Volume 145, number 2,3 PHYSICS LETTERS A 2 April 1990 zyxwvutsrq ONSET OF CHAOS IN AN EXTENSIBLE PENDULUM H.N. NtifiEZ-YEPEZ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Deparfamento de Fisica, Universidad Autdnoma Metropolitana-Iztapa!apa, Apartado Postal 55-534. Iztapalapa 09340 D.F.. Mexico A.L. SALAS-BRITO, C.A. VARGAS Departamento de Ciencias Brisicas. Universidad Autdnoma Metropolitana-Azcapotzalco. Apartado Postal 21- 726, Coyoacan 04000 D.F., Mexico and L. VICENTE Departamento de Fisica y Quimica Tedrica, Facultad de Quimicu. Universidad National Autdnoma de MPxico, M exico D. F., M exico Received 17 October 1989; revised manuscript received I3 December 1989; accepted for publication 22 December 1989 Communicated by A.R. Bishop A numerical study of the onset of chaos in an extensible pendulum at resonance is undertaken. We found that the system goes from regular to chaotic and back to regular behaviour as the total energy is increased. The existence of a localized region of negative curvature on the potential energy surface has been proposed to be related to this behaviour. We compare our results with the predictions of this proposal. A weight supported on a spring and free to move in a vertical plane, the so-called extensible pendu- lum, is one of the simplest realizations of a Hamil- tonian oscillatory system with two degrees of free- dom. This simple mechanical system has been studied extensively in the past as a paradigmatic example of a nonlinear system [ 1,2 1, as a classical analogue of the term producing the Fermi resonance in the in- frared and Raman spectra of CO* and similar mol- ecules [ 31, and in connection with problems in ce- lestial mechanics [ 41. In this paper we shall address the problem of chaotic motions in the extensible pendulum. In previous works particular attention has been given to the parametric instability in the system. The nonlinear coupling between the vertical spring mo- tion and the sideways pendulum motion becomes resonant when the ratio of the natural frequencies of these motions is an integer n>2 [ 21. But it seems that, despite the great interest that the study of Ham- iltonian systems has attained recently [ 5-81, very little attention has been paid to the possibility of chaotic motions in this system. The only exception we know is the work of Petrosky [ 91 on the kinetic description of the diffusion-like behaviour in the vi- cinity of the separatrix of a nonextensible pendulum with the same length. Now the growing interest in quantum chaos zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR [ 1 O-l 31 makes the classical transi- tion to chaos in this system worth of study. Besides, given the existence of a negative curvature region in the potential energy surface (PES) of the system and of a proposed stochastization scenario based on this fact, recently used to draw conclusions about the be- haviour of chaotic quantum systems [ 131, it offers the possibility of contrasting the predictions of such a scenario with numerical calculations. The system is depicted in fig. 1. It consists of a massless spring of length I= I,+mg/k and elastic 0375-9601/90/g 03.50 0 Elsevier Science Publishers B.V. (North-Holland ) 101