scholarly journals On the escape from potentials with two exit channels

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Juan F. Navarro
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbits ◽  
Unstable Periodic Orbits ◽  
Stable And Unstable Manifolds ◽  
Surface Of Section ◽  
Escape Dynamics ◽  
Unstable Manifolds

Abstract The aim of this paper is to investigate the escape dynamics in a Hamiltonian system describing the motion of stars in a galaxy with two exit channels through the analysis of the successive intersections of the stable and unstable manifolds to the main unstable periodic orbits with an adequate surface of section. We describe in detail the origin of the spirals shapes of the windows through which stars escape.

ALFVÉN INTERIOR CRISIS

2004 ◽  
Vol 14 (07) ◽  
pp. 2375-2380 ◽  
Author(s):  
F. A. BOROTTO ◽  
A. C.-L. CHIAN ◽  
E. L. REMPEL
Keyword(s):  
Periodic Orbits ◽  
Large Amplitude ◽  
Numerical Study ◽  
Alfven Wave ◽  
Unstable Periodic Orbits ◽  
Stable And Unstable Manifolds ◽  
Laboratory Plasmas ◽  
Derivative Nonlinear Schrödinger Equation ◽  
Unstable Manifolds ◽  
Low Dimensional

A numerical study of an interior crisis of a large-amplitude Alfvén wave described by the driven-dissipative derivative nonlinear Schrödinger equation, in the low-dimensional limit, is reported. An example of Alfvén interior crisis is characterized using the unstable periodic orbits and their associated invariant stable and unstable manifolds in the Poincaré plane. We suggest that this type of chaotic transition can be observed in space and laboratory plasmas.

scholarly journals CHAINS OF ROTATIONAL TORI AND FILAMENTARY STRUCTURES CLOSE TO HIGH MULTIPLICITY PERIODIC ORBITS IN A 3D GALACTIC POTENTIAL

2011 ◽  
Vol 21 (08) ◽  
pp. 2331-2342 ◽  
Author(s):  
M. KATSANIKAS ◽  
P. A. PATSIS ◽  
A. D. PINOTSIS
Keyword(s):  
Phase Space ◽  
Hamiltonian System ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Color Variation ◽  
Space Structures ◽  
Unstable Periodic Orbits ◽  
Unstable Periodic Orbit ◽  
Surface Of Section ◽  
Autonomous Hamiltonian System

This paper discusses phase space structures encountered in the neighborhood of periodic orbits with high order multiplicity in a 3D autonomous Hamiltonian system with a potential of galactic type. We consider 4D spaces of section and we use the method of color and rotation [Patsis & Zachilas, 1994] in order to visualize them. As examples, we use the case of two orbits, one 2-periodic and one 7-periodic. We investigate the structure of multiple tori around them in the 4D surface of section and in addition, we study the orbital behavior in the neighborhood of the corresponding simple unstable periodic orbits. By considering initially a few consequents in the neighborhood of the orbits in both cases we find a structure in the space of section, which is in direct correspondence with what is observed in a resonance zone of a 2D autonomous Hamiltonian system. However, in our 3D case we have instead of stability islands rotational tori, while the chaotic zone connecting the points of the unstable periodic orbit is replaced by filaments extending in 4D following a smooth color variation. For more intersections, the consequents of the orbit which started in the neighborhood of the unstable periodic orbit, diffuse in phase space and form a cloud that occupies a large volume surrounding the region containing the rotational tori. In this cloud the colors of the points are mixed. The same structures have been observed in the neighborhood of all m-periodic orbits we have examined in the system. This indicates a generic behavior.

scholarly journals Locating oscillatory orbits of the parametrically-excited pendulum

1996 ◽  
Vol 37 (3) ◽  
pp. 309-319 ◽  
Author(s):  
M. J. Clifford ◽  
S. R. Bishop
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Numerical Techniques ◽  
Stable And Unstable Manifolds ◽  
Parametrically Excited ◽  
Countable Infinity ◽  
Unstable Manifolds

AbstractA method is considered for locating oscillating, nonrotating solutions for the parametrically-excited pendulum by inferring that a particular horseshoe exists in the stable and unstable manifolds of the local saddles. In particular, odd-periodic solutions are determined which are difficult to locate by alternative numerical techniques. A pseudo-Anosov braid is also located which implies the existence of a countable infinity of periodic orbits without the horseshoe assumption being necessary.

scholarly journals THE STRUCTURE AND EVOLUTION OF CONFINED TORI NEAR A HAMILTONIAN HOPF BIFURCATION

2011 ◽  
Vol 21 (08) ◽  
pp. 2321-2330 ◽  
Author(s):  
M. KATSANIKAS ◽  
P. A. PATSIS ◽  
G. CONTOPOULOS
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbits ◽  
Transition Point ◽  
Initial Conditions ◽  
Unstable Periodic Orbits ◽  
Unstable Periodic Orbit ◽  
Surface Of Section ◽  
Autonomous Hamiltonian System ◽  
Hamiltonian Hopf Bifurcation ◽  
Stable Periodic

We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper, we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis & Zachilas, 1994]. We find that the consequents are contained in 2D "confined tori". Then, we investigate the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.

