"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.

- Period-doubling critical behavior.

Renormalization group analysis.*A.P.Kuznetsov, S.P.Kuznetsov, I.R.Sataev. A variety of period-doubling universality classes in multi-parameter analysis of transition to chaos. Physica***D109**, 1997, 91-112. - Critical behavior associated with quasiperiodicity at the golden-mean frequency
ratio.

Renormalization group analysis. - Critical behavior of conservative systems.
- Critical behavior of complex analytic iterative maps.

of theoretical nonlinear

dynamics

Хостинг от uCoz