Critical dynamics at the onset of chaos and renormalization group.
"Alphabet" of critical points

Scenario of transition to chaos means a sequence of bifurcations observed under slow variation of a control parameter in dynamical system on a way feom regular to chaotic behavior, for example, via period-doubling cascade, quasiperiodicity, intermittency. After the works of Feigenbaum it is clear that the dynamics at the chaos border often manifests scaling regularities associated with definite universality classes, or types of critical behavior. The first known universality class was discovered by Feigenbaum, latter other types of criticality were found and studied. The theoretical tool for analysis of the critical behavior is the renormalization group method (RG).

Generalizing the idea of "scenario" for a multi-parameter case, we should imagine some configuration in the parameter space, which includes domains of regular and chaotic dynamics. Critical behavior of a certain type may occur at some surfaces separating chaos and order, at curves bounding these surfaces, at points terminating these curves. Respectively, we speak on criticality of codimension one, two, three.

On this site we present a collection of types of criticality. Some of them are known from literature, others are revealed in our group in a course of the program of search of universality classes arising in multi-parameter analysis of transition to chaos in nonlinear systems. Note that in quasiperiodically forced systems critical behavior may be associated as well with a birth of strange nonchaotic attractors.

For each type of criticality we explain shortly in which situation does it occur, present basic results of RG analysis, including the universal scaling constants, a canonical model, which is the simplest representative of the universality class, charts of dynamical regimes in a vicinity of the critical point, illustrations of scaling in phase space and parameter space.

Saratov group
of theoretical nonlinear
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