Astrophysics: Dynamical Systems

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Definition of the Subject

Dynamical systems is a broad subject, where research has been very active in recent years. It deals with systems that change in time according to particular laws. Examples of such systems appear in astronomy and astrophysics, in cosmology, in various branches of physics and chemistry, and in all kinds of other applications, like meteorology, geodynamics, electronics, and biology. In this entry, we deal with the mathematical theory of dynamical systems from the point of view of dynamical astronomy. This theory is generic in the sense that its mathematics can be applied to diverse problems of physics and related sciences, ranging from elementary particles to cosmology.

Introduction

The theory of dynamical systems has close relations with astronomy and astrophysics, in particular with dynamical astronomy. Dynamical astronomy followed two quite different traditions until the middle of the twentieth century, namely, celestial mechanics and statistical mechanics.

Celestial mechanics was the prototype of order. The use of perturbation theories in the solar system was quite effective for predicting the motion of the planets, satellites, and comets. Despite the fact that the perturbation series were in most cases extremely long, their effectiveness was never doubted. One of the most painstaking developments in this field was the “Theory of the Motion of the Moon” of Delaunay (1867) and his successors. Today, the theory of the Moon has been extended to an unprecedented accuracy by computer algebra, giving the position of the Moon with an accuracy of a few centimeters (Chapront-Touzé and and Henrard 1980; Deprit et al. 1971).

Celestial mechanics had proved its effectiveness already by the nineteenth century with the discovery of the planet Neptune by Leverrier and Adams. Its subsequent development reached its culmination in the work of Poincaré: “Méthodes Nouvelles de la Mécanique Céleste” (Poincaré 1892). But at the same time, it reached its limits. Poincaré proved that most of the series of celestial mechanics are divergent. Therefore, their accuracy is limited. On the other hand, it was made clear that the most important problem of celestial mechanics, the N-body problem, cannot be solved analytically. The work of Poincaré contains the basic ideas of most of the recent theories of dynamical systems, in particular what we call today the “Theory of Chaos.”

A completely different approach of the theory of dynamical systems was based on statistical mechanics. This approach dealt with the N-body problem, but from a quite different point of view. Instead of a dominant central body (the Sun) and many small bodies (planets), it dealt with N-bodies of equal masses (or of masses of the same order) with large N.

Statistical mechanics, developed by Boltzmann, Gibbs, and others, dominated physics in general. In this approach, the notion of individual orbits of particles became not only secondary but even irrelevant. One might consider different types of orbits, but their statistical properties should be the same.

Nevertheless, a basic problem was present in the foundations of statistical mechanics. If the N-body systems are deterministic, how is it possible to derive the random properties of the ensembles of statistical mechanics? This question led to the development of a new theoretical approach that is called “ergodic” theory. If almost all orbits are ergodic, then the random behavior of the system can be proved, and not simply assumed. As an example, consider a room of gas of constant density. The constancy of the density may be considered as a probabilistic effect. But if the orbits of the molecules are ergodic, all particles stay on the average equal intervals of time in equal volumes inside this room. Thus, the constancy of density is due to the properties of the orbits of all particles.

In dynamical astronomy, there was a complete dichotomy between celestial mechanics and stellar dynamics. Celestial mechanics continued its work with perturbation series, and no chaos at all, while the new branch of stellar dynamics (and galactic dynamics in particular) was completely influenced by statistical considerations, as in statistical mechanics. In particular, one dealt with distribution functions of stellar velocities that were well described by a velocity ellipsoid. But it was realized that the time required for the establishment of a statistical equilibrium was much longer than the age of the universe. Thus, the stars had not enough time to settle into a statistical equilibrium. These problems led to a reconsideration of the foundations of galactic dynamics, namely, to the study of the individual orbits of the stars. Such studies started only in the 1960s. The first time when celestial mechanicians and galactic astronomers met at a common meeting was during the 1964 Thessaloniki Symposium on the “Theory of Orbits in the Solar System and in Stellar Systems” (Contopoulos 1966). At that time, a completely new tool was available in dynamical astronomy, namely, the use of computers.

The first computer experiment with an N-body system was the so-called Fermi-Pasta-Ulam paradox, in 1955. These authors considered particles along a line attracted by linear forces, plus a small nonlinear force. The energy was initially given to a few modes. The energy of each mode cannot change if the forces are only linear. But because of the nonlinearities, the energies of all the modes of the system were expected to change continually, in a way consistent with a statistical mechanics prediction. Instead of that, the computer results have shown that the energy changes were almost periodic and no tendency to a final statistical situation ever appeared. This was a paradox, a great surprise for people grown up with statistical mechanics.

A little later (1956), a computer was used by Per-Olof Lindblad to calculate orbits of stars in a plane galaxy, in order to find the formation of spiral arms (Lindblad 1960).

At about the same time, we calculated (Contopoulos 1958) two orbits of stars in three dimensions. Based on the prevalent assumptions of statistical mechanics, we expected these orbits to be ergodic and fill all the space inside the energy surface. Instead, we found that the orbits did not fill all the available space, but filled only curvilinear parallelograms, like deformed Lissajous figures (Fig. 1). Later it was shown that such orbits can be explained, qualitatively and quantitatively, by a formal third integral of motion (Contopoulos 1960).

Fig. 1

The first published orbits (1958) in the meridian plane of an axisymmetric galaxy inside the curve of zero velocity (CZV = equipotential)

The theory of the third integral goes back to the work of Birkhoff (1927) and Whittaker (1916, 1937). In particular, Whittaker found the so-called adelphic integral, i.e., an integral similar to the energy in simple dynamical systems. In the case of two harmonic oscillators, this integral is reduced to the energy of one oscillator only. In more general cases, higher-order terms have to be added to find a constant of motion. However, in resonant cases, the form of the third integral may be quite different. For example, in the case of two equal frequencies, it is a generalization of the angular momentum.

The series giving the third integral are in general divergent (Siegel 1956). But even so in numerical applications, the third integral is better conserved if it is truncated at higher and higher orders, up to a certain maximum order. The usefulness of these series was emphasized by Moser (1968).

The most important result of these studies was that, in general, dynamical systems are neither integrable nor ergodic. This was a surprise, because it was generally assumed that generic dynamical systems are either integrable or ergodic (Landau and Lifshitz 1960). This change of paradigm was emphasized by Lynden Bell (1998).

The existence of ordered domains in generic dynamical systems is the basic content of the famous Kolmogorov-Arnold-Moser (KAM) theorem. Kolmogorov (1954) announced a theorem proving the existence of invariant tori in dynamical systems. Such tori contain quasiperiodic motions with frequencies whose ratios are far from all rationals (see section “Formal Integrals: KAM and Nekhoroshev Theory”).

The details of the proof were given by Arnold (1961, 1964) in the analytical case and independently by Moser (1962, 1967) in sufficiently differentiable cases (with 333 derivatives!). More recently, the existence of such tori was proven for systems that are differentiable only a few times. Furthermore, invariant tori appear generically near stable periodic orbits. Such orbits appear even in systems with arbitrarily large perturbations. Thus, complete lack of ordered regions containing invariant tori is rather exceptional.

Classification

There are two main types of dynamical systems:

• 1. Maps (or mappings) of the form ${\overrightarrow{x}}_{n+1}=\overrightarrow{f}\left({\overrightarrow{x}}_n\right)$, where ${\overrightarrow{x}}_n$ is a vector of N-dimensions and $\overrightarrow{f}$ is a set of N-functions

• 2. Systems of differential equations of the form $\overrightarrow{x}=\overrightarrow{f}\left(\overrightarrow{x},t\right)$, where $\overrightarrow{x}$ is an N-dimensional vector, and the dot denotes derivative with respect to a continuous time t

Another separation of dynamical systems, both of maps and differential equations, is in conservative and dissipative systems. Conservative systems preserve the volume in phase space, while in dissipative systems, the volume decreases on the average. If we reverse the time direction, we have systems with increasing volume.

