Abstract
This chapter presents an overview of nonlinear dynamics and chaos. It starts with a background revision of dynamical systems. Concepts of equilibrium points, linearization, stability, and Poincaré maps are treated. Afterward, chaotic dynamics is explored. Horseshoe transformation is discussed in order to define the main aspects of chaos. Fractal characteristics are presented and discussed. Routes to chaos are investigated showing some definitions of bifurcation. Lyapunov exponents are defined presenting a diagnostic tool for chaos. The main concepts and tools are then presented by considering a case study related to a shape memory alloy system. Single and two degrees of freedom systems are treated using a polynomial constitutive model to describe the restitution forces.
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Acknowledgements
The author would also like to acknowledge the support of the Brazilian Research Agencies CNPq, CAPES and FAPERJ and through the INCT-EIE (National Institute of Science and Technology—Smart Structures in Engineering) the CNPq and FAPEMIG. The Air Force Office of Scientific Research (AFOSR) is also acknowledged.
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Savi, M.A. (2016). Nonlinear Dynamics and Chaos. In: Lopes Junior, V., Steffen Jr., V., Savi, M. (eds) Dynamics of Smart Systems and Structures. Springer, Cham. https://doi.org/10.1007/978-3-319-29982-2_5
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DOI: https://doi.org/10.1007/978-3-319-29982-2_5
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