Physics Letters A 311 (2003) 180–191 www.elsevier.com/locate/pla Wavelet statistical complexity analysis of the classical limit A.M. Kowalski a,b,c , M.T. Martin a , A. Plastino a , A.N. Proto b,c,∗ , O.A. Rosso d a La Plata Physics Institute, National University La Plata and Argentina National Research Council (CONICET), Casilla de Correo 727, (1900) La Plata, Argentina b Departamento de Computación, Facultad de Ingeniería, Universidad de Buenos Aires (UBA), Buenos Aires, Argentina c Buenos Aires Scientific Research Commission (CIC), Buenos Aires, Argentina d Instituto de Cálculo, Facultad de Ciencias Exactas y Naturales (UBA), Ciudad Universitaria, Pabellón II, 1428 Buenos Aires, Argentina Received 11 August 2002; received in revised form 29 November 2002; accepted 17 March 2003 Communicated by A.R. Bishop Abstract We introduce the notion of wavelet statistical complexity (WSC) and investigate the classical limit of the non-linear dynamics of two interacting harmonic oscillators. It is shown that a rather special relationship between entropy and chaos ensues that, using the WSC tool, sheds some light on the intricacies of the classical–quantum transition. The associated transition region is seen to consists of two sub-zones, each with quite different properties. In one of them, a solid–gas like (smooth) transition seems to take place.  2003 Elsevier Science B.V. All rights reserved. PACS: 03.65.Sq; 05.45.Mt Keywords: Semi-classical theories; Quantum chaos 1. Introduction During the last decade much effort has been ex- pended, both theoretically and experimentally, to shed light upon the transition from quantum to classical behavior, that has puzzled physicists for decades [1]. The emergence of classical chaos from quantum me- chanics is probably the key to unravel the quantum to classical transition conundrum [2]. In this respect, a rigorous quantifier of “quantum chaos” constitutes an essential ingredient. Recently, we have presented a * Corresponding author. E-mail address: aproto@conae.gov.ar (A.N. Proto). method to follow this transition [3–5], in a controlled way, by recourse to a semi-classical model that ex- hibits the semi-quantum chaos phenomenon (see, for instance, [3] and references therein). Our control para- meter is an invariant of the motion, I , closely related to the Uncertainty Principle. The invariant discrimi- nates between chaotic and non-chaotic orbits. The Li- ouville equation governs the whole transition process at all times and the limit ¯ h → 0 is reached in smooth fashion, as demonstrated in [3–5]. Although the sta- tus of the quantum–classical correspondence for non- linear dynamical systems is a bit unclear and maybe even controversial [6], we will consider here a case in which no ambiguity exists. 0375-9601/03/$ – see front matter  2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00470-5