A large class of systems of differential equations are the Hamiltonian systems, with conjugate variables $\overrightarrow{x}\left({x}_1,{x}_2,\dots, {x}_N\right)$ and $\overrightarrow{y}\left({y}_1,{y}_2,\dots, {y}_N\right)$ that satisfy equations of the form

$$ \overrightarrow{\overset{\cdotp }{x}}=\frac{\partial H}{\partial \overrightarrow{y}},\overrightarrow{\overset{\cdotp }{y}}=-\frac{\partial H}{\partial \overrightarrow{x}} $$

(1)

where H is the Hamiltonian function $H=H\left(\overrightarrow{x},\overrightarrow{y},t\right)$ and N is the number of degrees of freedom. The equations of motion (Eq. 1) are called canonical or Hamiltonian equations. The space of the variables $\overrightarrow{x}$ and $\overrightarrow{y}$ is called phase space. The phase space of a map and of a system of differential equations is N-dimensional, while the phase space of a Hamiltonian is 2N-dimensional. A change of variables $\left(\overrightarrow{x},\overrightarrow{y}\right)\to \left({\overrightarrow{x}}^{\mathit{\prime}},{\overrightarrow{y}}^{\mathit{\prime}}\right)$ is called canonical if the equations of motion in the new variables are also canonical.

The most important separation of dynamical systems is between integrable systems and chaotic systems.

The most simple integrable systems are the solvable systems, i.e., systems that can be solved explicitly, to give the variables as functions of time. This definition is too restricted, if by functions we mean known functions of time. A more general definition is in terms of single-valued functions of time (Wintner 1947), even if such functions can be given only numerically.

A more general definition of integrable systems is systems that have N-independent analytical integrals of motion in involution ${I}_i\left(\overrightarrow{x},t\right)={I}_i\left({\overrightarrow{x}}_0,{t}_0\right)$, i = 1,2,…,N. (By involution, we mean that any two such integrals have a zero Poisson bracket (see Eq. 6)).

As regards chaotic systems, their definition and classification are difficult tasks. The basic property of chaos is sensitive dependence on initial conditions. Namely, two orbits of a compact system starting very close to each other deviate considerably later on (exponentially fast in time).

A particular class of chaotic systems are the ergodic systems, in which the orbits go everywhere on a surface of constant energy. Namely, the orbits pass through the neighborhood of every point of the energy surface. In an ergodic system, the time average of a function is equal to its phase average, according to Birkhoff’s theorem (1931). An ergodic system is called “mixing” if two nearby particles on different (ergodic) orbits can go very far from each other.

A further distinction of mixing systems is related to the speed of deviation of nearby orbits. If the deviation is exponential in time, the system is called “Kolmogorov” or K-system. The deviation $\overrightarrow{\xi}$ of nearby orbits that are initially (at time t ₀) at an infinitesimal distance ${\overrightarrow{\xi}}_0$ is measured by the Lyapunov characteristic number (LCN) (or simply the Lyapunov exponent).

For almost all deviations ${\overrightarrow{\xi}}_0$ from the initial condition ${\overrightarrow{x}}_0$ of an orbit, the Lyapunov characteristic number is the same, equal to

$$ \mathrm{LCN}=\underset{t\to \infty }{ \lim}\frac{ \ln \Big|\overrightarrow{\xi}/{\overrightarrow{\xi}}_0\Big|}{t} $$

(2)

In a system of N degrees of freedom, one can define N Lyapunov characteristic numbers. If not specified explicitly otherwise, the term Lyapunov characteristic number means the maximal LCN. A system is said to have sensitive dependence on the initial conditions if its LCN is positive.

A simple system with positive LCN is given by ξ = ξ ₀ exp(qt). In this system, ln(ξ/ξ ₀) = qt; therefore, LCN = q (constant). A system in which the deviation ξ increases linearly in time (or as a power of time) is not Kolmogorov although it is mixing. For example, if ξ = ξ ₀ + αt, we have ln(ξ/ξ ₀) ≈ ln t, and the limit LCN = lim _t→∞ (ln t/t) is zero.

If in a Kolmogorov system the LCN for all orbits is between certain positive limits, i.e., if C ₁≤LCN≤C ₂, the system is called “Anosov” or C-system.

A C-system has infinite periodic orbits, but none of them is stable. Furthermore, the Anosov systems are hyperbolic, i.e., the stable and unstable manifolds of each unstable periodic orbit intersect transversally. The Anosov systems are structurally stable, i.e., a small perturbation of an Anosov system leads to another Anosov system.

The usual classification (Lichtenberg and and Lieberman 1992) of dynamical systems is the following: integrable systems, ergodic systems, mixing systems, Kolmogorov systems, and Anosov systems.

This classification represents the view that if a system is not integrable, it is at least ergodic. The first edition of the book of Landau and Lifshitz (1960) separated all dynamical systems into two classes, integrable and ergodic. (However, after the 3rd edition, this part was omitted.) This classification misses the most important property of chaos, the fact that in general chaos coexists with order in the same system.

In fact, in most systems that have been studied numerically up to now, one finds both chaotic and ordered orbits. Such systems are called systems with divided phase space. Chaotic orbits have a positive Lyapunov characteristic number, while ordered orbits have LCN = 0. Both chaotic and ordered orbits cover parts of the phase space. This is best seen in numerical studies of systems of two degrees of freedom. The ordered and chaotic domains are intricately mixed. However, there are regions where order is predominant, and other regions where chaos is predominant.

Systems of two degrees of freedom that are close to integrable have only small regions of chaos and most orbits are ordered. On the other hand, systems that are far from integrable may seem completely chaotic, but after a closer look, we usually find small islands of stability in them.

Completely integrable systems are exceptional. On the other hand, truly ergodic systems are also exceptional. For example, some piecewise linear maps, which are given modulo a constant, or systems with abrupt reflections, like the stadium, have been proven to be ergodic. But if we add generic nonlinear terms in these maps, the structure of phase space changes and islands of stability appear.

The present-day classification of dynamical systems is as shown below:

	Ordered	Chaotic		Random
Compact	Integrable	(General case) systems with divided phase space	(Limiting cases) ergodic, mixing, Kolmogorov and Anosov
Noncompact	Integrable with escapes	Nonintegrable with escapes	–

A particular class is the noncompact systems (i.e., systems with escapes). In such systems, some initial conditions lead to escapes, while others do not. These systems are either integrable or chaotic. In the latter case, chaos appears in the form of chaotic scattering. But properly speaking, chaos refers only to compact systems. Finally, the random systems are limiting cases of Anosov systems, because they have infinitely large Lyapunov characteristic numbers.

Integrable Systems

The simplest integrable systems can be solved explicitly. Autonomous Hamiltonian equations of the form Eq. 1 with one degree of freedom (N = 1) are always solvable.

Very useful sets of variables are the action-angle variables $\left(\overrightarrow{J},\overrightarrow{\theta}\right)$. In particular, the change of variables

$$ {x}_i=\sqrt{2{J}_i} \sin {\theta}_i,\kern0.5em {y}_i=\sqrt{2{J}_i} \cos {\theta}_i,\kern0.5em \left(i=1,2\dots N\right) $$

(3)

is canonical and the new variables $\left(\overrightarrow{J},\overrightarrow{\theta}\right)$ satisfy canonical equations of the form

$$ \overset{\cdotp }{\theta }=\frac{\partial H}{\partial J},\kern0.5em \overset{\cdotp }{J}=-\frac{\partial H}{\partial \theta } $$

(4)

In the particular case that H is a function of the actions only $H=H\left(\overrightarrow{J}\right)$, the actions $\overrightarrow{J}\left({J}_1,{J}_2,\dots {J}_N\right)$ are integrals of motion, because ${\overset{\cdotp }{J}}_i=0$, and consequently, the derivatives of H with respect to J _i are constant, i.e., $\partial H/\partial \overrightarrow{J}=\overrightarrow{\omega}\left(\overrightarrow{J}\right)$ where $\overrightarrow{\omega}$ represents the frequencies (ω ₁ ,ω ₂ ,…,ω _N ). Then the angles (θ ₁ ,θ ₂ ,…, θ _N ) are linear functions of the time θ _i = ω _i (t − t _i0). In such a case, the motion takes place on an N-dimensional torus.

