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period-doubling RG analysis

golden-mean RG analysis

Let we have an evolution operator of some dynamical system
on a definite time interval. Applying this operator several times one can construct
an evolution operator for larger interval. Let us suppose that it is possible
to adjust parameters of the original system in such way that the new operator
may be reduced by scale variable changes to the old one. The procedure is called
*RG transformation*, and the adjusted parameters define location of the
*critical point*. RG transformation may be applied again and again to obtain
a sequence of the evolution operators for larger and larger time intervals.

At the critical point the rescaled long-time evolution operators are determined
by structure of the RG transformation rather than by the nature of the original
dynamical system (*universality*). As the operators produced by multiple
application of the RG procedure coincide (up to the scale change), the system
manifests similar dynamics on different time scales, the property called *scaling*.

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Saratov groupof theoretical nonlinear

dynamics

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