Conserved quantities and adaptation to the edge of chaos.

Certain dynamical systems, such as the shift map and the logistic map, have an edge of chaos in their parameter spaces. On one side of this edge, the dynamics are chaotic for many parameter values, on the other side of the edge they are periodic. We find that discrete-time dynamical systems with wavelet filtered feedback from the dynamical variable to the parameters are attracted to a narrow parameter range near the edge of chaos, the periodic boundary regime. We show that the migration from the chaotic regime to the periodic boundary regime can be attributed to a conserved quantity, and find that such adaptation to the edge of chaos is accompanied by a depopulation of the chaotic regime. We use this conserved quantity to determine the location of the periodic boundary regime and show that its size is proportional to the size of the feedback. Further, we compute the dynamics of the probability density for the parameter for a specific example.