A function $I\left(\overrightarrow{x},\overrightarrow{y},t\right)$ in a Hamiltonian system is called an integral of motion if it remains constant along any orbit, i.e., its total derivative is zero. Then

$$ \frac{\mathrm{d}I}{\mathrm{d}t}=\frac{\partial I}{\partial \overrightarrow{x}}\frac{\mathrm{d}\overrightarrow{x}}{\mathrm{d}t}+\frac{\partial I}{\partial \overrightarrow{y}}\frac{\mathrm{d}\overrightarrow{y}}{\mathrm{d}t}+\frac{\partial I}{\partial t}=\left[I,H\right]+\frac{\partial I}{\partial t}=0 $$

(5)

where $\overrightarrow{x}$, $\overrightarrow{y}$ are additive and

$$ \left[I,H\right]\equiv \frac{\partial I}{\partial \overrightarrow{x}}\frac{\partial H}{\partial \overrightarrow{y}}-\frac{\partial I}{\partial \overrightarrow{y}}\frac{\partial H}{\partial \overrightarrow{x}} $$

(6)

is called the Poisson bracket between the functions I and H.

An important class of integrable systems is the Stäckel potentials (Eddington 1915; Stäckel 1890; Weinacht 1924). They are given in elliptical coordinates (λ, μ) by potentials of the form

$$ V=-\frac{\left[{F}_1\left(\lambda \right)-{F}_2\left(\mu \right)\right]}{\lambda -\mu } $$

(7)

where λ and μ are the roots of the second-degree equation in τ

$$ \frac{x^2}{\tau -{a}^2}+\frac{y^2}{\tau -{b}^2}=1 $$

(8)

with Cartesian coordinates x, y, and a ² ≥ b ². Equation 8 represents confocal ellipses and hyperbolas. From every point, (x, y) passes an ellipse (defined by λ) and a hyperbola (defined by μ). Both have the same foci at $y=\overline{a}=\pm {\left({a}^2-{b}^2\right)}^{1/2}$. The orbits in the plane (x, y) fill regions between such ellipses and hyperbolas (Fig. 2).

Fig. 2

Regions filled by the various types of orbits in a Stäckel potential

The Stäckel potentials are in general nonrotating. Up to now, only one rotating Stäckel potential has been found, besides the trivial case of a homogeneous ellipsoid, namely, the potential

$$ V=-\frac{k_1}{{\left[{\left({x}^2+{y}^2\right)}^2+2{\overline{a}}^2\left({x}^2-{y}^2\right)+{\overline{a}}^4\right]}^{1/2}}+\frac{1}{2}{\Omega}_{\mathrm{s}}^2{r}^2 $$

(9)

The orbits in the rotating case are in general tubes around both foci or tubes around one focus only (Contopoulos and and Vandervoort 1992) (Fig. 3).

Fig. 3

Orbits in a rotating Stäckel potential

Stäckel potentials in three degrees of freedom were considered by many authors like (de Zeeuw 1985; Hunter 1988; Lynden-Bell 1962). There are 32 possible forms of 3-D Stäckel orbits (Contopoulos 2002).

The Stäckel potentials have been used extensively in recent years in constructing self-consistent models of elliptical galaxies (Bishop 1987; de Zeeuw et al. 1987; Hunter 1990; Statler 1987). Such models represent well real galaxies.

Formal Integrals: KAM and Nekhoroshev Theory

If a system is close to an integrable one, we may use perturbation methods to find approximate integrals that are valid for long intervals of time, and sometimes for all times. Usually, such integrals are in the form of formal power series. Formal integrals for polynomial Hamiltonians were first derived by Whittaker (1916, 1937).

Another form of such integrals was derived by Birkhoff (1927) and by Cherry (1924). Formal integrals appropriate for galactic dynamics were derived by Contopoulos (1960, 1963). In the case of an axisymmetric galaxy, an integral of this form is called a third integral (the first integral being the energy and the second the z component of the angular momentum).

Birkhoff (1927) and Cherry (1924) use Hamiltonians of the form

$$ H=i{\omega}_1{\xi}_1{\eta}_1+i{\omega}_2{\xi}_2{\eta}_2+\dots +i{\omega}_N{\xi}_N{\eta}_N+{H}_3+{H}_4+\dots $$

(10)

where H _k is of degree k in (ξ _λ,η _λ), and ξ _λ,η _λ are complex variables (λ = 1, 2, … N).

By successive canonical changes of variables, the Hamiltonian takes the normal form $\overline{H}={\overline{H}}_2+{\overline{H}}_4+\dots$ where ${\overline{H}}_k$ is a function of the products ${\Omega}_{\lambda }=\left({\overline{\xi}}_{\lambda }{\overline{\eta}}_{\lambda}\right)$ only (in the nonresonant case); thus, it is an even function of the variables. Then the quantities Ω_λ are integrals of motion, and the equations of motion can be solved explicitly.

The third integral in a galaxy refers to a Hamiltonian of the form (Contopoulos 1960)

$$ H=\frac{1}{2}{\displaystyle \sum_{\lambda =1}^2\left({y}_{\lambda}^2+{\omega}_{\lambda}^2{x}_{\lambda}^2\right)}+{H}_3+{H}_4+\dots $$

(11)

where H _k is a function of degree k in x _λ,y _λ (λ = 1,2). This Hamiltonian represents a galactic potential on a plane of symmetry. If r = r ₀ represents a circular orbit on this plane, we have x ₁ = r − r ₀, x ₂ = z and ${y}_1={\overset{\cdotp }{x}}_1$, ${y}_2={\overset{\cdotp }{x}}_2$. Then the term H ₃ of the Hamiltonian takes the form

$$ {H}_3=-\upepsilon {x}_1{x}_2^2-\frac{\upepsilon^{\prime }}{3}{x}_1^3 $$

(12)

An integral of motion has its total derivative equal to zero. The successive terms of the third integral Φ can be found from the equation

$$ \frac{\mathrm{d}\varPhi }{\mathrm{d}t}=\left[\varPhi, H\right]=0 $$

(13)

(where Φ = Φ ₂ + Φ ₃ + …, and [Φ,H] is the Poisson bracket) by equating terms of equal degree. The Hamiltonian H ₂ has two independent integrals of motion ${\varPhi}_2=\frac{1}{2}\left[{y}_1^2+{\omega}_1^2{x}_1^2\right]$ and Φ ₂′ = H ₂ − Φ ₂. Then, we find

$$ \begin{array}{l}\kern5em \underset{3}{\varPhi }=-\frac{1}{\left({\omega}_1^2-4{\omega}_2^2\right)}\\ {}\left[\left({\omega}_1^2-2{\omega}_2^2\right){x}_1{x}_2^2-2{x}_1{y}_2^2+2{y}_1{x}_2{y}_2\right]\end{array} $$

(14)

etc. The expression (Eq. 14) and the higher-order terms Φ _k of the third integral contain denominators of the form (mω ₁ − nω ₂) that become zero if ω ₁/ω ₂ = n/m (rational). Then Φ contains secular terms. For example, the form (Eq. 14) of Φ ₃ cannot be valid if ω ₁ − 2ω ₂ = 0. It is not valid also if ω ₁ − 2ω ₂ is small (case of a “small divisor”) because then Φ ₃ is large, while the series expansion of Φ implies that the successive terms Φ ₂, Φ ₃, etc. become smaller as the order increases.

