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renormalization group method

period-doubling RG analysis

Let us consider quasiperiodic dynamics and assume that ratio of two basic frequencies
is equal to the golden mean, the irrational number .
As known, the convergent of continued fraction is defined as *F _{m}*

Let and
be evolution
operators of some 1D map over time intervals *F _{m}* and

(1)

With appropriate selection of the scaling factor () the fixed point of this equation is represented by a universal function with cubic inflection point at the origin. It is associated with the critical behavior of type GM discovered in works of Shenker, Feigenbaum-Kadanoff-Shenker, and Rand-Ostlund-Sethna-Siggia. It occurs e.g. in circle map at the threshold of destruction of quasiperiodic regime with the golden-mean rotation number. The perturbation of the fixed point gives rise to an equation

.(2)

It is linear in respect to *h _{m}*(

Let us turn to a generalization and assume that an instantaneous state of a
system is defined by two variables (*x*, *u*), and evolution on *F _{m
}*time steps is described by equations ,
. In terms
of renormalized functions
we obtain relation

(3)

that may be regarded as a generalization of Eq.(1). Two distinct fixed points of this equation are responsible for the criticality of type TF and TCT, and period-3 cycle for the type TDT. The mentioned types of critical behavior occur in quasiperiodically forced 1D maps and are found in our joint researches with Potsdam University group (Pikovsky, Feudel, Neumann).

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Saratov groupof theoretical nonlinear

dynamics

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