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Golden-mean RG analysis

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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-1/F_m, where F_m - are the Fibonacci numbers: F₀=0, F₁=1, F_m+1= F_m + F_m-1. The idea is to consider sequence of evilution operators over time intervals given by the Fibonacci numbers.

Let and be evolution operators of some 1D map over time intervals F_m and F_m-1, respectively. According to the definition of the Fibonacci numbers, . At each m-th step of the construction let us introduce normaluzation of the dynamical variable by factor , then the evolution operator for F_m time steps will be represented by the function . Recurrent equation for subsequent functions g_m reads

(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(x), so a solution may be searched as . It leads to an eigenvalue problem. There are two relevant eigenvalues larger than 1 in modulus, and they represent the scaling factors on the parameter plane.

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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