In such resonance cases, we can construct a third integral by a different method. If ω₁/ω₂ is rational, there are further isolating integrals of motion of the lowest-order Hamiltonian H ₂ beyond the partial energies Φ ₂, Φ ₂ ^′. For example, if ω ₁/ω ₂ = n/m, then H ₂ has also the integrals

$$ \begin{array}{l}\begin{array}{c}\hfill {S}_0\hfill \\ {}\hfill {C}_0\hfill \end{array}={\left(2{\varPhi}_2\right)}^{m/2}{\left(2{\varPhi}_2^{\prime}\right)}^{n/2}\begin{array}{c}\hfill \sin \hfill \\ {}\hfill \cos \hfill \end{array}\left[n{\omega}_2t-m{\omega}_1\left(t-{t}_0\right)\right]\\ {}\kern1.4em =\mathrm{constant}\end{array} $$

(15)

which can be written as polynomials of degree m + n in the variables x _λ,y _λ (Contopoulos 1963). The integrals S ₀,C ₀ can be used for constructing series of the form S = S ₀ + S ₁ + S ₂ + … , C = C ₀ + C ₁ + C ₂ + … . These series also contain secular terms. However, we can prove that there are combinations of the three integrals Φ ₂, Φ ₂ ^′, and S ₀ (or C ₀) that do not contain secular terms of any order; thus, they are formal integrals of motion.

The series of the third integral are in general not convergent. However, they may give very good applications, if truncated at an appropriate level. A simple example of such a series was provided by Poincaré (1892, Vol. II). The series

$$ {f}_1={\displaystyle \sum_{n=0}^{\infty}\frac{n!}{(1000)^n}} $$

(16)

is divergent (i.e., f ₁ is infinite), but if it is truncated at any large order n smaller than 1000, it gives approximately the same value, and it seems to converge, as n increases, to a finite value. On the other hand, the series

$$ {f}_2={\displaystyle \sum_{n=0}^{\infty}\frac{(1000)^n}{n!}} $$

(17)

is convergent, but in order to give an approximate value of f ₂, one has to take more than 1000 terms.

There is an important theorem establishing the existence of N-dimensional invariant tori in systems of N degrees of freedom. This is the celebrated KAM theorem, announced by Kolmogorov (1954). The details of the proof were given by Arnold (1961, 1964) and independently by Moser (1962, 1967).

The KAM theorem states that in an autonomous Hamiltonian system, close to an integrable system expressed in action-angle variables, there are N-dimensional invariant tori, i.e., tori containing quasiperiodic motions with frequencies ω _i , satisfying a Diophantine condition (after the ancient Greek mathematician Diophantus) $\left|\overrightarrow{\omega k}\right|>\gamma /{\left|k\right|}^{N+1},$ where $\overrightarrow{\omega k}={\omega}_1{k}_1+{\omega}_2{k}_2+\dots +{\omega}_N{k}_N$ with k ₁,k ₂,…,k _N integers |k| = |k ₁| + |k ₂| + … + |k _N | ≠ 0, while γ is a small quantity depending on the perturbation. The set of such tori has a measure different from zero if γ is small.

In the case of two degrees of freedom, the ratio ω₁/ω₂ is far from all resonances, if it satisfies a Diophantine condition |ω ₁/ω ₂ − n/m| > ϵ/m ³. If we exclude the sets of all irrationals that are near every rational n/m, on both its sides, the set of the excluded irrationals is smaller than

$$ {\displaystyle \sum_{m=1}^{\infty }{\displaystyle \sum_{n=1}^{m-1}\frac{2\upepsilon}{m^3}}}<2\upepsilon {\displaystyle \sum_{m=1}^{\infty}\frac{1}{m^2}}=2C\upepsilon $$

(18)

where C = π ²/6 ≈ 1.64. Therefore, the set of Diophantine numbers ω₁/ω₂ in the interval (0,1) has a measure larger than 1 −2Cϵ, and if ϵ is small, this is not only positive, but it can even be close to 1.

The formal integrals Φ are convergent only in integrable cases. In other cases, Φ are asymptotic series that may not represent particular functions. The question then arises for how long the formal integrals Φ = Φ ₂ + Φ ₃ + … are valid within a given approximation. The answer to this question is provided by the theory of Nekhoroshev (1977). This theory establishes that the formal integrals are valid over exponentially long times. Namely, if the deviation from an integrable system is of order ϵ, the Nekhoroshev time is of order

$$ t={t}_{\ast } \exp \left(\frac{M}{\upepsilon^m}\right) $$

(19)

where t _*, M, and m are positive constants. The formal integrals must be truncated at an optimal order, such that the variation of the truncated integrals is minimum. An analytical estimate of the optimal order of truncation was provided recently by Efthymiopoulos et al. (2004).

Let us consider the orbits in the phase space of an autonomous Hamiltonian system of two degrees of freedom H ≡ H(x, y, p _x , p _y ) = h. As the energy of the system, h, is an integral of motion, one variable, say p _y , can be derived from H if the other three variables are known. All orbits with fixed energy equal to h lie on the above 3-D surface, in the 4-D phase space. Thus, we may consider the orbits in the 3-dimensional phase space (x,y,p _x ).

If every orbit is intersected by a particular surface S(p _x ,p _y ,x,y) = const within every fixed interval of time T, then this surface is called a Poincaré surface of section. The intersections of the points of S are called consequents; they define a map on the surface of section S that is called a Poincaré map. In particular, a periodic orbit of multiplicity three is represented by three points (e.g., O ₁, O ₂, O ₃ in Fig. 4).

Fig. 4

Distribution of periodic orbits of various multiplicities in the potential $V=\frac{1}{2}\left({\omega}_1^2{x}_1^2+{\omega}_2^2{x}_2^2\right)-\upepsilon {x}_1{x}_2^2$ with ω ₁ ² = 1.6, ω ₂ ² = 0.9, ε = 0.3

Periodic Orbits

The most important orbits in a dynamical system are the periodic orbits. If the periodic orbits are stable, they trap around them a set of nonperiodic orbits with the same average topology. If they are unstable, they repel the orbits in their neighborhood. In integrable systems, the main unstable orbits separate the resonant orbits from the nonresonant orbits. In nonintegrable systems, the unstable periodic orbits introduce chaos in the system. Thus, the study of a dynamical system usually starts with an exploration of its periodic orbits.

In the case of an area preserving map x ₁ = f(x ₀, y ₀, ϵ), y ₁ = g(x ₀, y ₀, ϵ), where ϵ is a parameter, a periodic orbit is an invariant point x ₁ = x ₀, y ₁ = y ₀. Its stability is found by linearizing the equations around this point. We have Δx ₁ = aΔx ₀ + bΔy ₀, Δy ₁ = cΔx ₀ + dΔy ₀, where a = ∂f/∂x ₀, b = ∂f/∂y ₀, c = ∂g/∂x ₀, d = ∂g/∂y ₀. The conservation of areas implies ad − bc = 1.

In a Hamiltonian system of two degrees of freedom, the periodic orbits and the deviations $\overrightarrow{\xi}=\left(\Delta x,\Delta y\right)$ are defined in the 4-D phase space $\left(x,y,\overset{\cdotp }{x},\overset{\cdotp }{y}\right)$. Along particular directions in the 4-D phase space, the original deviation ${\overrightarrow{\xi}}_0=\left(\Delta {x}_0,\Delta {y}_0\right)$ becomes ${\overrightarrow{\xi}}_1=\left(\Delta {x}_1,\Delta {y}_1\right)$ after one iteration, where ${\overrightarrow{\xi}}_1$ is equal to ${\overrightarrow{\xi}}_0$ multiplied by a factor λ. Such vectors ${\overrightarrow{\xi}}_0$ are called eigenvectors (with arbitrary length) and the factors λ are called eigenvalues. The eigenvector forms an angle Φ with the x-axis, given by tan Φ = (λ − a)/b = c/(λ − d), that has two opposite solutions Φ and Φ + 180° for each real eigenvalue λ.

The two eigenvalues λ ₁,λ ₂ are found from the equation

$$ \left|\begin{array}{cc}\hfill a-\lambda \hfill & \hfill b\hfill \\ {}\hfill c\hfill & \hfill d-\lambda \hfill \end{array}\right|={\lambda}^2-\left(a+d\right)\lambda +1=0 $$

(20)

The roots λ ₁,λ ₂ of this equation are inverse. If |a + d| > 2, the roots λ are real (|λ ₁| > 1 and |λ ₂| < 1). Then, the orbit is unstable and the eigenvectors ${\overrightarrow{\xi}}_{01},{\overrightarrow{\xi}}_{02}$ form the angles Φ with the x-axis. On the other hand, if |a + d| < 2, the roots are complex conjugate with |λ _i | = 1 and the orbit is stable.

