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 Computing and Controlling Basins of Attraction in Multistability Scenarios

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
John Alexander Taborda ◽  
Fabiola Angulo
Keyword(s):  
Fixed Point ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Control Technique ◽  
Unstable Periodic Orbits ◽  
Control Effort ◽  
Coexistence Of Attractors ◽  
And Control

The aim of this paper is to describe and prove a new method to compute and control the basins of attraction in multistability scenarios and guarantee monostability condition. In particular, the basins of attraction are computed only using a submap, and the coexistence of periodic solutions is controlled through fixed-point inducting control technique, which has been successfully used until now to stabilize unstable periodic orbits. In this paper, however, fixed-point inducting control is used to modify the domains of attraction when there is coexistence of attractors. In order to apply the technique, the periodic orbit whose basin of attraction will be controlled must be computed. Therefore, the fixed-point inducting control is used to stabilize one of the periodic orbits and enhance its basin of attraction. Then, using information provided by the unstable periodic orbits and basins of attractions, the minimum control effort to stabilize the target periodic orbit in all desired ranges is computed. The applicability of the proposed tools is illustrated through two different coupled logistic maps.

scholarly journals Pattern recognition using chaotic neural networks

1998 ◽  
Vol 2 (4) ◽  
pp. 243-247 ◽  
Author(s):  
Z. Tan ◽  
B. S. Hepburn ◽  
C. Tucker ◽  
M. K. Ali
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Pattern Recognition ◽  
Dynamical Systems ◽  
Periodic Orbits ◽  
Basins Of Attraction ◽  
Limited Information ◽  
Unstable Periodic Orbits ◽  
Chaotic Neural Networks

Pattern recognition by chaotic neural networks is studied using a hyperchaotic neural network as model. Virtual basins of attraction are introduced around unstable periodic orbits which are then used as patterns. Search for periodic orbits in dynamical systems is treated as a process of pattern recognition. The role of synapses on patterns in chaotic networks is discussed. It is shown that distorted states having only limited information of the patterns are successfully recognized.

scholarly journals INSTABILITIES AND STICKINESS IN A 3D ROTATING GALACTIC POTENTIAL

2013 ◽  
Vol 23 (02) ◽  
pp. 1330005 ◽  
Author(s):  
M. KATSANIKAS ◽  
P. A. PATSIS ◽  
G. CONTOPOULOS
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Rotational Axis ◽  
Unstable Periodic Orbits ◽  
Chaotic Orbits ◽  
Asymptotic Curves ◽  
Axis Orbit ◽  
Diffusion Speed ◽  
Transition Points ◽  
Sticky Chaotic Orbits

We study the dynamics in the neighborhood of simple and double unstable periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic type. In order to visualize the four-dimensional spaces of section, we use the method of color and rotation. We investigate the structure of the invariant manifolds that we found in the neighborhood of simple and double unstable periodic orbits in 4D spaces of section. We consider orbits in the neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis (the rotational axis of our system). Close to the transition points from stability to simple instability, in the neighborhood of the bifurcated simple unstable x1v2 periodic orbits, we encounter the phenomenon of stickiness as the asymptotic curves of the unstable manifold surround regions of the phase space occupied by rotational tori existing in the region. For larger energies, away from the bifurcating point, the consequents of the chaotic orbits form clouds of points with mixing of color in their 4D representations. In the case of double instability, close to x1v2 orbits, we find clouds of points in the four-dimensional spaces of section. However, in some cases of double unstable periodic orbits belonging to the z-axis family we can visualize the associated unstable eigensurface. Chaotic orbits close to the periodic orbit remain sticky to this surface for long times (of the order of a Hubble time or more). Among the orbits we studied, we found those close to the double unstable orbits of the x1v2 family having the largest diffusion speed. The sticky chaotic orbits close to the bifurcation point of the simple unstable x1v2 orbit and close to the double unstable z-axis orbit that we have examined, have comparable diffusion speeds. These speeds are much slower than of the orbits in the neighborhood of x1v2 simple unstable periodic orbits away from the bifurcating point, or of the double unstable orbits of the same family having very different eigenvalues along the corresponding unstable eigendirections.

Periodic orbits and chaos around two fixed black holes. II

1991 ◽  
Vol 435 (1895) ◽  
pp. 551-562 ◽  
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Periodic Orbits ◽  
Satellite Orbits ◽  
Unstable Periodic Orbits ◽  
Zero Velocity ◽  
Stable Periodic

We study the orbits of particles (time-like geodesics) around two fixed black holes when the energy is elliptic, i. e. it does not allow the motion to extend to infinity. Most orbits are chaotic, but in many cases there are also ordered motions around stable periodic orbits. The orbits that fall into the first or the second black hole are separated by unstable periodic orbits. These are the satellite periodic orbits around the black holes when they exist. But for certain intervals of parameters there are no satellite orbits around the first or the second black hole. Then the limiting orbits are like arcs of hyperbolae, reaching the curve of zero velocity.

scholarly journals Cantori, Islands and Asymptotic Curves in The Stickiness Region

1999 ◽  
Vol 172 ◽  
pp. 221-230
Author(s):  
C. Efthymioulos ◽  
G. Contopoulos ◽  
N. Voglis
Keyword(s):  
Periodic Orbits ◽  
Unstable Periodic Orbits ◽  
Resonant Structure ◽  
Asymptotic Curves

AbstractThe resonant structure near a noble cantorus is found. Islands of stability are located near the gaps of the cantorus. The crossing of the gaps of the cantorus by the asymptotic curves of unstable periodic orbits is shown numerically (non-schematically). We discuss how these structures influence stickiness.

