scholarly journals Symmetric periodic orbits near a heteroclinic loop in formed by two singular points, a semistable periodic orbit and their invariant manifolds

2009 ◽  
Vol 238 (6) ◽  
pp. 699-705 ◽  
Author(s):  
Montserrat Corbera ◽  
Jaume Llibre ◽  
Marco Antonio Teixeira
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Singular Points ◽  
Heteroclinic Loop ◽  
Symmetric Periodic Orbits

scholarly journals GENERATION OF SYMMETRIC PERIODIC ORBITS BY A HETEROCLINIC LOOP FORMED BY TWO SINGULAR POINTS AND THEIR INVARIANT MANIFOLDS OF DIMENSIONS 1 AND 2 IN ℝ3

2007 ◽  
Vol 17 (09) ◽  
pp. 3295-3302 ◽  
Author(s):  
MONTSERRAT CORBERA ◽  
JAUME LLIBRE
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifold ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Main Tool ◽  
Singular Points ◽  
Dimensional Sphere ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Heteroclinic Loop

In this paper we will find continuous periodic orbits passing near infinity for a class of polynomial vector fields in ℝ3. We consider polynomial vector fields that are invariant under a symmetry with respect to a plane Σ and that possess a "generalized heteroclinic loop" formed by two singular points e+ and e- at infinity and their invariant manifolds Γ and Λ. Γ is an invariant manifold of dimension 1 formed by an orbit going from e- to e+, Γ is contained in ℝ3 and is transversal to Σ. Λ is an invariant manifold of dimension 2 at infinity. In fact, Λ is the two-dimensional sphere at infinity in the Poincaré compactification minus the singular points e+ and e-. The main tool for proving the existence of such periodic orbits is the construction of a Poincaré map along the generalized heteroclinic loop together with the symmetry with respect to Σ.

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.

Existence of Homoclinic Cycles and Periodic Orbits in a Class of Three-Dimensional Piecewise Affine Systems

2018 ◽  
Vol 28 (02) ◽  
pp. 1850024 ◽  
Author(s):  
Lei Wang ◽  
Xiao-Song Yang
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Spatial Location ◽  
Homoclinic Bifurcation ◽  
Piecewise Affine Systems ◽  
Affine Systems ◽  
Piecewise Affine

For a class of three-dimensional piecewise affine systems, this paper focuses on the existence of homoclinic cycles and the phenomena of homoclinic bifurcation leading to periodic orbits. Based on the spatial location relation between the invariant manifolds of subsystems and the switching manifold, a concise necessary and sufficient condition for the existence of homoclinic cycles is obtained. Then the homoclinic bifurcation is studied and the sufficient conditions for the birth of a periodic orbit are obtained. Furthermore, the sufficient conditions are obtained for the periodic orbit to be a sink, a source or a saddle. As illustrations, several concrete examples are presented.

scholarly journals Using Lagrangian Descriptors to Uncover Invariant Structures in Chesnavich’s Isokinetic Model with Application to Roaming

2020 ◽  
Vol 30 (05) ◽  
pp. 2050076
Author(s):  
Vladimír Krajňák ◽  
Gregory S. Ezra ◽  
Stephen Wiggins
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Unstable Periodic Orbits ◽  
Invariant Structures ◽  
Text Model ◽  
Model Subject ◽  
Lagrangian Descriptors

Complementary to existing applications of Lagrangian descriptors as an exploratory method, we use Lagrangian descriptors to find invariant manifolds in a system where some invariant structures have already been identified. In this case, we use the parametrization of a periodic orbit to construct a Lagrangian descriptor that will be locally minimized on its invariant manifolds. The procedure is applicable (but not limited) to systems with highly unstable periodic orbits, such as the isokinetic Chesnavich CH[Formula: see text] model subject to a Hamiltonian isokinetic theromostat. Aside from its low computational requirements, the method enables us to study the invariant structures responsible for roaming in the isokinetic Chesnavich CH[Formula: see text] model.

scholarly journals Twist Periodic Orbits for Continuous Maps of the Eight Space

2013 ◽  
Vol 6 ◽  
pp. 4-8
Author(s):  
Iftichar Mudhar Talb Al-Shraa
Keyword(s):  
Fixed Point ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Continuous Map ◽  
Continuous Maps

Let g be a continuous map from 8 to itself has a fixed point at (0,0), we prove that g has a twist periodic orbit if there is a rational rotation number.

scholarly journals Grazing-Sliding Bifurcations Creating Infinitely Many Attractors

2017 ◽  
Vol 27 (12) ◽  
pp. 1730042 ◽  
Author(s):  
David J. W. Simpson
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Simple Type ◽  
Asymptotically Stable ◽  
Sliding Bifurcation ◽  
Continuous Maps ◽  
Stable Periodic ◽  
Smooth System ◽  
Structurally Stable

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations, structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.

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