Ordered orbits lie on invariant tori that intersect a surface of section along invariant curves. Orbits that do not lie on closed invariant curves and do not escape to infinity are chaotic. On a surface of section, such orbits are represented by the irregular distribution of their consequents. Such chaotic orbits appear near every unstable periodic orbit of a nonintegrable system. The domains filled by the chaotic orbits surround the corresponding islands of stability of the same chaotic zone that contains the unstable periodic orbits. For example, the chaotic domains of the Hamiltonian (Eq. 11) with H ₃ = − ϵx ₁ x ²₂ are near the unstable points O ₁, O ₂, O ₃ of type n/m = 2/3 in Fig. 4. These chaotic domains surround also the islands around the stable periodic orbit $\left({\overline{O}}_1{\overline{O}}_2{\overline{O}}_3\right)$.

For relatively small perturbations, the various resonances are well separated by invariant curves that close around the central periodic orbit O; therefore, chaos is limited. But when the perturbation increases, the various chaotic domains increase and join each other, destroying the separating invariant curves and producing a large connected chaotic domain (Fig. 5) (for the same Hamiltonian but with ϵ = 4.5). This is a manifestation of a “resonance overlap” or “resonance interaction.”

Fig. 5

As in Fig. 4 for ϵ = 4.5

The asymptotic curves of a hyperbolic point of an integrable system join into one separatrix, unless these curves extend to infinity. The separatrix may be considered as an orbit (homoclinic orbit) or as an invariant curve containing the images of the initial points on this curve. An orbit starting very close to the origin along the unstable branch U will move far from O, but eventually, it will return close to O along the stable branch S.

However, in a nonintegrable system, there are no closed separatrices, and the unstable and stable asymptotic curves intersect at infinite points called homoclinic points.

All initial points on an asymptotic curve generate asymptotic orbits, i.e., orbits approaching the periodic orbit O either in the forward or in the backward direction of time. The homoclinic points define doubly asymptotic orbits because they approach the orbit O both for t → − ∞ and for t → ∞.

The intersecting asymptotic curves of an unstable periodic orbit form the so-called homoclinic tangle. If we start an orbit in this region, its consequents fill the region in a practically random way, forming the chaotic domain that we see near every unstable periodic orbit of a nonintegrable system.

Transition from Order to Chaos

Dissipative Systems: The Logistic Map

A simple one-dimensional dissipative system, where we can see the transition from order to chaos, is the logistic map

$$ {x}_{i+1}=f\left({x}_i\right)=4\lambda {x}_i\left(1-{x}_i\right) $$

(21)

This is the prototype of a quadratic map, and it is studied extensively in the pioneering article of May (1976) and in books on chaos (see, e.g., (Argyris et al. 1994; Collet and and Eckmann 1980)). Thus, we will only describe here its main properties. We consider only values of λ between 0 and 1 in order to have x always between 0 and 1. The logistic map has two simple periodic orbits on the intersection of the parabola (Eq. 21) with the diagonal x _{i + 1} = x _i , namely, the points x = 0 and $x={x}_0=1-\frac{1}{4\lambda }$.

A periodic orbit x ₀ is stable if the derivative f ′ = df/dx ₀ is absolutely smaller than 1. This happens if 1/4 < λ < Λ₁ = 3/4.

It is easily found that when the orbit x ₀ = 1 − (1/4λ) of period 1 becomes unstable, then a period −2 family of periodic orbits (x _a ,x _b ) bifurcates, which is stable if ${\Lambda}_1<\lambda <{\Lambda}_2=\left(1+\sqrt{6}\right)/4$.

There is an infinite set of period-doubling bifurcations (Fig. 6a). The intervals between successive bifurcations decrease almost geometrically at every period doubling, i.e.,

Fig. 6

Bifurcations and chaos: (a) in a dissipative system (the logistic map) and (b) in a conservative system

$$ \underset{n\to \infty }{ \lim}\frac{\Lambda_n-{\Lambda}_{n-1}}{\Lambda_{n+1}-{\Lambda}_n}=\delta =4.669201609\dots $$

(22)

This relation is asymptotically exact (i.e., for n → ∞) (Coullet and and Tresser 1978; Feigenbaum 1978). The number δ is universal, i.e., it is the same in generic dissipative systems.

The curves F ^m(λ), with m = 2ⁿ, starting at the nth period-doubling bifurcation, are similar to each other, decreasing by a factor α = 2.50… in size at each successive period-doubling bifurcation.

As the ratios (Eq. 22) decrease almost geometrically, the value Λ _n converges to the limiting value Λ_∞ = 0.893.

In 2-D maps, the areas are reduced at every iteration in the dissipative case and conserved in the conservative case. An example is provided by the Hénon map (Hénon 1969)

$$ {x}_{i+1}=1-K{x}_i^2+{y}_i,{y}_{i+1}=b{x}_i\kern0.5em \left( \mod 1\right) $$

(23)

(the original map was not given modulo 1). The Jacobian of the transformation is J = b. If 0 < b < 1, the system is dissipative and if b = 1, it is conservative.

In the dissipative case, there are attractors, i.e., manifolds, to which tend many orbits as t → ∞ (or i → ∞ in maps). These attractors may be points, curves (limit cycles), or strange attractors (Grassberger and and Procaccia 1983; Hénon 1976; Lorentz 1963), which are composed of infinite lines.

Other mechanisms of transition to chaos in dissipative systems are reviewed by Contopoulos (section 2.6.3 in Contopoulos 2002).

Conservative Systems

The first numerical study of the transition to chaos in a conservative system was provided by Hénon and Heiles (1964). The transition to chaos follows a number of scenarios.

Infinite Period-Doubling Bifurcations

This scenario is similar to the corresponding scenario of the dissipative case. In fact, a large degree of chaos is introduced after an infinite number of period-doubling bifurcations along the characteristics of the periodic orbits (x = x(λ)). However, there are three differences in the conservative case:

• 1. Infinite period-doubling bifurcations appear generically only in systems of two degrees of freedom.

• 2. The bifurcation ratio is different (δ≅8.72 in conservative systems versus δ≅4.67 in dissipative systems). Also, the scaling factors are different.

• 3. The introduction of chaos follows a different pattern. While in the dissipative case, chaos appears only after infinite bifurcations (Fig. 6a), in the conservative systems, some chaos appears around every unstable orbit. As the perturbation increases, the chaotic domains increase in size and they merge to form a large connected domain (Fig. 6b).

In some cases, the appearance of infinite bifurcations may be followed by their corresponding disappearance, as the perturbation increases further. Then the infinite unstable families terminate in the opposite way forming infinite bubbles.

Infinite Bifurcations from the Same Family

While in the infinite bifurcation scenario we have pitchfork bifurcations from the successive bifurcating families (Fig. 6), in the present scenario, the original family of periodic orbits becomes successively stable and unstable, an infinite number of times. For example, this happens in the case of the Hamiltonian

$$ H=\frac{1}{2}\left({\overset{\cdotp }{x}}^2+{\overset{\cdotp }{y}}^2+{x}^2+{y}^2\right)+x{y}^2=h $$

(24)

(Contopoulos and and Zikides 1980). The stable bifurcating families undergo further period-doubling bifurcations. The successive stable and unstable intervals along the original family have a limiting bifurcation ratio δ≅9.22. But this bifurcation ratio is not universal. It depends on the particular dynamical system considered. In the case (Eq. 24), it was proven analytically by Heggie (1983) that this ratio is $\delta = \exp \left(\pi /\sqrt{2}\right)$ but in other systems, it is different. In all cases near the limiting point, there is an infinity of unstable families that produce a large degree of chaos.

Infinite Gaps

In a dynamical system representing a rotating galaxy, the central periodic family, which consists of circular orbits in the unperturbed case, has two basic frequencies. These are the rotational frequency in the rotating frame (Ω − Ω_s) (where Ω is the angular velocity along a circular orbit and Ω_s the angular velocity of the system, e.g., of the spiral arms) and the epicyclic frequency κ of radial oscillations from the circular orbit.

