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Dynamics of Quasi-Stable Dissipative Systems

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Abstract

This chapter collects basic definitions, notions and also the simplest illustrating statements from the general theory of dynamical systems. We also describe all possible dynamical scenarios in 1D and 2D continuous systems and, by means of examples, discuss the principal bifurcation pictures. Our intention in the latter materials is to give the reader some feeling on what kind of dynamics can arise for low-dimensional (1 or 2) continuous time evolutions.

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Notes

  1. 1.

    We recommend omitting of this remark at the first reading.

  2. 2.

    The Schauder theorem (see, e.g., Zeidler [231, Volume I, Chapter 2]) states that any continuous mapping from a convex compact set in a Banach space into itself has a fixed point.

  3. 3.

    If the space X is locally compact, then the second condition can be omitted. See, e.g., Sibirsky [212, Theorem 2.8].

  4. 4.

    There are examples showing that the property \( \mathop{\mathrm{dist}}\nolimits (v,S_{t}w) \rightarrow 0 \) as \( t \rightarrow +\infty \) does not imply the Lyapunov stability. See, e.g., Teschl [217, p. 168] and also Example 1.9.6 with μ = 0 in Section 1.9.

  5. 5.

    This means that \( \vert x(t)\vert \rightarrow \infty \) as t → T ∗ from the left.

  6. 6.

    In the Hamiltonian case (μ = 0) every nonzero trajectory γ starting in Δ is periodic, i.e., \( \gamma = \omega (\gamma ) = \alpha (\gamma ) \).

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Chueshov, I. (2015). Basic Concepts. In: Dynamics of Quasi-Stable Dissipative Systems. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-22903-4_1

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