G. Contopoulos

Order and chaos

We review integrable and chaotic systems and the transition from Order to Chaos. We give the various types of orbits in Stiickel potentials of 2 and 3 degrees of freedom. Then we discuss integrable systems in general. In particular we consider Li ouville systems and the Painleve test of integrability. Formal integrals (like the “third” integral) are an important tool in galactic dynamics. We discuss their construction, and the limits of their applicability by means of Nehoroshevs theory. Then we enumerate the various routes to chaos in dissipative and conservative systems. The creation of large degree of chaos follows the destruction of the last KAM (Kolmogorov, Arnold, Moser) torus and the formation of cantori. We discuss -the structure of the chaotic regions, that are filled with the asymptotic curves of the unstable periodic orbits. In systems of 3 degrees of freedom we have 3 main new phenomena: Complex instability, Collisions of bifurcations, and Arnold diffusion. Finally we discuss applications of the theory of fractals on the forms of the orbits.

Invariant spectra of orbits in dynamical systems

We show that in deterministic dynamical systems any orbit is associated with an invariant spectrum of stretching numbers, i.e. numbers expressing the logarithmic divergences of neighbouring orbits within one period. The first moment of this invariant spectrum is the maximal Lyapunov characteristic number (LCN). In the case of a chaotic domain, a single invariant spectrum characterizes the whole domain. The invariance of this spectrum allows the estimation of the LCN by calculating, for short times, many orbits with initial conditions in the same chaotic region instead of calculating one orbit for extremely long times. However, if part of the initial conditions are in an ordered region, the average of the short-time calculations may deviate considerably from the LCN. Invariant spectra appear not only for conservative but also for dissipative systems. A few examples are given.

Fractal properties of escape from a two-dimensional potential

Abstract This paper summarizes a numerical investigation of the escape of particles from the two-dimensional potential V(x, y) = 1 2 x 2 + 1 2 y 2 - ϵx 2 y 2 , for variableϵ but fixed energy h = 0.12. For ϵ 1 ≡ 1 (4h) ≈ 2.08 , escape is impossible, but for ϵ≥ϵ1, particles will escape. For such larger ϵ, the final outcome, as characterized by the time and direction in which particles escape, is a smooth function of initial conditions in certain phase space regions. However, in other regions the evolution evidences an extremely sensitive dependence on initial conditions indicative of a complex microscopic evolution. For sufficiently large values of ϵ>ϵ2≈4.90±0.01, these sensitive regions appear to exhibit a fractal structure with simple scaling properties implying that, at a coarse-grained level, the late time behavior is quite regular. Specifically, the evolution is asymptotically Markovian in the sense that, for particles that have not yet escaped, at late times the coarse-grained escape probabilities per unit time tend toward time-independent values p∞ > 0 independent of the initial conditions. The asymptotic values p∞ and the rate of convergence towards these values were examined as a function of ϵ and the scale of the coarse-graining, and were observed to exhibit simple scaling behaviour. At least for values of ϵ≤5.7, p∞∞(ϵ−ϵ2)α, with a critical exponent α≈0.49±0.05. For a coarse-graining corresponding to a linear scale r=0.05, the time T required for convergence towards this value scales as T(ϵ)∞(ϵ−ϵ2-β, with β≈0.39+0.14-0.06. The difference γ≡α-β=0. 09+0.04-0.02. The value p∞ appears independent of r, but the convergence time scales as T(r)∞r-δ, with δ=0.08±0.03, seemingly independent of ϵ. To within statistical uncertainties, α − β − δ ≈0.

How to Observe a Non-Kerr Spacetime Using Gravitational Waves

We present a generic criterion which can be used in gravitational-wave data analysis to distinguish an extreme-mass-ratio inspiral into a Kerr background spacetime from one into a non-Kerr spacetime. We exploit the fact that when an integrable system, such as the system that describes geodesic orbits in a Kerr spacetime, is perturbed, the tori in phase space which initially corresponded to resonances disintegrate so as to form Birkhoff chains on a surface of section. The KAM curves of the islands in such a chain share the same ratio of frequencies, even though the frequencies themselves vary from one KAM curve to another inside an island. However the KAM curves, which do not lie in a Birkhoff chain, do not share this characteristic property. Such a temporal constancy of the ratio of frequencies during the evolution of the gravitational-wave signal will signal a non-Kerr spacetime.