The ratio of the two frequencies (rotation number) goes through infinite rational values κ/(Ω − Ω_s) = n/m (resonances) as the energy increases.

The most important resonances are those with m = 1. At every even resonance (n = 2n ₁, m = 1), we have a bifurcation of a resonant family of periodic orbits in the unperturbed case, which becomes a gap in the perturbed case (when a spiral, or a bar, is added to the axisymmetric background; Fig. 7a). The gap is formed by separating the characteristic of the main family into two parts and joining the first part with a branch of the resonant family and the second part containing the other branch of the resonant family.

Fig. 7

(a) Gaps appear along the characteristic of the family x ₁ at all even resonances 2n ₁ / m. (b) Spiral characteristics near corotation (L ₄)

As we approach corotation (where Ω = Ω_s), the rotation number tends to infinity. Therefore, an infinity of gaps is formed, followed by the appearance of an infinity of unstable periodic orbits, and this leads to a large degree of chaos (Contopoulos 1983).

Infinite Spirals

There are an infinite number of families of periodic orbits whose characteristics form spirals near corotation (Fig. 7b). These families contain an infinite number of unstable periodic orbits that interact with each other, producing chaos (Contopoulos 2002; Pinotsis 1988).

Resonance Overlap

In a nonintegrable Hamiltonian, we have several types of resonant islands, and their size can be found by means of the third integral, calculated for each resonance separately.

We consider a Hamiltonian of the form (Eq. 11) with ω₁/ω₂ near a resonance n/m. The maximum size D of an island of type n/m is of O(ϵ^(m+n−4)/2) (Contopoulos 1967). The same is true for the area covered by the islands of the resonance n/m.

If the islands of various nearby resonances are well separated, then there is no resonance overlap and no large degree of chaos. However, as the perturbation increases, the various islands increase in size and the theoretical islands overlap. As this overlapping is not possible in reality, what really happens is that the region between the islands becomes chaotic.

There are two ways to estimate the critical perturbation of the resonance overlap:

• 1. We calculate the areas of the various islands. When this quantity becomes equal to the total area of phase space on the surface of section, we start to have large degree of chaos (Contopoulos 1967).

• 2. We may find the critical value for the production of large chaos between two main resonances of the system, say n/m = 4/3 and n/m = 3/2. The positions of the triple and double periodic orbits are given by the corresponding forms of the third integral as functions of the perturbation ε (characteristics). The theoretical characteristics intersect for a certain value of ε. On the other hand, the real characteristics cannot intersect. Instead, between the two characteristics, a chaotic region is formed, due to the resonance overlap.

The resonance overlap criterion for large chaos was considered first by Rosenbluth et al. (1966) and Contopoulos (1967). It was later described in detail by Walker and Ford (1969), Chirikov et al. (1971), Chirikov (1979), and many others.

Arnold Diffusion

Nearly integrable autonomous Hamiltonian systems of N degrees of freedom have a 2N − 1-dimensional phase space of constant energy, and a large set of N-dimensional invariant surfaces (invariant tori), according to the KAM theorem. If N = 2, there are N = 2-dimensional surfaces separating the 2N − 1 = 3-dimensional phase space, and diffusion through these surfaces is impossible. But if N ≥3, the N-dimensional surfaces do not separate the 2N − 1-dimensional phase space, and diffusion is possible all over the phase space. This is called “Arnold diffusion” (Arnold 1964). Such a diffusion is possible even if the perturbation is infinitesimal.

In order to understand better this phenomenon, consider a system of three degrees of freedom, which has 3-D tori in a 5-D phase space of constant energy. If we reduce the number of both dimensions by two, we have 1-D tori (lines) in a 3-D space. In an integrable case, there is a line passing through every point of the space and all motions are regular. However, in a nonintegrable case, there are gaps between the lines and these gaps communicate with each other. Therefore, if a chaotic orbit starts in such a gap, it may go everywhere in the 3-D space.

Extensive numerical studies (e.g., (Laskar 1993)) in a 4-D map have shown that in the same dynamical system, there are both ordered domains, where diffusion is very slow (Arnold diffusion), and chaotic domains, where diffusion is dominated by resonance overlap.

As an example, we consider two coupled standard maps:

$$ \begin{array}{l}{x}_1^{\mathit{\prime}}={x}_1+{y}_1^{\mathit{\prime}},\\ {}{y}_1^{\mathit{\prime}}={y}_1+\frac{K}{2\pi } \sin 2\pi {x}_1-\frac{\beta }{\pi } \sin 2\pi \left({x}_2-{x}_1\right)\\ {}\kern28em \left( \mod 1\right)\\ {}{x}_2^{\mathit{\prime}}={x}_2+{y}_2^{\mathit{\prime}},\\ {}{y}_2^{\mathit{\prime}}={y}_2+\frac{K}{2\pi } \sin 2\pi {x}_2-\frac{\beta }{\pi } \sin 2\pi \left({x}_1-{x}_2\right)\end{array} $$

(25)

Here, K is the nonlinearity and β the coupling constant (Contopoulos and and Voglis 1996).

Taking the same initial conditions, we calculate the time of diffusion from an ordered region to the large chaotic sea. If the coupling β is larger than a critical value β = β _c ≅ 0.305, the diffusion time T increases exponentially with decreasing β. If, however, β < β _c, the time T increases superexponentially as β decreases. We can identify the exponential increase case as due to resonance overlap diffusion and the superexponential increase as due to Arnold diffusion.

Cantori

When the perturbation increases and a particular torus is destroyed, it becomes a cantorus, i.e., a Cantor set of points that is nowhere dense (Aubry 1978; Percival 1979). The cantorus has a countable infinity of gaps, but it contains a noncountable infinity of orbits. The most surprising property of the cantori is that their measure is zero and their fractal dimension is also zero.

In the case of the standard map, cantori with golden rotation number $rot=\left(\sqrt{5}-1\right)/2$ (expressed in continuous fraction representation as [1,1,1,…]) exist for all values of K larger than a critical value K _cr = 0.972 (Mather 1982).

The cantorus is formed when all the periodic orbits corresponding to the various truncations of the golden number become unstable. In the limit K = K _cr, the invariant curve (torus) is a fractal with self-similar structure on all scales (Greene et al. 1981; Shenker and and Kadanoff 1982).

What happens in general around an island of stability is the following. Before the destruction of the last KAM curve surrounding the island, there is a chaotic layer, just inside the last KAM curve, while outside the last KAM curve, there is a large chaotic sea. When the last KAM curve becomes a cantorus, the orbits from the inner chaotic layer can escape to the large chaotic sea, but only after a long time. Thus, the region just inside a cantorus with small holes is differentiated by the larger density of the points inside it (Fig. 8). This is the phenomenon of stickiness.

Fig. 8

Stickiness in the standard map for K = 5. The sticky region (dark) surrounds an island of stability and is surrounded by a large chaotic sea

Stickiness

This phenomenon was found numerically by Contopoulos (1971). Stickiness has been observed in practically all nonintegrable dynamical systems that contain islands (Menjuk 1985). In Fig. 8, the sticky zone (the dark region surrounding the island O ₁) is limited on its inner side by the outer boundary of the island O ₁ and on its outer side by a set of cantori, but mainly by the cantorus that has the smallest gaps. An orbit starting in the sticky zone requires a long time to escape outside. The details of the escape can be found if we calculate the asymptotic curves of unstable periodic orbits just inside the main cantorus.

As an example, consider an unstable asymptotic curve from the periodic orbit of period 215 in the case of the standard map for K = 5 (Fig. 9; Efthymiopoulos et al. 1997). If we start with a small interval along the asymptotic curve, we find an image of this curve that crosses the cantorus after two iterations along the same asymptotic curve. In Fig. 9, we see that the asymptotic curve starts from P by going first to the left and downward and then it moves to the right and upward, crossing the cantorus in one of its largest gaps (marked A in Fig. 9). But then the asymptotic curve makes several oscillations back and forth, entering again inside the cantorus several times, before going to a large distance in the chaotic sea. Then it moves around for a long time before coming back and getting trapped again in the sticky zone close to the island.

