IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design
Nonlinear dynamics has been enjoying a vast development for nearly four decades resulting in a range of well established theory, with the potential to significantly enhance performance, effectiveness, reliability and safety of physical systems as well as
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Abstract In this paper we study the dynamics of a large ring of unidirectionally coupled autonomous Duffing oscillators. We paid our attention to the role of unstable periodic solutions for the appearance of spatio-temporal structures and the Eckhaus effect. We provide an explanation for the fast transition to chaos showing that the parameter interval, where the transition from a stable periodic state to chaos occurs, scales like the inverse square of the number of oscillators in the ring. Keywords Coupled oscillators • Duffing • Spatio-temporal structures • Transition to chaos
1 Introduction In the last decade, one can observe a growing interest in the studies of the networks of coupled oscillators (Strogatz 2001). The knowledge of the dynamical behavior of such systems can lead to the understanding of fundamental dynamical features of physical, biological, engineering or economical coupled
P. Perlikowski • A. Stefanski • T. Kapitaniak () Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland e-mail: [email protected]; [email protected]; [email protected] S. Yanchuk Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany e-mail: [email protected] M. Wolfrum Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany e-mail: [email protected] M. Wiercigroch and G. Rega (eds.), IUTAM Symposium on Nonlinear Dynamics for Advanced Technologies and Engineering Design, IUTAM Bookseries (closed) 32, DOI 10.1007/978-94-007-5742-4 5, © Springer ScienceCBusiness Media Dordrecht 2013
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systems (Zillmer et al. 2006; Mosekilde et al. 2002). The most important question is how the specific properties of the individual behavior and the coupling architecture can give rise to different types of collective behavior (Pikovsky et al. 2001). The other problem, which is discussed here, is connected with the structure of the attractors in higher dimensional phase space and, in particular, with the occurrence of the hyperchaotic attractors. When a map is at least two-dimensional or a flow is at least four-dimensional, its evolution can take place on a hyperchaotic attractor. Such attractors are characterized by at least two positive Lyapunov exponents for typical trajectories on them. The first example of such a system with hyperchaotic attractor was presented by Rossler (1976) for a chemical reaction model. Later, hyperchaotic attractors have been found in electronic circuits and other chemical reactions (Baier and Klein 1991; Peinke et al. 1992). In the works Kapitaniak et al. (1994) it was shown that by a weakly coupling of N chaotic systems it is possible to obtain a hyperchaotic attractor with N positive Lyapunov exponents. The transition from chaos to hyperchaos has been studied in Kapitaniak and Steeb (1991), Kapitaniak (1993), and Harrison and Lai (1999). It was shown that at this transition the attractor’s dimension and the second Lyapunov exp
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