Bifurcations of Double Homoclinic Loops in Reversible Systems

2020 ◽  
Vol 30 (16) ◽  
pp. 2050246
Author(s):  
Yuzhen Bai ◽  
Xingbo Liu
Keyword(s):  
Coordinate System ◽  
Periodic Orbits ◽  
Homoclinic Orbit ◽  
Poincaré Map ◽  
Local Coordinate System ◽  
Reversible Systems ◽  
Bifurcation Equation ◽  
Bifurcation Phenomena ◽  
New Type ◽  
Symmetric Periodic Orbits

This paper is devoted to the study of bifurcation phenomena of double homoclinic loops in reversible systems. With the aid of a suitable local coordinate system, the Poincaré map is constructed. By means of the bifurcation equation, we perform a detailed study to obtain fruitful results, and demonstrate the existence of the R-symmetric large homoclinic orbit of new type near the primary double homoclinic loops, the existence of infinitely many R-symmetric periodic orbits accumulating onto the R-symmetric large homoclinic orbit, and the coexistence of R-symmetric large homoclinic orbit and the double homoclinic loops. The homoclinic bellow can also be found under suitable perturbation. The relevant bifurcation surfaces and the existence regions are located.

DEGENERATE BIFURCATIONS OF HETERODIMENSIONAL CYCLES WITH ORBIT FLIP

2013 ◽  
Vol 23 (05) ◽  
pp. 1350080 ◽  
Author(s):  
XINGBO LIU ◽  
JUNYING LIU ◽  
DEMING ZHU
Keyword(s):  
Coordinate System ◽  
Periodic Orbits ◽  
Bifurcation Analysis ◽  
Homoclinic Orbit ◽  
Poincaré Map ◽  
Local Coordinate System ◽  
Three Dimensional ◽  
Poincare Map ◽  
Orbit Flip

In this paper, nongeneric bifurcation analysis near heterodimensional cycles with orbit flip is investigated for three-dimensional systems. With the aid of a suitable local coordinate system, the Poincaré map is constructed. By means of the bifurcation equations, the existence, nonexistence, coexistence and uniqueness of homoclinic orbit, periodic orbits and the heterodimensional cycle are studied, the relevant bifurcation surfaces and their existing regions are given. Some known results are extended. An example is given to show the existence of the system which has a heterodimensional cycle with orbit flip.

CODIMENSION 3 HETEROCLINIC BIFURCATIONS WITH ORBIT AND INCLINATION FLIPS IN REVERSIBLE SYSTEMS

2008 ◽  
Vol 18 (12) ◽  
pp. 3689-3701 ◽  
Author(s):  
YANCONG XU ◽  
DEMING ZHU ◽  
FENGJIE GENG
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Homoclinic Orbit ◽  
Heteroclinic Orbit ◽  
Homoclinic Bifurcation ◽  
Reversible Systems ◽  
Reversible System ◽  
Bifurcation Diagrams ◽  
Symmetric Periodic Orbits ◽  
The Continuum

Heteroclinic bifurcations with orbit-flips and inclination-flips are investigated in a four-dimensional reversible system by using the method originally established in [Zhu, 1998; Zhu & Xia, 1998]. The existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic orbit, R-symmetric homoclinic orbit and R-symmetric periodic orbit are obtained. The double R-symmetric homoclinic bifurcation is found, and the continuum of R-symmetric periodic orbits accumulating into a homoclinic orbit is also demonstrated. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation diagrams are drawn.

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.

scholarly journals Resonant Homoclinic Flips Bifurcation in Principal Eigendirections

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Tiansi Zhang ◽  
Xiaoxin Huang ◽  
Deming Zhu
Keyword(s):  
Periodic Orbit ◽  
Coordinate System ◽  
Homoclinic Orbit ◽  
Principal Eigenvalue ◽  
Small Neighborhood ◽  
Homoclinic Bifurcation ◽  
Bifurcation Equation ◽  
Orbit Flip ◽  
Inclination Flip ◽  
Poincaré Return Map

A codimension-4 homoclinic bifurcation with one orbit flip and one inclination flip at principal eigenvalue direction resonance is considered. By introducing a local active coordinate system in some small neighborhood of homoclinic orbit, we get the Poincaré return map and the bifurcation equation. A detailed investigation produces the number and the existence of 1-homoclinic orbit, 1-periodic orbit, and double 1-periodic orbits. We also locate their bifurcation surfaces in certain regions.

scholarly journals Dynamics of Symmetric Dynamical Systems with Delayed Switching

2010 ◽  
Vol 16 (7-8) ◽  
pp. 1111-1140 ◽  
Author(s):  
J. Sieber ◽  
P. Kowalczyk ◽  
S.J. Hogan ◽  
M. Di Bernardo
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Poincaré Map ◽  
Invariant Tori ◽  
Poincare Map ◽  
Single Degree Of Freedom ◽  
Finite Dimensional ◽  
Symmetric Periodic Orbits ◽  
Full Reflection

We study dynamical systems that switch between two different vector fields depending on a discrete variable and with a delay. When the delay reaches a problem-dependent critical value, so-called event collisions occur. This paper classifies and analyzes event collisions, a special type of discontinuity-induced bifurcations, for periodic orbits. Our focus is on event collisions of symmetric periodic orbits in systems with full reflection symmetry, a symmetry that is prevalent in applications. We derive an implicit expression for the Poincaré map near the colliding periodic orbit. The Poincaré map is piecewise smooth, finite-dimensional, and changes the dimension of its image at the collision. In the second part of the paper we apply this general result to the class of unstable linear single-degree-of-freedom oscillators where we detect and continue numerically collisions of invariant tori. Moreover, we observe that attracting closed invariant polygons emerge at the torus collision.

BIFURCATION OF ROUGH HETEROCLINIC LOOP WITH ORBIT FLIPS

2012 ◽  
Vol 22 (11) ◽  
pp. 1250278 ◽  
Author(s):  
XINGBO LIU ◽  
ZHENZHEN WANG ◽  
DEMING ZHU
Keyword(s):  
Periodic Orbit ◽  
Coordinate System ◽  
Vector Fields ◽  
Poincaré Map ◽  
Local Coordinate System ◽  
Heteroclinic Orbit ◽  
Dimensional Vector ◽  
Homoclinic Loop ◽  
Heteroclinic Loop ◽  
Approximate Expressions

In this paper, heteroclinic loop bifurcations with double orbit flips are investigated in four-dimensional vector fields. We obtain the bifurcation equations by setting up a local coordinate system near the rough heteroclinic orbit and establishing the Poincaré map. By means of the bifurcation equations, we investigate the existence, coexistence and noncoexistence of periodic orbit, homoclinic loop and heteroclinic loop under some nongeneric conditions. The approximate expressions of corresponding bifurcation curves (or surfaces) are also given. An example of application is also given to demonstrate the existence of the heteroclinic loop with double orbit flips.

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