Fig. 9

An asymptotic curve of the periodic orbit P in the standard map with K = 5 makes several oscillations inside the cantorus (stars) and then escapes. We mark some islands inside the cantorus (on the right) and outside it (on the left)

Another type of stickiness refers to the unstable asymptotic curves that extend far into the chaotic sea (dark lines in Fig. 8). These asymptotic curves attract nearby orbits, and thus, thick dark lines are formed (Contopoulos and and Harsoula 2008).

Dynamical Spectra

Despite the importance of the Lyapunov characteristic numbers in distinguishing between order and chaos, their practical application is limited by the very long calculations that are usually required for their evaluation. In fact, if we calculate the finite time LCN

$$ \chi =\frac{ \ln \left|\xi /{\xi}_0\right|}{t} $$

(26)

the variation of χ is irregular for relatively small t and only for large t the value of χ of a chaotic orbit stabilizes and tends to a constant limiting value (curve 2 of Fig. 10), which is the Lyapunov characteristic number LCN = lim _{t → ∞} χ. If, on the other hand, χ varies approximately as 1/t, the LCN is zero (ordered orbit 1 of Fig. 10).

Fig. 10

The variation of χ(t) of (1) an ordered orbit and (2) a chaotic orbit

Of special interest is to find a finite time LCN after the shortest possible time. In the case of a map, the shortest time is one iteration (t = 1). Thus, one finds the quantities

$$ {a}_i= \ln \left({\xi}_{i+1}/{\xi}_i\right) $$

(27)

which are called Lyapunov indicators by Froeschlé et al. (1993) or stretching numbers by Voglis and Contopoulos (1994).

The distribution of successive values of the stretching numbers a _i along an orbit is called the “spectrum of stretching numbers.” Namely, the spectrum gives the proportion dN/N of the values of a _i in a given interval (a,a + da) divided by da, i.e.,

$$ S(a)=\mathrm{d}N/\left(N\mathrm{d}a\right) $$

(28)

where da is a small quantity and N a large number of iterations.

The main properties of the spectra of stretching numbers are (Voglis and Contopoulos 1994):

• 1. The spectrum is invariant along the same orbit.

• 2. The spectrum does not depend on the initial deviation ${\overrightarrow{\xi}}_0$ from the same initial point in the case of two-dimensional maps.

• 3. The spectrum does not depend on the initial conditions of orbits in the same connected chaotic domain.

• 4. In the case of ordered orbits in two-dimensional maps, the spectrum does not depend on the initial conditions on the same invariant curve.

The Lyapunov characteristic number is equal to LCN = ∫S(a) a da. The spectrum gives much more information about a system than LCN. For example, the standard map

$$ {x}^{\mathit{\prime}}=x+{y}^{\mathit{\prime}},\kern0.5em {y}^{\mathit{\prime}}=y+\frac{K}{2\pi } \sin 2\pi x\kern0.5em \left( \mod 1\right) $$

(29)

for K = 10 and the Hénon map (Eq. 23) for K′ = 7.407 and b = 1 give two apparently random distributions of points with the same LCN = 1.62. However, their spectra are quite different (Fig. 11).

Fig. 11

The spectra of stretching numbers: (a) of the standard map and (b) of the Hénon map

The use of the stretching numbers allows a fast distinction between ordered and chaotic orbits. For example, while the calculation of the LCN requires 10⁵ − 10⁶ iterations, in Fig. 12 we can separate the chaotic from the ordered orbits after only 20 iterations. In Fig. 12, the noise is large; nevertheless, the chaotic orbits have roughly the same value of LCN = 〈a〉 > 0, while the ordered orbits have ⟨a⟩ close in the to zero (gap). Further details about this topic can be found in the book of Contopoulos (2002).

Fig. 12

The average value of the stretching number, ⟨α⟩, for 10 iterations (beyond the first 10 transients) as a function of x for constant y

Application: Order and Chaos in Galaxies

Galactic Orbits

Galaxies are composed of stars, gas (including dust), and dark matter. The stars and the dark matter produce the main part of the galactic potential and force, while the contribution of the gas in the potential and force is small.

In the Hubble classification of galaxies, the elliptical galaxies (E) and the early-type spiral galaxies (Sa, Sb) and barred galaxies (SBa, SBb) have less gas than the late-type galaxies (Sc, SBc) and the irregular galaxies (I) (Fig. 13).

Fig. 13

The basic types of galaxies: (a) M59/NGC 4,621 elliptical [E], (b) M51/NGC 5,194 normal spiral [Sb], and c NGC1300 barred spiral [SBb]

Galaxies that have a well-developed spiral structure (usually two symmetric spiral arms) are called “grand design,” while galaxies with irregular and multiple fragments of spirals are called “flocculent.”

The study of stellar (or mainly stellar) galaxies is based on a systematic exploration of their orbits.

The exploration of orbits in galaxies is very important, because the orbits are needed in constructing self-consistent models of galaxies.

Self-consistency is a new type of problem that does not appear, e.g., in accelerators or in the solar system. It requires the construction of appropriate sets of orbits of stars, such that their superposition gives a response density that matches the imposed density of the model. Such a construction is done in many cases by taking a grid of initial conditions and calculating the orbits numerically in a computer and then populating these orbits with stars. But it is much more helpful and illuminating if one studies the types of orbits that form the building blocks of the galaxies.

Thus, the first step in understanding the structure and dynamics of a galaxy is to calculate its orbits, periodic, quasiperiodic, and chaotic.

The most important orbits in a galactic system are the periodic orbits. The stable orbits trap around them sets of quasiperiodic orbits, which give the main features of the galaxy. On the other hand, the unstable orbits separate the various types of ordered orbits and also characterize the chaotic domains in a galaxy.

The simplest galactic orbits are the circular periodic orbits on the plane of symmetry of an axisymmetric galaxy. The circular orbits in such a galaxy are in general stable.

Orbits close to the stable circular periodic orbits are called epicyclic orbits. Such orbits fill circular rings in the plane of symmetry of the galaxy and are called “rosette” orbits (Fig. 14a). In a frame of reference rotating with the angular velocity (frequency) Ω of the center of the epicycle (epicenter, or guiding center), these orbits are closed and they are approximately ellipses around the center of the epicycle (Fig. 14b).

Fig. 14

(a) A rosette orbit in an axisymmetric galaxy. (b) The corresponding epicyclic orbit

If V ₀ is the axisymmetric potential, we have Ω ² = V ₀′/r and κ ² = V ₀ ^′′ + 3V ₀′/r. The frequency κ along the epicycle is called “epicyclic frequency.”

In most cases (spiral or barred galaxies), the frame of reference is not inertial, but rotates with angular velocity Ω _s. This is called “pattern velocity.” In this rotating frame, the form of the galaxy is stationary. Then the two basic frequencies of a moving star are (Ω − Ω _s) and κ. If the ratio κ/(Ω − Ω _s) = (n/m) is rational, then we have a resonant periodic orbit in the rotating system. Two most important resonances are the Lindblad resonances κ/(Ω − Ω _s) = ±2/1 (+inner Lindblad resonance [ILR], − outer Lindblad resonance [OLR]). A third most important resonance is corotation (or particle resonance), where Ω = Ω _s. The angular velocity Ω of an axisymmetric family is a function of the distance r. Ω is a monotonically decreasing function of r (Fig. 15).

Fig. 15

The curves Ω and Ω − κ/2 in two cases: (1) with only one ILR (dotted) and (2) two ILRs (solid), for two different values of the pattern velocity Ω_s

Corotation and the inner and outer Lindblad resonances appear at the intersections of the line Ω = Ω _s (Fig. 15) with the curves Ω, Ω − κ/2 and Ω + κ/2. There is only one distance corresponding to corotation and one distance corresponding to the outer Lindblad resonance (OLR). However, depending on the model used, we may have one inner Lindblad resonance, or two ILRs, or no ILR at all.

The study of nonperiodic orbits on the plane of symmetry of a galaxy is realized by means of a surface of section.

The invariant curves of the ordered nonperiodic orbits surround either the point x ₁ or the point x ₄, which represent direct or retrograde periodic orbits that are reduced to circular orbits in the circular case.

