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Astrophysics, Chaos and Complexity in

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

Astronomy is the science that deals with the origin, evolution, composition, distance to, and motion of all bodies and scattered matter in theuniverse. It includes astrophysics, which is usually considered to be the theoretical part of astronomy, and as such focuses onthe physical properties and structure of cosmic bodies, scattered matter and the universe as a whole. Astrophysics exploits the knowledge acquired inphysics and employs the latter's methods, in an effort to model astronomical systems and understand the processes taking place in them.

In recent decades a new approach to nonlinear dynamical systems (DS) has beenintroduced and applied to a variety of disciplines in which DS are used as mathematicalmodels. The theory of chaos and complexity, as this approach is often called, evolved from thestudy of diverse DS which behave unpredictably and exhibit complex characteristics, despitetheir seeming simplicity and deterministic nature. The complex behavior...

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Abbreviations

Accretion disk:

A flat gaseous structure into which matter endowed with angular momentum settles when it is being gravitationally attracted to a relatively compact object. Dissipative processes, which are needed to allow for the extraction of angular momentum from the orbiting gas so as to allow for it to be accreted by the star, heat the disk and make it observable by the outgoing radiation.

Attractor:

A geometrical object in the state space of a dynamical system to which the system's trajectories tend for large times (or iteration numbers). Attractors of dissipative systems have a dimension that is lower than that of the state space and when this dimension is fractal, the attractor is called strange.

Bifurcation:

A qualitative change in the essential behavior of a mathematical system (e. g., algebraic equations, iterated maps, differential equations), as a parameter is varied. Usually involves an abrupt appearance and/or disappearance of solutions.

Close binary stars:

Two stars in a Keplerian orbit around each other, having a small enough separation, so that they may affect each other during their evolution (for example by mass transfer). When the pair contains, in addition to a normal unevolved star, a compact object (in cataclysmic binaries it is a white dwarf and in X-ray binaries a neutron star or black hole) the system may be observationally prominent. Mass transfer from the normal star via an accretion disk onto the compact object may result in violent phenomena, such as nova explosions or X-ray bursts.

Dynamical system (DS):

A set of rules by application of which the state of a physical (or some other, well defined) system can be found, if an initial state is known. Iterated maps (IM) in which the evolution proceeds in discrete steps, i. e., one defined by appropriate iterative formula, and differential equations, both ordinary (ODE) and partial (PDE) are typical DS.

DS flow:

All trajectories (or orbits) of a DS in its state space, which is spanned by the relevant state variables. The flow is fully determined by the DS and may serve as its geometrical representation.

Hubble time:

Estimate for the age of the universe obtained from taking the inverse of the Hubble constant, the constant ratio between the recession velocity of galaxies and the distance to them (Hubble law). Current estimates give the Hubble time as \( { \sim 14\,\text{Gyr} } \).

Interstellar medium (ISM):

Diffuse matter of low density, filling the space between stars in a galaxy. The average number density of the ISM is typically \( { n\sim 1\,\text{cm}^{-3} } \) but its distribution is highly non‐uniform, with the densest clouds reaching \( { n \sim 10^6\,\text{cm}^{-3} } \).

Fractal:

A set whose (suitably defined) geometrical dimension is non‐integral. Typically, the set appears self‐similar on all scales. A number of geometrical objects associated with chaos (e. g. strange attractors) are fractals.

Integrable DS:

A DS whose exact solution may be calculated analytically in terms of elementary or special functions and possibly quadratures. In integrable Hamiltonian DS the motion proceeds on a set of invariant tori in phase space.

Integral of motion:

A function of a DS's state variables which remains constant during the evolution. The existence of integrals of motion allows for a dimensional reduction of the DS. A sufficient number of independent integrals of motion guarantees integrability.

KAM theorem:

An important theorem by Kolmogorov, Arnold and Moser (KAM), elucidating the loss of integrability and transition to chaos in Hamiltonian DS by the breakdown of invariant tori.

Liapunov exponents:

A quantitative measure of the local divergence of nearby trajectories in a DS. If the largest such exponent, Λ, is positive, trajectories diverge locally exponentially, giving rise to chaotic behavior. In that case a measure for the time that chaos is manifested, the Liapunov time, is \( { \propto 1/\Lambda } \).

Poincaré section:

A geometrical construction with the help of which the behavior of a multi‐dimensional DS can be examined by means of a two‐dimensional IM. The IM is obtained by finding the system's trajectory successive crossings of a given surface cutting the state space of the DS (surface of section). Essential features of the dynamics can be captured if the surface is placed in an appropriate position.

Pulsating stars:

Stars that, due to an instability, exhibit radial or non‐radial pulsation. Most have periodic variations, with periods ranging from a few days to years, but some are irregular. The classical Cepheids, which are the most common regular pulsators, were instrumental in determining cosmological distances by using their known period‐luminosity relation.

Shilnikov scenario:

One of the known mathematical structures that guarantee chaos in dissipative DS. Let an unstable fixed point exist in the DS state space and have a fast unstable direction and a stable subspace of trajectories that slowly spiral in. If a homoclinic (connecting the point to itself) orbit exists then there are chaotic orbits around it.

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Author information

Authors and Affiliations

  1. Department of Physics, Technion–Israel – Institute of Technology, Haifa, Israel

    Oded Regev

  2. Department of Astronomy, Columbia University, New York, USA

    Oded Regev

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  1. Oded Regev
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Regev, O. (2009). Astrophysics, Chaos and Complexity in. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_26

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