Observable signature of a background deviating from the Kerr metric

By detecting gravitational wave signals from extreme mass ratio inspiraling sources (EMRIs) we will be given the opportunity to check our theoretical expectations regarding the nature of supermassive bodies that inhabit the central regions of galaxies. We have explored some qualitatively new features that a perturbed Kerr metric induces in its geodesic orbits. Since a generic perturbed Kerr metric does not possess all the special symmetries of a Kerr metric, the geodesic equations in the former case are described by a slightly nonintegrable Hamiltonian system. According to the

Transition spectra of dynamical systems

The spectra of ‘stretching numbers’ (or ‘local Lyapunov characteristic numbers’) are different in the ordered and in the chaotic domain. We follow the variation of the spectrum as we move from the centre of an island outwards until we reach the chaotic domain. As we move outwards the number of abrupt maxima in the spectrum increases. These maxima correspond to maxima or minima in the curve a(θ), where a is the stretching number, and θ the azimuthal angle. We explain the appearance of new maxima in the spectra of ordered orbits. The orbits just outside the last KAM curve are confined close to this curve for a long time (stickiness time) because of the existence of cantori surrounding the island, but eventually escape to the large chaotic domain further outside. The spectra of sticky orbits resemble those of the ordered orbits just inside the last KAM curve, but later these spectra tend to the invariant spectrum of the chaotic domain. The sticky spectra are invariant during the stickiness time. The stickiness time increases exponentially as we approach an island of stability, but very close to an island the increase is super exponential. The stickiness time varies substantially for nearby orbits; thus we define a probability of escape Pn(x) at time n for every point x. Only the average escape time in a not very small interval Δx around each x is reliable. Then we study the convergence of the spectra to the final, invariant spectrum. We define the number of iterations, N, needed to approach the final spectrum within a given accuracy. In the regular domain N is small, while in the chaotic domain it is large. In some ordered cases the convergence is anomalously slow. In these cases the maximum value of ak in the continued fraction expansion of the rotation number a = [a0,a1,... ak,...] is large. The ordered domain contains small higher order chaotic domains and higher order islands. These can be located by calculating orbits starting at various points along a line parallel to the q-axis. A monotonic variation of the sup {q}as a function of the initial condition q0 indicates ordered motions, a jump indicates the crossing of a localized chaotic domain, and a V-shaped structure indicates the crossing of an island. But sometimes the V-shaped structure disappears if the orbit is calculated over longer times. This is due to a near resonance of the rotation number, that is not followed by stable islands.

STICKINESS IN CHAOS

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.

Stickiness and cantori

We study the phenomenon of stickiness in the standard map. The sticky regions are limited by cantori. Most important among them are the cantori with noble rotation numbers, that are approached by periodic orbits corresponding to the successive truncations of the noble numbers. The size of an island of stability depends on the last KAM torus. As the perturbation increases, the size of the KAM curves increases. But the outer KAM curves are gradually destroyed and in general the island decreases. Higher-order noble tori inside the outermost KAM torus are also destroyed and when the outermost KAM torus becomes a cantorus, the size of an island decreases abruptly. Then we study the crossing of the cantori by asymptotic curves of periodic orbits just inside the cantorus. We give an exact numerical example of this crossing (non-schematic) and we find how the asymptotic curves, after staying for a long time near the cantorus, finally extend to large distances outwards. Finally, we find the relation between the forms of the sticky region and asymptotic curves.

Painleve analysis for the mixmaster universe model

We show that the mixmaster universe, or Bianchi IX, model passes the Painleve test in the form of the Ablowitz-Ramani-Segur algorithm, i.e. the solutions of the equations of motion do not have movable critical points. Thus this system is probably integrable and therefore non-chaotic.

Comparison of stellar and gasdynamics of a barred galaxy

The stellar and gas dynamics of several models of barred galaxies were studied, and results for some representative cases are reported for galaxies in which the stars and gas respond to the same potentials. Inside corotation there are two main families of periodic orbits, designated x1 and 4/1. Close to the center, the x1 orbits are like elongated ellipses. As the 4/1 resonance is approached, these orbits become like lozenges, with apices along the bar and perpendicular to it. The family 4/1 consists of orbits like parallelograms which produce the boxy component of the bar. The orbits in spirals outside corotation enhance the spiral between the outer -4/1 resonance and the outer Lindblad resonance. Between corotation and the -4/1 resonance in strong spirals, the orbits are mostly stochastic and fill almost circular rings. A spiral field must be added to gasdynamical models to obtain gaseous arms extending from the end of a bar. 38 refs.