STICKINESS EFFECTS IN CONSERVATIVE SYSTEMS

2010 ◽  
Vol 20 (07) ◽  
pp. 2005-2043 ◽  
Author(s):  
G. CONTOPOULOS ◽  
M. HARSOULA
Keyword(s):  
Periodic Orbits ◽  
Unstable Periodic Orbits ◽  
Chaotic Orbits ◽  
Asymptotic Curves ◽  
Abrupt Changes ◽  
Long Time ◽  
New Type ◽  
Barred Spiral Galaxies ◽  
Sticky Chaotic Orbits ◽  
Stickiness Effects

Stickiness refers to chaotic orbits that stay in a particular region for a long time before escaping. For example, stickiness appears near the borders of an island of stability in the phase space of a 2-D dynamical system. This is pronounced when the KAM tori surrounding the island are destroyed and become cantori (see [Contopoulos, 2002]). We find the time scale of stickiness along the unstable asymptotic curves of unstable periodic orbits around an island of stability, that depends on several factors: (a) the largest eigenvalue |λ| of the asymptotic curve. If λ > 0 the orbits on the unstable asymptotic manifold in one direction (fast direction) escape faster than the orbits in the opposite direction (slow direction) (b) the distance from the last KAM curve or from the main cantorus (the cantorus with the smallest gaps) (c) the size of the gaps of the main cantorus and (d) the other cantori, islands and asymptotic curves. The most important factor is the size of the gaps of the main cantorus. Then we find when the various KAM curves are destroyed. The distance of the last KAM curve from the center of an island gives the size of the island. When the central periodic orbit becomes unstable, chaos is also formed around it, limited by a first KAM curve. Between the first and the last KAM curves there are still closed invariant curves. The sizes of the islands as functions of the perturbation, have abrupt changes at resonances. These functions have some universal features but also some differences. A new type of stickiness appears near the unstable asymptotic curves of unstable periodic orbits that extend far into the large chaotic sea. Such a stickiness lasts for long times, increasing the density of points close to the unstable asymptotic curves. However after a much longer time, the density becomes almost equal everywhere outside the islands of stability. We consider also stickiness near the asymptotic curves from new periodic orbits, and stickiness in Anosov systems and near totally unstable orbits. In systems that allow escapes to infinity the stickiness delays the escapes. An important astrophysical application is the case of barred-spiral galaxies. The spiral arms outside corotation consist mainly of sticky chaotic orbits. Stickiness keeps the spiral forms for times longer than a Hubble time, but after a much longer time most of the chaotic orbits escape to infinity.

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.

Periodic orbits and chaos around two black holes

1990 ◽  
Vol 431 (1881) ◽  
pp. 183-202 ◽  
Keyword(s):  
Black Holes ◽  
Periodic Orbits ◽  
Initial Conditions ◽  
Classical Problem ◽  
Period Doubling ◽  
Satellite Orbits ◽  
Unstable Periodic Orbits ◽  
Newtonian Case ◽  
Stable Periodic ◽  
Escaping Orbits

We calculate orbits of photons and particles in the relativistic problem of two extreme Reissner-Nordström black holes (fixed). In the case of photons there are three types of non-periodic orbits, namely orbits falling into the black holes M 1 and M 2 (types I and II), and orbits escaping to infinity (III). The various types of orbits are separated by orbits asymptotic to the three main types of (unstable) periodic orbits: ( a ) around M 1 , ( b ) around M 2 and ( c ) around both M 1 and M 2 . Between two non-periodic orbits of two different types there are orbits of the third type. The initial conditions of the three types of orbits form three Cantor sets, and this fact is a manifestation of chaos. The role of higher-order periodic orbits is explained. In the case of particles the situation is similar in the parabolic and hyperbolic cases. However, in the elliptic case the orbits are contained inside a curve of zero velocity, and there are no escaping orbits. Instead we have orbits trapped around stable periodic orbits (IV) and stochastic orbits not falling on the black holes M 1 and M 2 (V). The stable periodic orbits become unstable by period doubling, and infinite period doublings lead to chaos. The newtonian limit is an integrable problem (essentially the same as the classical problem of two fixed centres). The periodic orbits of type ( c ) are topologically similar to the corresponding relativistic orbits, but they differ considerably numerically. We prove that in the newtonian case there are no satellite orbits around M 1 or M 2 (of type ( a ) or ( b )). The post-newtonian case is non-integrable. In this case there are in general orbits of all three types ( a ), ( b ) and ( c ).

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.

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.

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