A nonlinear version of the Liouville equation based on terminal attractors is part of a mathematical formalism for describing postinstability motions of dynamical systems characterized by exponential divergences of trajectories leading to chaos (including turbulence as a form of chaos). The formalism can be applied to both conservative systems (e.g., multibody systems in celestial mechanics) and dissipative systems (e.g., viscous fluids).

This formalism at an earlier stage of development was reported in “Extension of Liouville Formalism to Postinstability Dynamics” (NPO-30393), NASA Tech Briefs, Vol. 27, No. 9 (September 2003), page 56. To recapitulate: The problem is to predict the postinstability motions of a dynamic system governed by a system of nonlinear equations and subject to initial conditions. The formalism of nonlinear dynamics does not afford means to discriminate between stable and unstable motions: an additional stability analysis is necessary for such discrimination. However, an additional stability analysis does not suggest any modifications of a mathematical model that would enable the model to describe postinstability

motions efficiently. The most important type of instability that necessitates a postinstability description is associated with positive Lyapunov exponents. Such an instability leads to exponential growth of small errors in initial conditions or, equivalently, exponential divergence of neighboring trajectories.

The development of the present formalism was undertaken in an effort to remove positive Lyapunov exponents. The means chosen to accomplish this is coupling of the governing dynamical equations with the corresponding Liouville equation that describes the evolution of the flow of error probability. The underlying idea is to suppress the divergences of different trajectories that correspond to different initial conditions, without affecting a target trajectory, which is one that starts with prescribed initial conditions.

This formalism applies to a system of n first-order ordinary differential equations in n unknown dynamical (state) variables:

where i is an integer between 1 and n, xi is one of the unknown dynamical variables, the overdot signifies differentiation with respect to time, x is the vector of all the dynamical variables (x1,x2,...xn), and t is time. The prescribed initial conditions are given by

The corresponding Liouville equation for the evolution of the probability distribution, P(x1,x2,...xn,t ), of errors in the initial conditions is

where f is the vector of all the forcing functions ( f1,f2,...fn). It is assumed that this probability distribution peaks at zero error (representing the prescribed initial conditions).

Fictitious control (stabilizing) forces [F = (F1,F2,...Fn)] are added to the system of differential equations. The form of these forces differs from that of the fictitious stabilizing force described in the cited previous article: Whereas previously, the fictitious stabilizing force was proportional to the gradient of the probability density in the space of the dynamical variables, the present fictitious control forces are functions of the differences between expected and actual values of the dynamical variables xi:

where ϒi is a positive constant and xi is the expected value of xi, as given by

The control forces have two important properties:

  • Because they vanish as x 〈 x 〉, they do not affect the target trajectory; and
  • Because the magnitudes of their derivatives approach as x 〈 x 〉, they make the target trajectory infinitely stable. In other words, the target trajectory becomes a terminal attractor.

The resulting modified system of dynamical equations is


The corresponding modified Liouville equation is

wherein the terminal attractors act as nonlinear sinks of probability.

At the limit as xi xi 〉, one can neglect the real force fi as being much smaller than the control force fi, making it possible to decompose the Liouville equation into n independent equations and to express P as a product of n probabilities Pi :

By use of these equations, it can be shown that the control forces create a powerful terminal attractor in probability space that corresponds to occurrence of the target trajectory with probability one (see figure). In configuration space (space in the sense in which “space” is understood in casual conversation), the effect of the control forces is to suppress exponential divergence of close neighboring trajectories without affecting the target trajectory. As a result, the post-instability motion is represented by a set of functions that describe the evolution of such statistical invariants such as expectations, variances, and higher moments of the statistics of the state variables xi as functions of time.

This work was done by Michail Zak of Caltech for NASA’s Jet Propulsion Laboratory. For further information, access the Technical Support Package (TSP) free online at www.techbriefs.com/tsp under the Mechanics category.

The software used in this innovation is available for commercial licensing. Please contact Karina Edmonds of the California Institute of Technology at (818) 393-2827. Refer to NPO-41519.

This Brief includes a Technical Support Package (TSP).
Stochastic Representation of Chaos Using Terminal Attractors

(reference NPO-41519) is currently available for download from the TSP library.

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This article first appeared in the May, 2006 issue of NASA Tech Briefs Magazine.

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