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 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.

scholarly journals STICKINESS IN CHAOS

2008 ◽  
Vol 18 (10) ◽  
pp. 2929-2949 ◽  
Author(s):  
G. CONTOPOULOS ◽  
M. HARSOULA
Keyword(s):  
Degrees Of Freedom ◽  
Outer Boundary ◽  
Characteristic Number ◽  
Heteroclinic Orbits ◽  
Escape Time ◽  
Unstable Periodic Orbits ◽  
Time Direction ◽  
Asymptotic Curves ◽  
Long Time ◽  
Stickiness Effects

We distinguish two types of stickiness in systems of two degrees of freedom: (a) stickiness around an island of stability, and (b) stickiness in chaos, along the unstable asymptotic curves of unstable periodic orbits. In fact, there are asymptotic curves of unstable orbits near the outer boundary of an island that remain close to the island for some time, and then extend to large distances into the surrounding chaotic sea. But later the asymptotic curves return close to the island and contribute to the overall stickiness that produces dark regions around the islands and dark lines extending far from the islands. We have studied these effects in the standard map with a rather large nonlinearity K = 5, and we emphasized the role of the asymptotic curves U , S from the central orbit O (x = 0.5, y = 0), that surround two large islands O 1 and O ′1, and the asymptotic curves U + U - S + S - from the simplest unstable orbit around the island O 1. This is the orbit 4/9 that has 9 points around the island O 1 and 9 more points around the symmetric island O ′1. The asymptotic curves produce stickiness in the positive time direction ( U , U +, U -) and in the negative time direction ( S , S +, S -). The asymptotic curves U +, S + are closer to the island O 1 and make many oscillations before reaching the chaotic sea. The curves U -, S - are further away from the island O 1 and escape faster. Nevertheless all curves return many times close to O 1 and contribute to the stickiness near this island. The overall stickiness effects of U +, U - are very similar and the stickiness effects along S +, S - are also very similar. However, the stickiness in the forward time direction, along U +, U -, is very different from the stickiness in the opposite time direction along S +, S -. We calculated the finite time LCN (Lyapunov characteristic number) χ( t ), which is initially smaller for U +, S + than for U -, S -. However, after a long time all the values of χ( t ) in the chaotic zone approach the same final value LCN = lim t → ∞ χ(t). The stretching number (LCN for one iteration only) varies along an asymptotic curve going through minima at the turning points of the asymptotic curve. We calculated the escape times (initial stickiness times) for many initial points outside but close to the island O 1. The lines that separate the regions of the fast from the slow escape time follow the shape of the asymptotic curves S +, S -. We explained this phenomenon by noting that lines close to S + on its inner side (closer to O 1) approach a point of the orbit 4/9, say P 1, and then follow the oscillations of the asymptotic curve U +, and escape after a rather long time, while the curves outside S + after their approach to P 1 follow the shape of the asymptotic curves U - and escape fast into the chaotic sea. All these curves return near the original arcs of U +, U - and contribute to the overall stickiness close to U +, U -. The isodensity curves follow the shape of the curves U +, U - and the maxima of density are along U +, U -. For a rather long time, the stickiness effects along U +, U - are very pronounced. However, after much longer times (about 1000 iterations) the overall stickiness effects are reduced and the distribution of points in the chaotic sea outside the islands tends to be uniform. The stickiness along the asymptotic curve U of the orbit O is very similar to the stickiness along the asymptotic curves U +, U - of the orbit 4/9. This is related to the fact that the asymptotic curves of O and 4/9 are connected by heteroclinic orbits. However, the main reason for this similarity is the fact that the asymptotic curves U , U +, U - cannot intersect but follow each other.

scholarly journals THE STRUCTURE OF INVARIANT TORI IN A 3D GALACTIC POTENTIAL

2011 ◽  
Vol 21 (02) ◽  
pp. 467-496 ◽  
Author(s):  
M. KATSANIKAS ◽  
P. A. PATSIS
Keyword(s):  
Phase Space ◽  
Periodic Orbits ◽  
Invariant Tori ◽  
Galactic Dynamics ◽  
Chaotic Orbits ◽  
Rotation Method ◽  
Galactic Potential ◽  
Long Time ◽  
Stable Periodic ◽  
Sticky Chaotic Orbits

