A mathematical formalism has been developed for predicting the postinstability motions of a dynamic system governed by a system of nonlinear equations and subject to initial conditions. Previously, there was no general method for prediction and mathematical modeling of postinstability behaviors (e.g., chaos and turbulence) in such a system.

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 variables:

*x _{i}* =

*f*[

_{i}**x**(

*t*),

*t*],

where *i* is an integer between 1 and *n*, *x _{i}* is one of the unknown dynamical variables, the overdot signifies differentiation with respect to time,

**x**is the vector of all the dynamical variables (

*x*), and

_{1}, x_{2}, . . . x_{n}*t*is time. The prescribed initial conditions are given by

*x _{i}*(0) =

*x*

_{i}^{0}

The corresponding Liouville equation for the evolution of the probability distribution of errors in the initial conditions is

∂ρ/∂*t* + ∇ · (ρ**f**) = 0

where **f** is the vector of all the forcing functions (*f _{1}, f_{1}, . . . f_{n}*). It is assumed that this probability distribution peaks at zero error (representing the prescribed initial conditions). A fictitious stabilizing force proportional to the gradient of the probability density in the space of the dynamical variables is added to the system of differential equations, yielding the following system of modified dynamical equations:

*x* = *f _{i}* +

*h*

_{0}∂ρ/∂

*x*,

_{i}where ρ(**x**(*t*)) is the probability distribution and *h*_{0} is an arbitrary factor of proportionality. The corresponding modified Liouville equation is

∂ρ/∂t + ∇ · [ρ(**f** + *h*_{0}∇ρ)],

The stabilizing potential *h*_{0}ρ creates a powerful attractor that corresponds to the occurrence of the target trajectory with probability one.

Because the modified Liouville equation does not depend on the modified dynamical equations, the modified Liouville equation can be solved in advance, so that the stabilizing force becomes a known function. The modified Liouville equation is solved subject to a normalization constrain and to an initial condition (an initial probability distribution) that can be specified somewhat arbitrarily. The initial condition can be, for example, a product of analysis of errors in previous dynamical computations.

An application of this formalism to Hamiltonian dynamics leads to a demonstration of a formal similarity between the stabilizing potential and a quantum potential that appears in the Madelung form of the Schroedinger equation of a single particle. Although physical meaning of the quantum potential is not completely understood, loosely speaking, it can be interpreted as a mechanism for enforcement of the uncertainty relationship that bounds the precision with which positions and velocities can be observed.

*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 on-line at www.techbriefs.com/tsp under the Mechanics category. NPO-30393.*

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##### NASA Tech Briefs Magazine

This article first appeared in the September, 2003 issue of *NASA Tech Briefs* Magazine.

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