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Bifurcations and Chaos in Dynamical Systems

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Complex and Adaptive Dynamical Systems

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Abstract

Complex systems theory deals with dynamical systems containing often large numbers of variables. It extends dynamical systems theory, which treats dynamical systems containing a few variables. A good understanding of dynamical systems theory is therefore a prerequisite when studying complex systems.

In this chapter we introduce core concepts, like attractors and Lyapunov exponents, bifurcations, and deterministic chaos from the realm of dynamical systems theory. An introduction to catastrophe theory will be provided together with the notion of rate-induced tipping and colliding attractors.

Most of the chapter will be devoted to ordinary differential equations and maps, the traditional focus of dynamical systems theory, venturing however towards the end into the intricacies of time delay dynamical systems.

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Notes

  1. 1.

    Liouville’s theorem will be discussed in Sect. 3.1.1 of Chap. 3.

  2. 2.

    Invariant manifolds are touched upon further in in Sect. 3.2.1 of Chap. 3.

  3. 3.

    The Landau-Ginzburg theory of phase transitions will be treated at length in the context of self-organized criticality, in Sect. 6.1. of Chap. 6.

  4. 4.

    In complex plane, with \(z=x+iy\), Eq. (2.28) takes the form of a Stuart–Landau oscillator, \(\dot {z}=i\omega z+d(\varGamma -|z|{ }^2)z\), with \(\omega =1\).

  5. 5.

    Per definition, adaption implies that a system may both dissipate energy and increase its own reservoir, as discussed further in Sect. 3.2 of Chap. 3.

  6. 6.

    This formulation parallels the Kuramoto model, which is treated detail in Sect. 9.1 of Chap. 9.

  7. 7.

    See Chap. 8.

  8. 8.

    More realistic car-following models are discussed in Sect. 4.4 of Chap. 4.

References

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Authors and Affiliations

  1. Institute for Theoretical Physics, Goethe University Frankfurt, Frankfurt/Main, Germany

    Claudius Gros

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  1. Claudius Gros
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Gros, C. (2024). Bifurcations and Chaos in Dynamical Systems. In: Complex and Adaptive Dynamical Systems. Springer, Cham. https://doi.org/10.1007/978-3-031-55076-8_2

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