Biomedical subjects

Sergey P Kuznetsov

Publications and source records attributed to Sergey P Kuznetsov.

6 recordsLinked to original sources

PubMed2006

Arnold's cat map dynamics in a system of coupled nonautonomous van der Pol oscillators.

An example of a flow system is presented with an attractor concentrated mostly at a surface of a two-dimensional torus, the dynamics on which is governed by the Arnold cat map. The system is composed of four coupled nonautonomous van der Pol oscillators. Three of them have equal characteristic frequencies, and in the other one the frequency is twice as large. The parameters controlling excitation of the two pairs of oscillators are forced to undergo a slow counterphase periodic modulation in time. At the end of the active stage for one pair of the oscillators, the excitation is passed to another pair, than back, and so on. In terms of a stroboscopic Poincaré section, the respective eight-dimensional (8D) mapping, due to strong phase volume compression, reduces approximately to a 2D map for the phases of one pair of the oscillators that corresponds approximately to the Arnold cat map. The largest two Lyapunov exponents (one positive and one negative) are close to those predicted with the cat map model. Estimates for the fractal dimension of the attractor of the Poincaré map are close to 2.

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PubMed2005

Example of a physical system with a hyperbolic attractor of the Smale-Williams type.

A simple and transparent example of a nonautonomous flow system with a hyperbolic strange attractor is suggested. The system is constructed on the basis of two coupled van der Pol oscillators, the characteristic frequencies differ twice, and the parameters controlling generation in both oscillators undergo a slow periodic counterphase variation in time. In terms of stroboscopic Poincaré sections, the respective 4D mapping has a hyperbolic strange attractor of the Smale-Williams type. Qualitative reasoning and quantitative data of numerical computations are presented and discussed, e.g., Lyapunov exponents and their parameter dependencies. A special test for hyperbolicity based on analysis of distributions of angles between stable and unstable subspaces of a chaotic trajectory is performed.

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PubMed2005

Effect of noise on the dynamics at the torus-doubling terminal point in a quadratic map under quasiperiodic driving.

Scaling regularities are examined associated with the effect of additive noise on a system driven by an external quasiperiodic force with the golden-mean frequency ratio near the terminal point of the torus-doubling bifurcation curve (TDT point). This point was studied in the context of the problem of the onset of a strange nonchaotic attractor on the basis of renormalization group (RG) analysis [Kuznetsov, Phys. Rev. E 57, 1585 (1998)] and observed in experiments with a quasiperiodically driven resistor-inductor-diode circuit [Bezruchko, Phys. Rev. E 62, 7828 (2000)]. The method implemented in the present paper is based on a generalization of the RG approach of Crutchfield [Phys. Rev. Lett. 46, 933 (1981)] and Shraiman [Phys. Rev. Lett., 46, 935 (1981)], originally developed for the period-doubling transition to chaos in the presence of noise. At the TDT point, a constant determining the rescaling rule for the intensity of noise is found to be gamma = 20.048 637 7 . It means that a decrease of the noise amplitude by this factor ensures the possibility of observing one more level of the fractal-like structure of the dynamics, with increase of the characteristic time scale by [(square root (5) + 1)/2]3. Numeric results demonstrating evidence of the expected scaling are presented, e.g., portraits of the noisy attractors and Lyapunov charts on the parameter plane in different scales.

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PubMed2004

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. E 64, 055201 (2001)], 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 (1981)] and Shraiman et al. [Phys. Rev. Lett. 46, 935 (1981)], 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 gamma=12.206 640 9 em leader. Numerical evidence of the expected scaling is demonstrated.

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PubMed2003

Characterization of the noise effect on weak synchronization.

We investigate the noise effect on weak synchronization in two coupled identical one-dimensional (1D) maps. Due to the existence of positive local transverse Lyapunov exponents, the weakly stable synchronous chaotic attractor (SCA) becomes sensitive with respect to the variation of noise intensity. To quantitatively characterize such noise sensitivity, we introduce a quantifier, called the noise sensitivity exponent (NSE). For the case of bounded noise, the values of the NSE are found to be the same as those of the exponent characterizing a parameter sensitivity of the weakly stable SCA in presence of a parameter mismatch between the two 1D maps. Furthermore, it is found that the scaling exponent for the average time spent near the diagonal for both the bubbling and riddling cases occurring in the regime of weak synchronization is given by the reciprocal of the NSE, as in the parameter-mismatching case. Consequently, both the noise and parameter mismatch have the same effect on the scaling behavior of the average characteristic time.

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PubMed2002

Torus fractalization and intermittency.

The bifurcation transition is studied for the onset of intermittency analogous to the Pomeau-Manneville mechanism of type I, but generalized for the presence of a quasiperiodic external force. The analysis is concentrated on the torus-fractalization (TF) critical point that occurs at some critical amplitude of driving. (At smaller amplitudes the bifurcation corresponds to a collision and subsequent disappearance of two smooth invariant curves, and at larger amplitudes it is a touch of attractor and repeller at some fractal set without coincidence.) For the TF critical point, renormalization group (RG) analysis is developed. For the golden mean rotation number a nontrivial fixed-point solution of the RG equation is found in a class of fractional-linear functions with coefficients depending on the phase variable. Universal constants are computed that are responsible for scaling in phase space (alpha=2.890 053... and beta= -1.618 034...) and in parameter space (delta(1)=3.134 272... and delta(2)=1.618 034...). An analogy with the Harper equation is outlined, which reveals important peculiarities of the transition. For amplitudes of driving less than the critical value the transition leads (in the presence of an appropriate reinjection mechanism) to intermittent chaotic regimes; in the supercritical case it gives rise to a strange nonchaotic attractor.

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