scholarly journals Heteroclinic orbits and chaotic invariant sets for monotone twist maps

2003 ◽  
Vol 9 (1) ◽  
pp. 69-95 ◽  
Author(s):  
Tifei Qian ◽  
◽  
Zhihong Xia ◽  
Keyword(s):  
Heteroclinic Orbits ◽  
Invariant Sets ◽  
Twist Maps

HETEROCLINIC ORBITS FOR MONOTONE TWIST MAPS THROUGH A VARIATIONAL PRINCIPLE

2001 ◽  
Vol 11 (09) ◽  
pp. 2451-2461
Author(s):  
TIFEI QIAN
Keyword(s):  
Variational Principle ◽  
Variational Method ◽  
Fixed Points ◽  
Heteroclinic Orbit ◽  
Geometric Method ◽  
Heteroclinic Orbits ◽  
Constructive Approach ◽  
Connecting Orbits ◽  
Relative Ease ◽  
Twist Maps

The variational method has shown many advantages over the geometric method in proving the existence of connecting orbits since it requires much weaker hyperbolicity and less smoothness. Many results known to be difficult to obtain by the geometric method can now be obtained by a variational principle with relative ease. In particular, a variational principle provides a constructive approach to the existence of heteroclinic orbits. In this paper a variational principle is used to construct a heteroclinic orbit between an adjacent minimal pair of fixed points for monotone twist maps on (ℝ/ℤ) × ℝ. Application of our results to a standard map is also given.

scholarly journals Global Dynamics of the Hastings-Powell System

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Luis N. Coria
Keyword(s):  
Numerical Simulations ◽  
Equilibrium Points ◽  
Heteroclinic Orbits ◽  
Invariant Sets ◽  
Chaotic Attractors ◽  
Global Dynamics ◽  
State Variables ◽  
System Parameters ◽  
First Order ◽  
Extremum Conditions

This paper studies the problem of bounding a domain that contains all compact invariant sets of the Hastings-Powell system. The results were obtained using the first-order extremum conditions and the iterative theorem to a biologically meaningful model. As a result, we calculate the bounds given by a tetrahedron with excisions, described by several inequalities of the state variables and system parameters. Therefore, a region is identified where all the system dynamics are located, that is, its compact invariant sets: equilibrium points, periodic-homoclinic-heteroclinic orbits, and chaotic attractors. It was also possible to formulate a nonexistence condition of the compact invariant sets. Additionally, numerical simulations provide examples of the calculated boundaries for the chaotic attractors or periodic orbits. The results provide insights regarding the global dynamics of the system.

Applications of the Melnikov method to twist maps in higher dimensions using the variational approach

1997 ◽  
Vol 17 (2) ◽  
pp. 445-462 ◽  
Author(s):  
HECTOR E. LOMELI
Keyword(s):  
Fixed Point ◽  
Infinite Series ◽  
Variational Approach ◽  
Melnikov Method ◽  
Higher Dimensions ◽  
Heteroclinic Orbits ◽  
Local Description ◽  
Twist Maps ◽  
Hyperbolic Fixed Point ◽  
Stable And Unstable Manifold

We work with symplectic diffeomorphisms of the $n$-annulus ${\Bbb{A}}^n=T^*({\Bbb{R}}^n/{\Bbb{Z}}^n)$. Using the variational approach of Aubry and Mather, we are able to give a local description of the stable (and unstable) manifold for a hyperbolic fixed point. We use this in order to get a Melnikov-like formula for exact symplectic twist maps. This formula involves an infinite series that could be computed in some specific cases. We apply our formula to prove the existence of heteroclinic orbits for a family of twist maps in ${\Bbb{R}}^4$.

