M. Harsoula

Orbital structure in N‐body models of barred‐spiral galaxies

We study the orbital structure in a series of self-consistent

STICKINESS IN CHAOS

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Asymptotic orbits in barred spiral galaxies

-body configurations simulating rotating barred galaxies with spiral and ring structures. We perform frequency analysis in order to measure the angular and the radial frequencies of the orbits at two different time snapshots during the evolution of each

STICKINESS EFFECTS IN CONSERVATIVE SYSTEMS

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Stability and instability in the anisotropic Kepler problem

-body system. The analysis is done separately for the regular and the chaotic orbits. We thereby identify the various types of orbits, determine the shape and percentages of the orbits supporting the bar and the ring/spiral structures, and study how the latter quantities change during the secular evolution of each system. Although the frequency maps of the chaotic orbits are scattered, we can still identify concentrations around resonances. We give the distributions of frequencies of the most important populations of orbits. We explore the phase space structure of each system using projections of the 4D surfaces of section. These are obtained via the numerical integration of the orbits of test particles, but also of the real

RECURRENCE OF ORDER IN CHAOS

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DIFFUSION OF CHAOTIC ORBITS IN BARRED SPIRAL GALAXIES

-body particles. We thus identify which domains of the phase space are preferred and which are avoided by the real particles. The chaotic orbits are found to play a major role in supporting the shape of the outer envelope of the bar as well as the rings and the spiral arms formed outside corotation.

Chaos in the case of two fixed black holes

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 O1 and O′1, and the asymptotic curves U+U-S+S- from the simplest unstable orbit around the island O1. This is the orbit 4/9 that has 9 points around the island O1 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 O1 and make many oscillations before reaching the chaotic sea. The curves U-, S- are further away from the island O1 and escape faster. Nevertheless all curves return many times close to O1 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 = limt → ∞ 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 O1. 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 O1) approach a point of the orbit 4/9, say P1, 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 P1 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.

Convergence regions of the Moser normal forms and the structure of chaos

We study the formation of the spiral structure of barred spiral galaxies, using an N-body model. The evolution of this N-body model in the adiabatic approximation maintains a strong spiral pattern for more than 10 bar rotations. We find that this longevity of the spiral arms is mainly due to the phenomenon of stickiness of chaotic orbits close to the unstable asymptotic manifolds originated from the main unstable periodic orbits, both inside and outside corotation. The stickiness along the manifolds corresponding to different energy levels supports parts of the spiral structure. The loci of the disc velocity minima (where the particles spend most of their time, in the configuration space) reveal the density maxima and therefore the main morphological structures of the system. We study the relation of these loci with those of the apocentres and pericentres at different energy levels. The diffusion of the sticky chaotic orbits outwards is slow and depends on the initial conditions and the corresponding Jacobi constant.

Systems with Escapes

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.