Adaptation to the Edge of Chaos and Critical Scaling in Self-adjusting Dynamical Systems

We present a mechanism for adaptation in dynamical systems. Systems which have this mechanism are called called self-adjusting systems. The control parameters in a self-adjusting system are slowly varying, rather than constant. The dynamics of the control parameters are governed by a lowpass filtered feedback from the dynamical variables. We apply this model to several systems, numerically, analytically, and experimentally, and examine the behavior of the control parameters. We observe a high probability of finding the parameter at the boundary between periodicity and chaos. We therefore find that self-adjusting systems adapt to the edge of chaos. In addition, we find that noise in the system drives the parameter away from the edge of chaos on very long timescales so that chaos is suppressed in the system. We show that, with the presence of noise, the parameter can re-enter the chaotic regime. This is called a chaotic outbreak in the system and we find that the distribution of outbreaks is a power-law with the duration of the outbreak. We then study the robustness of adaptation to the edge of chaos by examining the effect of a control force being applied to the parameter. We find the behavior to be very robust, except for very large control forces. Finally, we look at systems of coupled maps and show that adaptation to the edge of chaos occurs in systems of higher dimensions, as well.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Physics

Year dc:date

2012

Author dc:creator

• Melby, Paul Christian

Contributors dc:contributor

• Hubler, Alfred W.

• dynamical systems
• self-adjusting systems
• edge of chaos

Statement dc:rights

• ©2002 Melby

Language dc:language

en

Identifier

4591801

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/31343