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Birth of a new class of period-doubling scaling behavior as a result of bifurcation in the renormalization equation. (English) Zbl 1140.82023

The authors study a bifurcation in the renormalization group (RG) equation describing the period-doubling route to chaos in two-dimensional map. This bifurcation gives rise to a cycle of period-2 developing flow a fixed point of the RG equation. It generates a new universal type of critical behavior at the border of chaos. Two-dimensional generalization of the Feigenbaum-Cvitanovic equation is described. Critical behavior is considered, which may be regarded as an analog of the trivial fixed point in the phase transition theory. The model system consists of two elements represented by one-dimensional maps with unidirectional coupling. It is assumed that the first map relates to the class manifesting the Feigenbaum period-doubling universality at the chaos threshold. The second map is supposed to be a generator of an integral characteristic: it produces a sum of a function of the dynamical variable of the first subsystem in the course of the dynamics. Then the RG transformation is modified to account the degree of the extremum point in the first map as a continuous parameter rather that a constant \(\kappa = 2\). It appears that the fixed point of the RG equation undergoes a bifurcation at \(\kappa _c = 1,984396\). As \(\kappa \) exceeds this value the bifurcation gives rise to a new period-2 stationary solution. Further the critical behavior associated with this solution is analyzed and find out that at \(\kappa = 2\) it corresponds to the universality class defected in RG irreversible maps and denoted as the C-type critical behavior. Asymptotic expansion of the period-2 solution in powers of \((\kappa - \kappa _c )^{1/2}\) is obtained. Bifurcation approach to the analyses of RG equation solutions is presented as analog of the \(\varepsilon \)-expansion method in phase transitions theory.

MSC:

82C28 Dynamic renormalization group methods applied to problems in time-dependent statistical mechanics
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
82C27 Dynamic critical phenomena in statistical mechanics
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