The periodic orbits of type x ₁ in the case of a strong spiral are like ellipses close to the inner Lindblad resonance (Fig. 16) but become like squares near the 4/1 resonance. These orbits and the nonperiodic orbits around them support the spiral up to the 4/1 resonance. However, beyond the 4/1 resonance, these orbits are out of phase and do not support the spiral (Fig. 16). Thus, self-consistent spirals should terminate near the 4/1 resonance (Contopoulos 1985). On the other hand, in stronger bars, we may have spirals outside corotation (see section “Chaotic Orbits”). More details about orbits in galaxies can be found in (Ollongren 1965) and (Contopoulos 2002).

Fig. 16

Periodic orbits in a spiral galaxy support the spiral up to the 4/1 resonance, but not beyond it

Integrals of Motion in Spiral and Barred Galaxies

The Hamiltonian on the plane of symmetry of a galaxy (spiral or barred) rotating with angular velocity Ω_s is

$$ H=\frac{{\overset{\cdotp }{z}}^2}{2}+\frac{J_0^2}{2{r}^2}+{V}_0(r)-{\varOmega}_{\mathrm{s}}{J}_0+{V}_1 $$

(30)

where $\overset{\cdotp }{r}$ is the radial velocity, J ₀ the angular momentum, V ₀(r) the axisymmetric potential, and V ₁ the spiral (or barred) perturbation. The value of H is conserved (Jacobi constant). The Hamiltonian in action-angle variables is of the form H = H ₀ + V ₁, where

$$ {H}_0={\omega}_1{I}_1+{\omega}_2{I}_2+a{I}_1^2+2b{I}_1{I}_2+c{I}_2^2+\dots $$

(31)

and

$$ \begin{array}{l}{V}_1=\mathrm{Re}\left\{A(r) \exp \left[i\left(\varPhi (r)-2\theta \right)\right]\right\}\\ {}\kern1.5em =\mathrm{Re}{\displaystyle \sum_{mn}{V}_{mn}}\left({I}_1,{I}_2\right) \exp \left[i\left(m{\theta}_1-n{\theta}_2\right)\right]\end{array} $$

(32)

The expansion in action-angle variables is found by expanding A, Φ, and θ in terms of two angles, θ ₁ (epicyclic angle) and θ ₂ (azimuth of the epicenter measured from a certain direction), and r − r _c = (2I ₁/ω ₁)^1/2, where r _c is the radius of a circular orbit that has the same Jacobi constant as the real orbit.

The main terms of V ₁ are of order A and contain the trigonometric terms cos(θ ₁ − 2θ ₂), cos(θ ₁ + 2θ ₂), and cos 2θ ₂. Away from the main resonances of the galaxy (inner and outer Lindblad resonances and corotation), it is possible to eliminate these terms by means of a canonical transformation that gives V ₁ as a function of new action variables I ₁ ^*, I ₂ ^*. Thus, in this case, the Hamiltonian H is a function of I ₁ ^*, I ₂ ^* only, i.e., I ₁ ^* and I ₂ ^* are integrals of motion.

The usual density wave theory of spiral structure deals with the forms of the integrals I ₁ ^*, I ₂ ^*, truncated after the first-order terms.

However, the transformation from I _i to I _i ^* contains denominators of the forms ω ₁ − 2ω ₂, ω ₁ + 2ω ₂ or ω ₂, which tend to zero close to the inner and outer Lindblad resonances and corotation, respectively. For example, near the inner Lindblad resonance, we cannot eliminate the term with cos(θ ₁ − 2θ ₂), because the transformation would have the denominator ω ₁ − 2ω ₂ which is close to zero. In this case, we eliminate only the terms with cos(θ ₁ + 2θ ₂) and cos 2θ ₂. Instead of eliminating cos(θ ₁ − 2θ ₂), we write this term as cos ψ ₁, introducing now resonant action-angle variables J ₁ = I ₁ ^∗, J ₂ = I ₂ ^∗ + 2I ₁ ^∗, ψ ₁ = θ ₁ ^∗ − 2θ ₂ ^∗, ψ ₂ = θ ₂ ^∗. Then H is expressed in terms of J ₁, J ₂, and ψ ₁, but does not contain ψ ₂. Therefore, the conjugate action J ₂ is an integral of motion. Thus, we have again two integrals of motion, namely, J ₂ and the Jacobi constant H.

The resonant form of H explains the forms of the orbits near the inner Lindblad resonance. In particular, near this resonance, we have not one but two periodic orbits roughly perpendicular to each other.

The resonant theory is an extension of the linear density wave theory and it is applicable near the ILR. In a similar way, we find the forms of the integrals near the outer Lindblad resonance and near corotation.

The three resonant forms of the Hamiltonian tend to the nonresonant form away from all resonances. In fact, if the amplitude of the spiral, or the bar, is small, the resonant effects are restricted in small regions around each resonance. However, if the amplitude of a spiral, or a bar, is large, we have an overlapping of resonances and the second integral is no more applicable.

Chaotic Orbits

Chaos in galaxies is always present near corotation, because this region contains an infinite number of higher-order resonances that interact with each other.

Besides corotation, chaos may appear near the center of the galaxy when there is a large mass concentration (e.g., a central black hole). Furthermore, large-scale chaos appears when the amplitude of the spiral, or the bar, is large. Appreciable chaos has been found in N-body simulations of galaxies (Contopoulos et al. 2000; Merritt and and Valluri 1999; Schwarzschild 1993).

Chaos in barred galaxies near corotation appears around the unstable Lagrangian points L ₁ and L ₂. Around these points, there are short-period unstable periodic orbits for values of the Jacobi constant larger than the value H(L ₁) appropriate for L ₁. Every unstable periodic orbit, PL ₁, PL ₂, of this family has two stable(S,SS or S′, SS′) and two unstable(U,UU or U′,UU′) manifolds is attached to it. Orbits near L ₁ or L ₂ follow the unstable manifolds of the corresponding orbits PL ₁ and PL ₂ and form trailing spiral arms along U and U ^′ (Fig. 17), which are clearly seen in strong bars (Contopoulos 2008; Voglis et al. 2006). Thus, in strong bars, chaos is important in explaining the outer spiral arms.

Fig. 17

Orbits starting close to the unstable orbits PL ₁, PL ₂ move toward PL ₁, PL ₂ along the stable asymptotic curves S, SS and S ^′, SS ^′ and away from PL ₁, PL ₂ along the unstable asymptotic curves U, UU and U ^′, UU ^′

Future Directions

The subject of dynamical systems in astrophysics is progressing along three main directions: (1) exploration of further basic phenomena, especially in systems of more than two degrees of freedom, (2) rigorous mathematical theorems, and (3) applications on various problems of astrophysics.

• 1. Much exploratory work exists now on systems of two degrees of freedom, but relatively little work has been done with systems of three or more degrees of freedom. In particular, it is important to find the applicability of Arnold diffusion in various cases. A better separation of ordered and chaotic domains in phase space is also necessary. Extensions of this work to find ordered and chaotic domains in quantum mechanical systems are also important ((Efthymiopoulos and and Contopoulos 2006) and references therein). Finally, N-body problems with large N are also of great interest, especially as regards the validity and applications of statistical mechanics. In particular, problems connected with the evolution of dynamical systems are quite important.

• 2. Many problems that have been explored numerically up to now require rigorous mathematical proofs. For example, the use of formal integrals of motion; the application of basic theorems, like the KAM and the Nekhoroshev theorems; and the applicability of many statistical methods require a better mathematical foundation.

• 3. Presently, much is being done with astrophysical applications of the theory of dynamical systems. For example, galactic dynamics has experienced an important expansion in recent years, with the exploration of the role of chaos in galaxies, especially in the formation and evolution of the outer spiral arms and in the evolution of the central black holes. There are many other astrophysical problems where order and chaos play an important role, like the structure and evolution of stars, stellar variability, solar and stellar activity, the role of magnetic fields in stars and galaxies, the properties of extrasolar planetary systems, and the evolution of the whole Universe.