arXiv subjects
Publications and source records attributed to O. B. Isaeva.
The model system manifesting phenomena peculiar to complex analytic maps is offered. The system is a non-autonomous ring cavity with nonlinear elements and filters,
A feasible model is introduced that manifests phenomena intrinsic to iterative complex analytic maps (such as the Mandelbrot set and Julia sets). The system is composed of two coupled alternately excited oscillators (or self-sustained oscillators). The idea is based on a turn-by-turn transfer of the excitation from one subsystem to another (S.P.~Kuznetsov, Phys.~Rev.~Lett. \bf 95 \rm, 2005, 144101) accompanied with appropriate nonlinear transformation of the complex amplitude of the oscillations in the course of the process. Analytic and numerical studies are performed. Special attention is paid to an analysis of the violation of the applicability of the slow amplitude method with the decrease in the ratio of the period of the excitation transfer to the basic period of the oscillations. The main effect is the rotation of the Mandelbrot-like set in the complex parameter plane; one more effect is the destruction of subtle small-scale fractal structure of the set due to the presence of non-analytic terms in the complex amplitude equations.
It is shown that critical phenomena associated with Siegel disk, intrinsic to 1D complex analytical maps, survives in 2D complex invertible dissipative Hénon map. Special numerical method of estimation of the Siegel disk scaling center position (for 1D maps it corresponds to extremum) for multi-dimensional invertible maps are developed.
The possibility of realization of the phenomena of complex analytic dynamics for the realistic physical models are investigated. Observation of the Mandelbrot and Julia sets in the parameter and phase spaces both for the discrete maps and non-autonomous continuous systems is carried out. For these purposes, the method, based on consideration of coupled systems, demonstrating period-doubling cascade is suggested. Novel mechanism of synchronization loss in coupled systems corresponded to the dynamical behavior intrinsic to the complex analytic maps is offered.
According to the method, suggested in our previous work (nlin/0509012) and based on the consideration of the specially coupled systems, the possibility of physical realization of the phenomena of complex analytic dynamics (such as Mandelbrot and Julia sets) is discussed. It is shown, that unlike the case of discrete maps or differential systems with periodic driving, investigated in mentioned work, there are some difficulties in attempts to obtain the Mandelbrot set for the coupled autonomous continuous systems. A system of coupled autonomous Rössler oscillators is considered as an example.
Accumulation point of period-tripling bifurcations for complexified Henon map is found. Universal scaling properties of parameter space and Fourier spectrum intrinsic to this critical point is demonstrated.
Universal regularities of the Fourier spectrum of signal, generated by complex analytic map at the period-tripling bifurcations accumulation point are considered. The difference between intensities of the subharmonics at the values of frequency corresponding to the neighbor hierarchical levels of the spectrum is characterized by a constant $γ=21.9$ dB?, which is an analogue of the known value $γ_F=13.4$ dB, intrinsic to the Feigenbaum critical point. Data of the physical experiment, directed to the observation of the spectrum at period-tripling accumulation point, are represented.
Analogy between an approximate version of Feigenbaum renormalization group analysis in complex domain and the phase transition theory of Yang-Lee (based on consideration of formally complexified thermodynamic values) is discussed. It is shown that the Julia sets of the renormalization transformation correspond to the approximation of Mandelbrot set of the original map. New aspects of analogy between the theory of dynamical systems and the phase transition theory are uncovered.
As shown recently (O.B.Isaeva et al., Phys.Rev E64, 055201), the phenomena intrinsic to dynamics of complex analytic maps under appropriate conditions may occur in physical systems. We study scaling regularities associated with the effect of additive noise upon the period-tripling bifurcation cascade generalizing the renormalization group approach of Crutchfield et al. (Phys.Rev.Lett., 46, 933) and Shraiman et al. (Phys.Rev.Lett., 46, 935), originally developed for the period doubling transition to chaos in the presence of noise. The universal constant determining the rescaling rule for the intensity of the noise in period-tripling is found to be $γ=12.2066409...$ Numerical evidence of the expected scaling is demonstrated.
We suggest an approach to constructing physical systems with dynamical characteristics of the complex analytic iterative maps. The idea follows from a simple notion that the complex quadratic map by a variable change may be transformed into a set of two identical real one-dimensional quadratic maps with a particular coupling. Hence, dynamical behavior of similar nature may occur in coupled dissipative nonlinear systems, which relate to the Feigenbaum universality class. To substantiate the feasibility of this concept, we consider an electronic system, which exhibits dynamical phenomena intrinsic to complex analytic maps. Experimental results are presented, providing the Mandelbrot set in the parameter plane of this physical system.
The accumulation point of the period-tripling bifurcation cascade in complex quadratic map was discovered by Golberg, Sinai, and Khanin (Russ.Math.Surv. 38:1, 1983, 187), and independently by Cvitanovic and Myrheim (Phys.Lett. A94:8, 1983, 329). As we argue, in the extended parameter space of smooth maps, not necessary satisfying the Cauchy - Riemann equations, the scaling properties associated with the period-tripling are governed by two relevant universal complex constants. The first one is $δ_1 = 4.6002-8.9812i$ (in accordance with the mentioned works), while the other one, responsible for the violation of the analyticity, is found to be $δ_2 = 2.5872+1.8067i$. It means that in the extended parameter space the critical behaviour associated with the period-tripling cascade is a phenomenon of codimension four. Scaling properties of the parameter space are illustrated by diagrams in special local coordinates. We emphasize a necessity for a nonlinear parameter change to observe the parameter space scaling.