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Lyapunov Exponent (Term)

Lyapunov Exponent — Tue, 24 Jan 2012 07:37:18 -0600

In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectory. Quantitatively, two trajectories in phase space with initial separation \delta \mathbf{Z}_0 diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by

: | \delta\mathbf{Z}(t) | \approx e^{\lambda t} | \delta \mathbf{Z}_0 | \,

where \lambda is the Lyapunov exponent.

The rate of separation can be different for different orientations of initial separation vector. Thus, there is a spectrum of Lyapunov exponents— equal in number to the dimensionality of the phase space. It is common to refer to the largest one as the Maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. A positive MLE is usually taken as an indication that the system is chaotic (provided some other conditions are met, e.g., phase space compactness). Note that an arbitrary initial separation vector will typically contain some component in the direction associated with the MLE, and because of the exponential growth rate, the effect of the other exponents will be obliterated over time.

The exponent is named after Aleksandr Lyapunov.
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Control of Chaos (Term)

Control of Chaos — Tue, 24 Jan 2012 07:17:22 -0600

Based on the theory that any chaotic attractor contains an infinite number of unstable periodic orbits. Chaotic dynamics then consists of a motion where the system state moves into one of these orbits, but because of their instability swiftly shifts to another one, and so forth. Control of chaos then is based on applying certain small perturbations to the system, making sure that it stays within one particular orbit and doesn't shift to another one.
The two most popular approaches are OGY (Ott, Grebogi and Yorke) method, and Pyragas continuous control.
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Interplanetary Transport Network (Term)

Interplanetary Transport Network — Sun, 22 Jan 2012 06:52:26 -0600

The Interplanetary Transport Network (ITN) is a collection of gravitationally determined pathways through the solar system that require very little energy for an object to follow. The ITN makes particular use of Lagrange points as locations where trajectories through Outer space are redirected using little or no energy. These points have the peculiar property of allowing objects to orbit around them, despite the absence of any material object therein. While they use little energy, the transport can take a very long time.
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