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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
L.F.P. Franca, M.A. Savi, Evaluating noise sensitivity on the time series determination of Lyapunov exponents applied to the nonlinear pendulum. Shock Vib. 10(1), 37–50 (2003)
J. Gleick, Caos (Heinemann, London, 1987)
C. Grebogi, E. Ott, S. Pelikan, J. Yorke, Strange attractors that are not chaotic. Phys. D 13, 261–268 (1984)
J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, 1983)
R.C. Hilborn, Chaos and Nonlinear Dynamics (Oxford Press, Oxford, 1994)
M.W. Hirsch, S. Smale, R.L. Devaney, Differential Equations, Dynamical Systems & An Introduction to Chaos (Elsevier, Amsterdam, 2004)
H. Kantz, T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, Cambridge, 1997)
T. Kapitaniak, Chaotic Oscillations in Mechanical Systems (Manchester University Press, Manchester, 1991)
E. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)
L.G. Machado, M.A. Savi, P.M.C.L. Pacheco, Nonlinear dynamics and chaos in coupled shape memory oscillators. Int. J. Solids Struct. 40(19), 5139–5156 (2003)
L.G. Machado, M.A. Savi, P.M.C.L. Pacheco, Bifurcations and crises in a shape memory oscillator. Shock Vib. 11(2), 67–80 (2004)
F.C. Moon, Chaotic and Fractal Dynamics (Wiley, New York, 1992)
L.H.A. Monteiro, Dynamical Systems. Editora Livraria da FÃsica (in Portuguese) (2002)
T. Mullin, The Nature of Chaos (Oxford University Press, Oxford, 1993)
A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations (Wiley, New York, 1979)
E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993)
A. Paiva, M.A. Savi, An overview of constitutive models for shape memory alloys. Math. Probl. Eng. 2006, 1–30 (2006), ID56876
T.S. Parker, L.O. Chua, Practical Numerical Algorithms for Chaotic Systems (Springer, New York, 1989)
M.A. Savi, Nonlinear Dynamics and Chaos. Editora E-papers (in Portuguese) (2006)
M.A. Savi, Rhythms of Nature. Editora E-papers (in Portuguese) (2014)
M.A. Savi, Nonlinear dynamics and chaos in shape memory alloy systems. Int. J. Non Linear Mech. 70, 2–19 (2015). doi:10.1016/j.ijnonlinmec.2014.06.001
M.A. Savi, A.M.B. Braga, Chaotic vibration of an oscillator with shape memory. J. Braz. Soc. Mech. Sci. 15(1), 1–20 (1993)
M.A. Savi, P.M.C.L. Pacheco, Chaos and hyperchaos in shape memory systems. Int. J. Bifurcat. Chaos 12(3), 645–657 (2002)
I. Stewart, Does God Play Dice? Jorge Zahar Editor (in Portuguese) (1991)
S.H. Strogatz, Nonlinear Dynamics and Chaos (Perseus, Cambridge, 1994)
J.M.T. Thompsom, H.B. Stewart, Nonlinear Dynamics and Chaos (Wiley, Chichester, 1986)
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, New York, 1990)
A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Determining Lyapunov exponents from a time series. Phys. D 16, 285–317 (1985)
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-29982-2_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-29981-5
Online ISBN: 978-3-319-29982-2
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)