LIAPOUNOV EXPONENT
| Collection | International Encyclopedia of Systems and Cybernetics |
|---|---|
| Year | 2004 |
| Vol. (num.) | 2(1) |
| ID | ◀ 1963 ▶ |
| Object type | Discipline oriented, Methodology or model |
The exponent 1/tau of the function e1/tau established by LIAPOUNOV.
LIAPOUNOV's exponents express the rate of separation or convergence of a dynamic system's trajectories. “And if either of these numbers is positive, indicating a positive rate of separation of the initial points in a given direction, we say that the system is chaotic” (J. CASTI, 1990, p.100).
- “A negative exponent means limit cycles and at least a somewhat regular behavio r. A positive exponent means chaos ”(C. ZIMMER, 2000, p.84)
J. GLEICK explains: “The Lyapunov exponents in a system provide a way of measuring the conflicting effects of stretching, contracting and folding in the phase space of an attractor. They give a picture of all the properties of a system that lead to stability or instability. An exponent greater than zero means stretching — nearby points would separate. An exponent smaller than zero means contraction… For a fixed-point attractor, all the Lyapunov exponents are negative, since the direction of pull is inward toward the final steady state. An attractor in the form of a periodic orbit has one exponent of exactly zero and other exponents… negative. A strange attractor… has… at least one positive Lyapunov exponent” (1987, p.252).
In I. PRIGOGINE and I. STENGERS words: “A behavior is chaotic if trajectories starting from points as close as wanted in the phase space, move away from each other in time in an exponential way. The distance between any two points belonging to such trajectories grows thus in proportion to a function e1/tau , in which 1/tau , positive by definition for chaotic systems is Lyapunov's exponent and tau Lyapunov's time… After an evolution time long in relation to Lyapunov's time, the knowledge that we possessed from the initial state of the system has lost its relevance and does not permit us anymore to determine its trajectory” (1992, p.77).
Thus, when tau is positive, it defines irreversibility in time, in conformity with the 2nd. Law of thermodynamics.