In this paper is presented a relationship between the synchronization and the topological entropy. We obtain the values for the coupling parameter, in terms of the topological entropy, to achieve synchronization of two unidirectional and bidirectional coupled piecewise linear maps. In addition, we prove a result that relates the synchronizability of two m-modal maps with the synchronizability of two conjugated piecewise linear maps. An application to the unidirectional and bidirectional coupled identical chaotic Duffing equations is given. We discuss the complete synchronization of two identical double-well Duffing oscillators, from the point of view of symbolic dynamics. Working with Poincaré cross-sections and the return maps associated, the synchronization of the two oscillators, in terms of the coupling strength, is characterized.

Dedicated to the memory of J. Sousa Ramos a b s t r a c tThe purpose of this paper is to study the symmetry effect on the kneading theory for symmetric unimodal maps and for symmetric bimodal maps. We obtain some properties about the kneading determinant for these maps, that implies some simplifications in the usual formula to compute, explicitly, the topological entropy. As an application, we study the chaotic behaviour of the two-well Duffing equation with forcing.Ó 2009 Elsevier Ltd. All rights reserved. Motivation and introductionThe Duffing equation has been used to model the nonlinear dynamics of special types of mechanical and electrical systems. This differential equation has been named after the studies of Duffing in 1918 [1], has a cubic nonlinearity and describes an oscillator. It is the simplest oscillator displaying catastrophic jumps of amplitude and phase when the frequency of the forcing term is taken as a gradually changing parameter. It has drawn extensive attention due to the richness of its chaotic behaviour with a variety of interesting bifurcations, torus and Arnold's tongues. The main applications have been in electronics, but it can also have applications in mechanics and in biology. For example, the brain is full of oscillators at micro and macro level [15]. There are applications in neurology, ecology, secure communications, cryptography, chaotic synchronization, and so on. Due to the rich behaviour of these equations, recently there has been also several studies on the synchronization of two coupled Duffing equations [13,14]. The most general forced form of the Duffing equation is In [2], Xie et al. used symbolic dynamics to study the behaviour of chaotic attractors and to analyze different periodic windows inside a closed bifurcation region in the parameter plane. Symbolic dynamics is a rigorous tool to understand chaotic motions in dynamical systems.In this work, we will use techniques of kneading theory due essentially to Milnor and Thurston [8] and Sousa Ramos [5][6][7], applying these techniques to the study of the chaotic behaviour. We will use kneading theory to evaluate the topological entropy, which measures the chaoticity of the system. Generally, the graphic of the return map is a very complicated set of points. Therefore, in order to be able to apply these techniques, we show, in Section 2, regions in the parameter plane where 0960-0779/$ -see front matter Ó

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In this work we discuss the complete synchronization of two identical double-well Duffing oscillators unidirectionally coupled, from the point of view of symbolic dynamics. Working with Poincaré cross-sections and the return maps associated, the synchronization of the two oscillators, in terms of the coupling strength, is characterized. We obtained analytically the threshold value of the coupling parameter for the synchronization of two unimodal and two bimodal piecewise linear maps, which by semi-conjugacy, under certain conditions, gives us information about the synchronization of the Duffing oscillators.