ATTRACTORS IN CONSERVATIVE SYSTEMS

2006 ◽  
Vol 16 (06) ◽  
pp. 1795-1807 ◽  
Author(s):  
G. CONTOPOULOS
Keyword(s):  
Black Holes ◽  
Hamiltonian System ◽  
Periodic Orbits ◽  
Basins Of Attraction ◽  
Unstable Periodic Orbits ◽  
Asymptotic Curves ◽  
Conservative Systems ◽  
Finite Distance

Normally, conservative systems do not have attractors. However, in a system with escapes, the infinity acts as an attractor. Furthermore, attractors may appear as singularities at a finite distance. We consider the basins of escape in a particular Hamiltonian system with escapes and the rates of escape for various values of the parameters. Then we consider the basins of attraction of a system of two fixed black holes, with particular emphasis on the asymptotic curves of its unstable periodic orbits.

scholarly journals Locating and Stabilizing Unstable Periodic Orbits Embedded in the Horseshoe Map

2021 ◽  
Vol 31 (04) ◽  
pp. 2150110
Author(s):  
Yuu Miino ◽  
Daisuke Ito ◽  
Tetsushi Ueta ◽  
Hiroshi Kawakami
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Ad Hoc ◽  
Computation Method ◽  
Control Input ◽  
Unstable Periodic Orbits ◽  
Stable And Unstable Manifolds ◽  
Controlling Chaos ◽  
Dimensional Dynamical System ◽  
The Given

Based on the theory of symbolic dynamical systems, we propose a novel computation method to locate and stabilize the unstable periodic points (UPPs) in a two-dimensional dynamical system with a Smale horseshoe. This method directly implies a new framework for controlling chaos. By introducing the subset based correspondence between a planar dynamical system and a symbolic dynamical system, we locate regions sectioned by stable and unstable manifolds comprehensively and identify the specified region containing a UPP with the particular period. Then Newton’s method compensates the accurate location of the UPP with the regional information as an initial estimation. On the other hand, the external force control (EFC) is known as an effective method to stabilize the UPPs. By applying the EFC to the located UPPs, robust controlling chaos is realized. In this framework, we never use ad hoc approaches to find target UPPs in the given chaotic set. Moreover, the method can stabilize UPPs with the specified period regardless of the situation where the targeted chaotic set is attractive. As illustrative numerical experiments, we locate and stabilize UPPs and the corresponding unstable periodic orbits in a horseshoe structure of the Duffing equation. In spite of the strong instability of UPPs, the controlled orbit is robust and the control input retains being tiny in magnitude.

USING COLOR AND ROTATION FOR VISUALIZING FOUR-DIMENSIONAL POINCARÉ CROSS-SECTIONS: WITH APPLICATIONS TO THE ORBITAL BEHAVIOR OF A THREE-DIMENSIONAL HAMILTONIAN SYSTEM

1994 ◽  
Vol 04 (06) ◽  
pp. 1399-1424 ◽  
Author(s):  
P.A. PATSIS ◽  
L. ZACHILAS
Keyword(s):  
Hamiltonian System ◽  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Cross Sections ◽  
Dimensional Space ◽  
Three Dimensional ◽  
New Method ◽  
Unstable Periodic Orbits ◽  
The Family

The problems encountered in the study of three-dimensional Hamiltonian systems by means of the Poincare cross-sections are reviewed. A new method to overcome these problems is proposed. In order to visualize the four-dimensional “space” of section we introduce the use of color and rotation. We apply this method to the case of a family of simple periodic orbits in a three-dimensional potential and we describe the differences in the orbital behavior between regions close to stable and unstable periodic orbits. We outline the differences between the transition from stability to simple instability and the transition from stability to complex instability. We study the changes in the structure of the 4D “spaces” of section, which occur when the family becomes complex unstable after a DU →Δ or a S →Δ transition. We conclude that the orbital behavior after the transition depends on the orbital behavior before it.

Study on the Geometrical Properties and Existence of Orbit Homoclinic to a Saddle Point in n-Dimensional Autonomous Vector Field with Polynomials

2018 ◽  
Vol 28 (14) ◽  
pp. 1850169
Author(s):  
Lingli Xie
Keyword(s):  
Vector Field ◽  
Saddle Point ◽  
Equilibrium Point ◽  
Necessary Conditions ◽  
Continuous System ◽  
Homoclinic Bifurcation ◽  
Stable And Unstable Manifolds ◽  
Geometrical Properties ◽  
Unstable Manifolds

According to the theory of stable and unstable manifolds of an equilibrium point, we firstly find out some geometrical properties of orbits on the stable and unstable manifolds of a saddle point under some brief conditions of nonlinear terms composed of polynomials for [Formula: see text]-dimensional time continuous system. These properties show that the orbits on stable and unstable manifolds of the saddle point will stay on the corresponding stable and unstable subspaces in the [Formula: see text]-neighborhood of the saddle point. Furthermore, the necessary conditions of existence for orbit homoclinic to a saddle point are exposed. Some examples including homoclinic bifurcation are given to indicate the application of the results. Finally, the conclusions are presented.

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