Onset of spatiotemporal chaos in damped anharmonically driven sine-Gordon systems R. Chaco ´n * Departamento de Fı ´sica Aplicada, Escuela de Ingenierı ´as Industriales, Universidad de Extremadura, E-06071, Badajoz, Spain Accepted 25 September 2006 Communicated by Prof. Mohamed Saladen El Naschie Abstract The onset is demonstrated of spatiotemporal chaos arising from the competition between sine-Gordon-breather and kink-antikink-pair solitons by reshaping of a sinusoidal force. After introducing soliton collective coordinates, Melni- kov’s method is applied to the resulting effective equation of motion to deduce the parameter-space regions of the ac force where chaotic instabilities are induced. The analysis reveals that the chaotic threshold amplitude when altering solely the pulse shape presents a minimum when the transmitted impulse is maximal, the remaining parameters being held constant. The universality of the results is shown by studying the behaviour of the Lyapunov exponent from a simple recursion relation which models an unstable limit cycle. Computer simulations of the sine-Gordon system show good agreement with the theoretical predictions. Additionally, it is found that the reshaping-induced order M chaos route is especially rich, including transitions from a two-breather state to a spatially uniform, periodic oscillatory state. The appearance of this spatially uniform state is explained by means of geometrical resonance analysis. Ó 2006 Elsevier Ltd. All rights reserved. 1. Introduction Dissipative nonautonomous dynamical systems, subjected to forces given by a periodic string of pulses, describe an enormous variety of physical, biological, chemical, and neuronal phenomena [1–3], to cite only a few. Real-world pulses exhibit a great diversity of waveforms as well as many complex transitions between them as the system’s parameters change. In general, to consider periodic strings of pulses with arbitrary waveform implies extending the amplitude-period parameter space to include the parameters, a i , that control the pulse waveform. In physical terms this means that, for fixed period and amplitude, the parameters a i are responsible for the temporal rate at which energy is transferred from the excitation mechanism to the system. Since the choice of a specific mathematical function to model a given real-world pulse determines, to a great extent, which range of phenomena it could suitably characterize, one would like to use a pulse function which generates a great diversity of waveforms with few parameters a i . Note that, in the general case, the param- eters a i are the (normalized) Fourier coefficients of the periodic function. Thus, one is forced to take into account an infin- ity of additional parameters, the resulting increase of the system’s complexity being highly discouraging. In this regard, 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.09.084 * Fax: +34 2423 6766. E-mail address: rchacon@unex.es Available online at www.sciencedirect.com Chaos, Solitons and Fractals 37 (2008) 902–911 www.elsevier.com/locate/chaos