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S. P. Kuznetsov

Publications and source records attributed to S. P. Kuznetsov.

17 recordsLinked to original sources
arXiv2019

Lyapunov exponents of quantum trajectories beyond continuous measurements

Quantum systems interacting with their environments can exhibit complex non-equilibrium states that are tempting to be interpreted as quantum analogs of chaotic attractors. Yet, despite many attempts, the toolbox for quantifying dissipative quantum chaos remains very limited. In particular, quantum generalizations of Lyapunov exponent, the main quantifier of classical chaos, are established only within the framework of continuous measurements. We propose an alternative generalization which is based on the unraveling of a quantum master equation into an ensemble of so-called 'quantum jump' trajectories. These trajectories are not only a theoretical tool but a part of the experimental reality in the case of quantum optics. We illustrate the idea by using a periodically modulated open quantum dimer and uncover the transition to quantum chaos matched by the period-doubling route in the classical limit.

quant-ph↗
arXiv2018

Smale-Williams Solenoids in a System of Coupled Bonhoeffer-van der Pol Oscillators

The principle of constructing a new class of systems with hyperbolic chaotic attractors is proposed. It is based on using oscillators, the transfer of excitation between which is provided resonantly due to the difference in the frequencies of small and large oscillations by an integer number of times being accompanied by phase transformation according to Bernoulli nap. We consider a system with Smale-Williams attractor, which is based on two coupled Bonhoeffer-van der Pol oscillators. The oscillators manifest activity and suppression turn by turn. With appropriate selection of the modulation, relaxation oscillations occur at the end of each activity stage, the frequency of which is by an integer factor $M = 2,3,4,\ldots$ smaller than that of small oscillations. When the partner oscillator enters the activity stage, the oscillations start being stimulated by the M-th harmonic of the relaxation oscillations, so that the transformation of phase during the modulation period corresponds to the M-fold Bernoulli map. In the state space of the Poincaré map this corresponds to Smale-Williams attractor, constructed with M-fold increase in the number of turns of the winding at each step of the mapping. The results of numerical studies confirming the occurrence of the hyperbolic attractors in certain parameter domains are presented, including the portraits of attractors, diagrams illustrating the phase transformation according to the Bernoulli map, Lyapunov exponents, and charts of regimes in parameter planes. The hyperbolic nature of the attractors is verified by numerical tests that confirm absence of tangencies of stable and unstable manifolds for trajectories on the attractor (criterion of angles). An electronic circuit is proposed that implements this principle of obtaining the hyperbolic chaos and its functioning is demonstrated using the software package Multisim.

nlin.CD↗
arXiv2017

The influence of noise on two- and three-frequency quasi-periodicity in a simple model system

We discuss the effect of noise on a system with a quasi-periodicity of different dimensions. As the basic model of our research we use the simplest three-dimensional map with two-frequency and three-frequency quasi-periodicity. Modification of the dynamical regimes at the influence of noise is considered with the help of Lyapunov chart method. The transformation of Lyapunov exponents plots characteristic for the quasi-periodic Hopf bifurcation of 3-torus birth at the presence of noise is examined.

nlin.CD↗
arXiv2014

Robust chaos in autonomous time-delay system

We consider an autonomous system constructed as modification of the logistic differential equation with delay that generates successive trains of oscillations with phases evolving according to chaotic maps. The system contains two feedback loops characterized by two generally distinct retarding time parameters. In the case of their equality, chaotic dynamics is associated with the Smale-Williams attractor that corresponds to the double-expanding circle map for the phases of the carrier of the oscillatory trains. Alternatively, at appropriately chosen two different delays attractor is close to torus with Anosov dynamics on it as the phases are governed by the Fibonacci map. In both cases the attractors manifest robustness (absence of regularity windows under variation of parameters) and presumably relate to the class of structurally stable hyperbolic attractors.

nlin.CD↗
arXiv2010

Phenomena of complex analytic dynamics in the systems of alternately excited coupled non-autonomous oscillators and self-sustained oscillators

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.

nlin.CD↗
arXiv2010

Chaotic dynamics of the Hunt model, an artificially constructed flow system with a hyperbolic attractor

We study numerically chaotic behavior associated with a hyperbolic strange attractor of Plykin type in the model of Hunt, an artificially constructed dynamical system with continuous time. There are presented portraits of the attractor, plots of realizations for chaotic signal generated by the system, illustrations of the sensitive dependence on initial conditions for the trajectories on the attractor. Quantitative characteristics of the attractor are estimated, including the Lyapunov exponents and the attractor dimension. We discuss symbolic dynamics on the attractor, find out and analyze some unstable periodic orbit belonging to the attractor.

nlin.CD↗
arXiv2008

Siegel disk for complexified Henon map

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.

nlin.CD↗
arXiv2005

On possibility of realization of the phenomena of complex analytic dynamics in physical systems. Novel mechanism of the synchronization loss in coupled period-doubling systems

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.

nlin.CD↗
arXiv2005

On possibility of realization of the Mandelbrot set in coupled continuous systems

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.

nlin.CD↗
arXiv2005

Approximate Description of the Mandelbrot Set. Thermodynamic Analogy

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.

nlin.CD↗
arXiv2004

The effect of noise on the dynamics of a complex map at the period-tripling accumulation point

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.

nlin.CD↗
arXiv2002

Scaling at the onset of chaos in a network of logistic maps with two types of global coupling

We study a network of logistic maps with two types of global coupling, inertial and dissipative. Features of the clusterization process are revealed and compared, which are associated with presence of each type of couplings. For the parameter region near the onset of chaos we outline scaling properties in dynamical behavior of the network and illustrate them on the Kaneko phase diagrams.

nlin.CD↗
arXiv2002

Generalized dimensions of Feigenbaum's attractor from renormalization-group functional equations

A method is suggested for the computation of the generalized dimensions of fractal attractors at the period-doubling transition to chaos. The approach is based on an eigenvalue problem formulated in terms of functional equations, with a coefficient expressed in terms of Feigenbaum's universal fixed-point function. The accuracy of the results is determined only by precision of the representation of the universal function.

nlin.CD↗
arXiv2001

Mandelbrot set in coupled logistic maps and in an electronic experiment

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.

nlin.CD↗
arXiv2001

Scaling properties in dynamics of non-analytic complex maps near the accumulation point of the period-tripling cascade

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.

nlin.CD↗