Abstract
We investigate the average response to small external perturbations for discrete dynamical systems with chaotic attractors. The average linear response satisfies a fluctuation theorem, and in general diverges exponentially in the long-time limitt→∞. It vanishes identically for allt>0 only in a number of special cases including the logistic model with bifurcation parameter α=4. The nonlinear response turns out to be crucial. Its average is analyzed for a time-localized (pulse) perturbation. Near the onset of chaos it exhibits universal scaling behaviour expressed by two critical exponents. For static perturbations the resulting dynamics is extremely sensitive to the perturbation strength.
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Work supported by the Swiss National Science Foundation
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Geisel, T., Heldstab, J. & Thomas, H. Linear and nonlinear response of discrete dynamical systems II: Chaotic attractors. Z. Physik B - Condensed Matter 55, 165–178 (1984). https://doi.org/10.1007/BF01420569
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DOI: https://doi.org/10.1007/BF01420569