 
Summary: J. Phys. A: Math. Gen. 32 (1999) 431442. Printed in the UK PII: S03054470(99)951119
On the relation between the maximal LCN and the width of
the stochastic layer in a driven pendulum
K Tsiganis, A Anastasiadis and H Varvoglis
Section of Astrophysics, Astronomy and Mechanics, Department of Physics, University of
Thessaloniki, GR54006 Thessaloniki, Greece
Received 15 June 1998, in final form 21 September 1998
Abstract. We examine whether the macroscopically measured diffusion rate in the chaotic region
ofatimeperturbedclassicalpendulumdependsonthevalueofthemaximalLyapunovcharacteristic
number, . In this respect we calculate the functions (l), w(l), ( ) and w( ), where l denotes the
physical length of the pendulum, the strength of the perturbation and w the width of the stochastic
layer around the separatrix. We find that all these functions follow power laws. In particular, both
(l) and w(l) scale as the Lyapunov exponent and the width of the resonance of the unperturbed
system, i.e. as l1/2 and l3/2, respectively. It follows that the width of the stochastic layer is
proportional to 3 so that, for sufficiently small values of l, stochastic diffusion is restricted to a
thin layer and, therefore, practically does not depend on .
1. Introduction
In a series of papers by Lecar and coworkers (Soper et al 1990, Lecar et al 1992a, b, Franklin
et al 1993, Murison et al 1994), numerical evidence was presented on the possible existence
of a law pertaining to the motion of asteroids in the outer asteroidal belt. According to these
