In the dynamical systems theory, "edge of chaos" is a metaphor describing the transition point between order and chaos. For example, in the context of the logistic map the edge of chaos corresponds to a = 5.569945.., with a the bifurcation parameter. But inside the large chaotic region there are a lot of islands of periodicity characterized by a period-doubling cascade to chaos. Do we have more edges of chaos? What happens for Tinkerbell map, where there is no obvious road from period-doubling bifurcation to chaos? Where is the edge of chaos?
Hi Dumitru,
If you define the logistic map as x_n+1 = a x_n (1-x_n), then from 3
Mr. Sanjuan , thank you so much, but my question still „survive”. I found the syntagm „edge of chaos” in a paper of Gottwald and Melbourne, entitled „Comment on the reliability of 0 – 1 test”, concerning the performances of their test. They applied the test for logistic map in a neighborhood of the Feigenbaum point a = 3.5699.... Because the test do not work properly in this area, I want to check how other tools for chaos, excepting Lyapunov exponent, behave for the same problem (more precisely, in the transition area between order and chaos). I am not so sure that the construction „edge of chaos” is accepted in the literature for the above-mentioned situation.
Kind regards,
D. Deleanu.
The "edge of chaos" term refers to the front of the process in which a nonlinear dynamic system continuously returns to the state before catastrophe constantly experiencing of it.
Most likely, we wrongly perceive these chaotic regions. All behavior of system is ordered, but we not always are able to fully trace these regularities within nonlinear system.
I always took the "edge of chaos" to mean a weakly chaotic system (one with a relatively small Lyapunov exponent). It is not a well-defined mathematical concept. Many complex systems in nature appear to spontaneously self-organize to such a state since it provides an optimal level of fitness. The Solar System is probably on the edge of chaos, as is your brain. In that sense, the logistic map has infinitely many such regions, all but a few of which are confined to a tiny range of the parameter. A similar thing occurs in the Tinkerbell map and to most systems that exhibit a period doubling route to chaos. However, the term "edge of chaos" is usually applied to high-dimensional (complex) systems rather than to low-dimensional maps.
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