCOMPANIED BY TRANSCRITICAL BIFURCATION

2008 ◽  
Vol 18 (04) ◽  
pp. 1069-1083 ◽  
Author(s):  
FENGJIE GENG ◽  
DAN LIU ◽  
DEMING ZHU
Keyword(s):  
Periodic Orbit ◽  
Heteroclinic Orbit ◽  
Transcritical Bifurcation ◽  
Homoclinic Loop ◽  
Heteroclinic Loop ◽  
Higher Dimensional ◽  
Hyperbolic Saddle

The bifurcations of generic heteroclinic loop with one nonhyperbolic equilibrium p1and one hyperbolic saddle p2are investigated, where p1is assumed to undergo transcritical bifurcation. Firstly, we discuss bifurcations of heteroclinic loop when transcritical bifurcation does not happen, the persistence of heteroclinic loop, the existence of homoclinic loop connecting p1(resp. p2) and the coexistence of one homoclinic loop and one periodic orbit are established. Secondly, we analyze bifurcations of heteroclinic loop accompanied by transcritical bifurcation, namely, nonhyperbolic equilibrium p1splits into two hyperbolic saddles [Formula: see text] and [Formula: see text], a heteroclinic loop connecting [Formula: see text] and p2, homoclinic loop with [Formula: see text] (resp. p2) and heteroclinic orbit joining [Formula: see text] and [Formula: see text] (resp. [Formula: see text] and p2; p2and [Formula: see text]) are found. The results achieved here can be extended to higher dimensional systems.

Bifurcation Complexity from Orbit-Flip Homoclinic Orbit of Weak Type

2016 ◽  
Vol 26 (04) ◽  
pp. 1650059 ◽  
Author(s):  
Qiuying Lu ◽  
Vincent Naudot
Keyword(s):  
Homoclinic Orbit ◽  
Weak Type ◽  
Blow Up ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Unperturbed System ◽  
Orbit Flip ◽  
Inclination Flip ◽  
Three Dimensional Vector Field ◽  
Degenerate Version

In this paper, we study the unfolding of a three-dimensional vector field having an orbit-flip homoclinic orbit of weak type. Such a homoclinic orbit is a degenerate version of the so-called orbit-flip homoclinic orbit. We show the existence of inclination-flip homoclinic orbits of arbitrary higher order bifurcating from the unperturbed system. Our strategy consists of using the local moving coordinates method and blow up in the parameter space. In addition, the numerical existence of the orbit-flip homoclinic orbit of weak type is presented based on the truncated Taylor expansion and the numerical computation for both the strong stable manifold and unstable manifold.

Bifurcation Analysis in a Class of Piecewise Nonlinear Systems with a Nonsmooth Heteroclinic Loop

2018 ◽  
Vol 28 (02) ◽  
pp. 1850026
Author(s):  
Yuanyuan Liu ◽  
Feng Li ◽  
Pei Dang
Keyword(s):  
Limit Cycles ◽  
Polynomial Systems ◽  
Melnikov Function ◽  
Heteroclinic Orbits ◽  
Phase Portraits ◽  
Unperturbed System ◽  
Piecewise Polynomial ◽  
Heteroclinic Loop ◽  
First Order ◽  
Nilpotent Critical Point

We consider the bifurcation in a class of piecewise polynomial systems with piecewise polynomial perturbations. The corresponding unperturbed system is supposed to possess an elementary or nilpotent critical point. First, we present 17 cases of possible phase portraits and conditions with at least one nonsmooth periodic orbit for the unperturbed system. Then we focus on the two specific cases with two heteroclinic orbits and investigate the number of limit cycles near the loop by means of the first-order Melnikov function, respectively. Finally, we take a quartic piecewise system with quintic piecewise polynomial perturbation as an example and obtain that there can exist ten limit cycles near the heteroclinic loop.

ESTIMATION OF CHAOTIC THRESHOLDS FOR THE RECENTLY PROPOSED ROTATING PENDULUM

2013 ◽  
Vol 23 (04) ◽  
pp. 1350074 ◽  
Author(s):  
N. HAN ◽  
Q. J. CAO ◽  
M. WIERCIGROCH
Keyword(s):  
Nonlinear System ◽  
Nonlinear Behavior ◽  
Melnikov Method ◽  
Original System ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Chaotic Attractors ◽  
Viscous Damping ◽  
Approximate System ◽  
Smoothness Parameter

In this paper, we investigate the nonlinear behavior of the recently proposed rotating pendulum which is a cylindrically nonlinear system with irrational type having smooth and discontinuous characteristics depending on the value of a smoothness parameter. We introduce a cylindrical approximate system whose analytical solutions can be obtained successfully to reflect the nature of the original system without the barrier of irrationalities. Furthermore, Melnikov method is employed to detect the chaotic thresholds for the homoclinic orbits of the second-type, a pair of homoclinic orbits of the first and second-type and the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing within the smooth regime. Numerical simulations show the efficiency of the proposed method and the results presented herein this paper demonstrate the predicated chaotic attractors of pendulum-type, SD-type and their mixture depending on the coupling of the nonlinearities.

Exact Torus Knot Periodic Orbits and Homoclinic Orbits in a Class of Three-Dimensional Flows Generated by a Planar Cubic System

2017 ◽  
Vol 27 (13) ◽  
pp. 1750205 ◽  
Author(s):  
Tonghua Zhang ◽  
Jibin Li
Keyword(s):  
Vector Field ◽  
Periodic Orbits ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Numerical Examples ◽  
Torus Knot ◽  
Cubic System ◽  
Theoretical Results

This paper considers a class of three-dimensional systems constructed by a rotating planar symmetric cubic vector field. To study its periodic orbits including homoclinic orbits, which may be knotted in space, we classify the types of periodic orbits and then calculate their exact parametric representations. Our study shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on three families of invariant tori. Numerical examples of [Formula: see text]-torus knot periodic orbits have also been provided to illustrate our theoretical results.

PERIODIC ORBITS NEAR A HETEROCLINIC LOOP FORMED BY ONE-DIMENSIONAL ORBIT AND A TWO-DIMENSIONAL MANIFOLD: APPLICATION TO THE CHARGED COLLINEAR THREE-BODY PROBLEM

2007 ◽  
Vol 17 (06) ◽  
pp. 2175-2183
Author(s):  
JAUME LLIBRE ◽  
DANIEL PAŞCA
Keyword(s):  
Periodic Orbits ◽  
Degrees Of Freedom ◽  
Body Problem ◽  
Negative Energy ◽  
Dimensional Manifold ◽  
Three Body Problem ◽  
Heteroclinic Loop ◽  
One Dimensional ◽  
Three Body ◽  
Total Collision

This paper is devoted to the study of a type of differential systems which appear usually in the study of the Hamiltonian systems with two degrees of freedom. We prove the existence of infinitely many periodic orbits on each negative energy level. All these periodic orbits pass near to the total collision. Finally we apply these results to study the existence of periodic orbits in the charged collinear three-body problem.

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