Abstract
Most dynamical systems are not isolated, but interacting with an embedding environment that may add stochastic components to the evolution equations. The internal dynamics slows down when energy is dissipated to the outside world, approaching attracting states which may be regular, such as fixpoints or limit cycle, or irregular, such as chaotic attractors. Adaptive systems alternate between phases of energy dissipation and uptake, until a balance between these two opposing processes is achieved.
In this chapter an introduction to adaptive, dissipative and stochastic systems will be given together with important examples from the realm of noise controlled dynamics, like diffusion, random walks and stochastic escape and resonance. We will discuss to which extent chaos, a regular guest of adaptive systems, may remain predictable.
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Notes
- 1.
For references, Chap. 2 is devoted to determinstic dynamics.
- 2.
The following derivations are informative, but somewhat advanced. In case, the reader may skip directly to the result, Eq. (3.18).
- 3.
The harmonic oscillator is resonant only when the frequency of the perturbation matches the internal frequency. Non-harmonic oscillators may however unstable against rational frequency ratios, as discussed in Sect. 2.1 of Chap. 2 in the context of the KAM theorem, with regard to the gaps in the Saturn rings.
- 4.
The term isocline stands for “equal slope” in ancient Greek.
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- 8.
Note that \(\int \mathrm {e}^{-x^2/a}\mathrm {d} x= \sqrt {a\pi }\), together with \(\lim _{a\to 0} \exp (-x^2/a)/\sqrt {a\pi }=\delta (x)\).
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The Lévy distribution \(\sqrt {c/(2\pi )}\exp (-\frac {c}{2t})/t^{3/2}\) is normalized on the interval \(t\in [0,\infty ]\).
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Gros, C. (2024). Dissipation, Noise and Adaptive Systems. In: Complex and Adaptive Dynamical Systems. Springer, Cham. https://doi.org/10.1007/978-3-031-55076-8_3
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DOI: https://doi.org/10.1007/978-3-031-55076-8_3
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