LYAPUNOV EXPONENT CALCULATION Andr´es L. Granados M. Department of Mechanics SIMON BOLIVAR UNIVERSITY Valle de Sartenejas, Estado Miranda Apdo.89000, Caracas 1080A, Venezuela. e-mail: agrana@usb.ve Abstract This paper introduces a relative simple method to calculate the Lyapunov exponent for a system of ordinary differential equations. It is based on the comparison of Taylor series for the Lyapunov Exponent definition and the Taylor series for the evolution function. Derivatives of different orders for this function are evaluated with the Fa` a di Bruno’s formula. PRELIMINARS Functions Let f (y): R N → R N a function that defines a system of ordinary differential equations of first order and y(x): R → R N the path of evolution of this dynamical system with respect to the variable x. System of Equations The type of system of first order differential equations we are going to deal with is holonomic (do not depends explicitly on x) and also it is an initial value problem started at x = x o dy dx = f (y) y o = y(x o ) (1) All the initial conditions are at this unique point. Thus (1) conforms a dynamical system in “time” x and “space” y, with “velocity” f . LYAPUNOV EXPONENT Definition The Lyapunov characteristic exponent of a dynamical system (1) is a quantity that charectizes the rate of separation of infinitesimal close trajectories to a stationary solution. Quantitatively, two trajectories in phase space with initial separation δy o diverge (provided the divergence can be treated within the linearized approximation) at a rate given by [6] δy ≈ e Λ(x−xo) .δy o (2) This analysis is justified because, almost locally, the system of equation have an integration factor that is an exponential y = y o e −λ(x−xo) d dx (y e λ(x−xo) )= 0 (3) with a particular exponent λ. 1