Abstract
Dynamical systems are generically not isolated but interact with the embedding environment and one speaks of noise whenever the impact of the dynamics of the environment cannot be predicted. The dynamical flow slows down when energy is dissipated to the environment, approaching attracting states which may be regular, such as fixpoints or limit cycle, or irregular, such as chaotic attractors. Adaptive systems alternate between phases when they dissipate energy and times when energy is taken up from the environment, with the steady state being characterized by a balance between these two opposing processes. In this chapter an introduction to adaptive, dissipative and stochastic systems will be given together with important examples from the real of noise controlled dynamics, like diffusion, random walks and stochastic escape and resonance.
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Notes
- 1.
The term isocline stands for “equal slope” in ancient Greek.
- 2.
Note: \(\int e^{-x^{2}/a }\mathrm{d}x = \sqrt{a\pi }\) and \(\lim _{a\rightarrow 0}\exp (-x^{2}/a)/\sqrt{a\pi } =\delta (x)\).
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Gros, C. (2015). Dissipation, Noise and Adaptive Systems. In: Complex and Adaptive Dynamical Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-16265-2_3
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