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Chaos Theory

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Lectures on Nonlinear Dynamics

Part of the book series: Understanding Complex Systems ((UCS))

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Abstract

This chapter presents an overview of chaos theory. It starts from the background of dynamical systems, presenting the mathematical representation and the concept of stability. Afterward, chaotic dynamics is explored from the horseshoe transformation, establishing that it is a consequence of the contraction-expansion-fold process. The main aspects of chaotic behavior are then discussed defining chaotic and fractal attractors. Routes to chaos are investigated showing some definitions of bifurcation, treating local and global bifurcations. Lyapunov exponents are defined in order to present a diagnostic tool for chaos.

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Acknowledgements

The author would also like to acknowledge the support of the Brazilian Research Agencies CNPq, CAPES and FAPERJ. The help of Guilherme V. Rodrigues with the Figures is also acknowledged.

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Authors and Affiliations

  1. Center for Nonlinear Mechanics—Department of Mechanical Engineering—COPPE—Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Marcelo A. Savi

Authors
  1. Marcelo A. Savi
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Correspondence to Marcelo A. Savi .

Editor information

Editors and Affiliations

  1. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    José Roberto Castilho Piqueira

  2. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    Carlos Eduardo Nigro Mazzilli

  3. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    Celso Pupo Pesce

  4. Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil

    Guilherme Rosa Franzini

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Savi, M.A. (2024). Chaos Theory. In: Castilho Piqueira, J.R., Nigro Mazzilli, C.E., Pesce, C.P., Franzini, G.R. (eds) Lectures on Nonlinear Dynamics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-031-45101-0_10

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  • DOI: https://doi.org/10.1007/978-3-031-45101-0_10

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-45100-3

  • Online ISBN: 978-3-031-45101-0

  • eBook Packages: EngineeringEngineering (R0)

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