Dynamical Analysis of Raychaudhuri Equations Based on the Localization Method of Compact Invariant Sets

2014 ◽  
Vol 24 (11) ◽  
pp. 1450136 ◽  
Author(s):  
Alexander P. Krishchenko ◽  
Konstantin E. Starkov
Keyword(s):  
Equilibrium Points ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Invariant Sets ◽  
Dynamical Analysis ◽  
Two Dimensional ◽  
Main Attention ◽  
Raychaudhuri Equations ◽  
Omega Limit Set

In this paper, we examine the localization problem of compact invariant sets of Raychaudhuri equations with nonzero parameters. The main attention is attracted to the localization of periodic/homoclinic orbits and homoclinic cycles: we prove that there are neither periodic/homoclinic orbits nor homoclinic cycles; we find heteroclinic orbits connecting distinct equilibrium points. We describe some unbounded domain such that nonescaping to infinity positive semitrajectories which are contained in this domain have the omega-limit set located in the boundary of this domain. We find a locus of other types of compact invariant sets respecting three-dimensional and two-dimensional invariant planes. Besides, we describe the phase portrait of the system obtained from the Raychaudhuri equations by the restriction on the two-dimensional invariant plane.

Unstable sets, heteroclinic orbits and generic quasi-convergence for essentially strongly order-preserving semiflows

2009 ◽  
Vol 52 (3) ◽  
pp. 797-807 ◽  
Author(s):  
Taishan Yi ◽  
Qingguo Li
Keyword(s):  
Invariant Set ◽  
Unstable Equilibrium ◽  
Heteroclinic Orbits ◽  
Invariant Sets ◽  
Limit Set

AbstractWe present conditions guaranteeing the existence of non-trivial unstable sets for compact invariant sets in semiflows with certain compactness conditions, and then establish the existence of such unstable sets for an unstable equilibrium or a minimal compact invariant set, not containing equilibria, in an essentially strongly order-preserving semiflow. By appealing to the limit-set dichotomy for essentially strongly order-preserving semiflows, we prove the existence of an orbit connection from an equilibrium to a minimal compact invariant set, not consisting of equilibria. As an application, we establish a new generic convergence principle for essentially strongly order-preserving semiflows with certain compactness conditions.

scholarly journals Some remarks on Birkhoff and Mather twist map theorems

1982 ◽  
Vol 2 (2) ◽  
pp. 185-194 ◽  
Author(s):  
A. Katok
Keyword(s):  
Recent Result ◽  
Periodic Orbits ◽  
Invariant Sets ◽  
Classical Theorem ◽  
Twist Map ◽  
Twist Maps ◽  
Birkhoff's Theorem ◽  
Birkhoff’S Theorem

AbstractA recent result of J. Mather [1] about the existence of quasi-periodic orbits for twist maps is derived from an appropriately modified version of G. D. Birkhoff's classical theorem concerning periodic orbits. A proof of Birkhoff's theorem is given using a simplified geometric version of Mather's arguments. Additional properties of Mather's invariant sets are discussed.

scholarly journals Orbital dynamics on invariant sets of contact Hamiltonian systems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Qihuai Liu ◽  
Pedro J. Torres
Keyword(s):  
Hamiltonian System ◽  
Equilibrium Points ◽  
Differential Invariant ◽  
Domain Of Attraction ◽  
Heteroclinic Orbits ◽  
Invariant Sets ◽  
Zero Energy ◽  
Contact Phase ◽  
Hamiltonian Flows ◽  
Geometric Conditions

<p style='text-indent:20px;'>In this paper, we shall give new insights on dynamics of contact Hamiltonian flows, which are gaining importance in several branches of physics as they model a dissipative behaviour. We divide the contact phase space into three parts, which are corresponding to three differential invariant sets <inline-formula><tex-math id="M1">\begin{document}$ \Omega_\pm, \Omega_0 $\end{document}</tex-math></inline-formula>. On the invariant sets <inline-formula><tex-math id="M2">\begin{document}$ \Omega_\pm $\end{document}</tex-math></inline-formula>, under some geometric conditions, the contact Hamiltonian system is equivalent to a Hamiltonian system via the Hölder transformation. The invariant set <inline-formula><tex-math id="M3">\begin{document}$ \Omega_0 $\end{document}</tex-math></inline-formula> may be composed of several equilibrium points and heteroclinic orbits connecting them, on which contact Hamiltonian system is conservative. Moreover, we have shown that, under general conditions, the zero energy level domain is a domain of attraction. In some cases, such a domain of attraction does not have nontrivial periodic orbits. Some interesting examples are presented.</p>

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