We study in detail the structure of phase space in the neighborhood of stable periodic orbits in a rotating 3D potential of galactic type. Using the color and rotation method to investigate the properties of the invariant tori in 4D spaces of the section, we compare our results with those of previous works and we describe the morphology of the rotational, as well as of the tube tori in 4D space. We find sticky chaotic orbits in the immediate neighborhood of sets of invariant tori surrounding 3D stable periodic orbits. Particularly useful for galactic dynamics is the behavior of chaotic orbits trapped for long time between 4D invariant tori. We find that they support during this time the same structure as the quasi-periodic orbits around the stable periodic orbits, contributing however to a local increase of the dispersion of velocities. Finally, we find that the tube tori do not appear in the 3D projections of the spaces of the section in the axisymmetric Hamiltonian we examined.

Synchronization of chaotic orbits: The influence of unstable periodic orbits

1993 ◽  
Vol 48 (3) ◽  
pp. R1620-R1623 ◽  
Author(s):  
Neelima Gupte ◽  
R. E. Amritkar
Keyword(s):  
Periodic Orbits ◽  
Unstable Periodic Orbits ◽  
Chaotic Orbits

scholarly journals Time averaged properties along unstable periodic orbits and chaotic orbits in two map systems

2008 ◽  
Vol 15 (4) ◽  
pp. 675-680 ◽  
Author(s):  
Y. Saiki ◽  
M. Yamada
Keyword(s):  
Fluid Dynamics ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Space Physics ◽  
Unstable Periodic Orbits ◽  
Unstable Periodic Orbit ◽  
Geophysical Fluid Dynamics ◽  
Chaotic Orbits ◽  
Low Dimensional ◽  
Complex Phenomena

Abstract. Unstable periodic orbit (UPO) recently has become a keyword in analyzing complex phenomena in geophysical fluid dynamics and space physics. In this paper, sets of UPOs in low dimensional maps are theoretically or systematically found, and time averaged properties along UPOs are studied, in relation to those of chaotic orbits.

scholarly journals Outer spiral structure in disk galaxies

2016 ◽  
Vol 11 (S321) ◽  
pp. 123-123
Author(s):  
P.A. Patsis
Keyword(s):  
Periodic Orbits ◽  
Spiral Galaxies ◽  
Spiral Structure ◽  
Disk Galaxies ◽  
Spiral Arms ◽  
Chaotic Orbits ◽  
Grand Design ◽  
Pattern Speed ◽  
Spiral Structures ◽  
Barred Spiral Galaxies

AbstractIn several grand design barred-spiral galaxies it is observed a second, fainter, outer set of spiral arms. Typical examples of objects of this morphology can be considered NGC 1566 and NGC 5248. I suggest that such an overall structure can be the result of two dynamical mechanisms acting in the disc. The bar and both spiral systems rotate with the same pattern speed. The inner spiral is reinforced by regular orbits trapped around the stable, elliptical, periodic orbits of the central family, while the outer system of spiral arms is supported by chaotic orbits. Chaotic orbits are also responsible for a rhomboidal area surrounding the inner barred-spiral region. In general there is a discontinuity between the two spiral structures at the corotation region.

DIFFUSION OF CHAOTIC ORBITS IN BARRED SPIRAL GALAXIES

2011 ◽  
Vol 21 (08) ◽  
pp. 2221-2233 ◽  
Author(s):  
M. HARSOULA ◽  
C. KALAPOTHARAKOS ◽  
G. CONTOPOULOS
Keyword(s):  
Spiral Structure ◽  
Initial Conditions ◽  
Unstable Periodic Orbits ◽  
Chaotic Orbits ◽  
The Galaxy ◽  
Barred Spiral Galaxy ◽  
Age Of The Universe ◽  
Barred Spiral Galaxies ◽  
The Universe ◽  
Asymptotic Manifolds

We study the diffusion of chaotic orbits in an N-body model simulating a barred spiral galaxy. Chaotic orbits with initial conditions outside corotation support the spiral structure of the galaxy due to the phenomenon of stickiness close and along the unstable asymptotic manifolds of the unstable periodic orbits. These orbits are diffused outwards after about 13 rotations of the bar. During this time, the spiral structure is clearly visible and then it fades out gradually. The diffusion time for the majority of the chaotic orbits with initial conditions inside corotation is much longer than the age of the Universe. These orbits support mainly the outer parts of the bar. However, a part of the chaotic orbits inside corotation are diffused outwards fast and support the spiral structure.

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.

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