Abstract
It was recently shown that self-adjusting systems adapt to the edge of chaos. We study the robustness of that adaptation with respect to a controlling force. We first use numerical simulations in a modified logistic map. With these, we find that, if the controlling force has a target value of the parameter that leads to periodic dynamics, the control is successful, even for very small controlling forces. We also find, however, that if the target value for the parameter leads to chaotic dynamics, the parameter resists the control and adaptation to the edge of chaos is still observed. When the controlling force is very strong, adaptation to the edge of chaos is weaker, but still present in the system. We also perform experiments with a self-adjusting Chua circuit and find the same behavior. We quantify these results with a measurement of the robustness of the adaptation as a function of the strength of the controlling force. The control used can be expressed either as a parametric control or as an additive, closed-loop control.
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