Alexander P. Kuznetsov

TWO-PARAMETER STUDY OF TRANSITION TO CHAOS IN CHUA'S CIRCUIT: RENORMALIZATION GROUP, UNIVERSALITY AND SCALING

A complex fine structure in the topography of regions of different dynamical behavior near the onset of chaos is investigated in a parameter plane of the 1D Chuas map, which describes approximately the dynamics of Chuas circuit. Besides piecewise-smooth Feigenbaum critical lines, the boundary of chaos contains an infinite set of codimension-2 critical points, which may be coded by itineraries on a binary tree. Renormalization group analysis is applied which is a generalization of Feigenbaums theory for codimension-2 critical points. Multicolor high-resolution maps of the parameter plane show that in regions near critical points having periodic codes, the infinitely intricate topography of the parameter plane reveals a property of self-similarity.

On the road towards multidimensional tori

Abstract The problem of persistence of four-frequency tori is considered in models represented by the coupled periodically driven self-oscillators. We show that the adding the third oscillator gives rise to destruction of the three-frequency tori, with appearance of regions of either chaotic attractors or four-frequency tori. As the coupling strength decreases, the four-frequency tori dominate, and the amplitude threshold of their occurrence vanishes. Also, for three oscillators, a domain of complete synchronization of the system by the external driving can disappear.

A structure of the oscillation frequencies parameter space for the system of dissipatively coupled oscillators

Abstract A structure of the oscillation frequencies parameter space for three and four dissipatively coupled van der Pol oscillators is discussed. Situations of different codimension relating to the configuration of the full synchronization area as well as a picture of different modes in its neighborhood are revealed. An organization of quasi-periodic areas of different dimensions is considered. The results for the phase model and for the original system are compared.

DYNAMICAL SYSTEMS OF DIFFERENT CLASSES AS MODELS OF THE KICKED NONLINEAR OSCILLATOR

Using the nonlinear dissipative kicked oscillator as an example, the correspondence between the descriptions provided by model dynamical systems of dierent classes is discussed. A detailed study of the approximate 1D map is undertaken: the period doubling is examined and the possibility of non-Feigenbaum period doubling is shown. Illustrations in the form of bifurcation diagrams and sets of iteration diagrams are given, the scaling properties are demonstrated, and the tricritical points (the terminal points of the Feigenbaum critical curves) in parameter space are found. The congruity with the properties of the corresponding 2D map, the Ikeda map, is studied. A description in terms of tricritical dynamics is found to be adequate only in particular areas of parameter space.

About Landau–Hopf scenario in a system of coupled self-oscillators

Abstract The conditions are discussed for which an ensemble of interacting oscillators may demonstrate the Landau–Hopf scenario of successive birth of multi-frequency quasi-periodic motions. A model is proposed that is a network of five globally coupled oscillators characterized by controlled degree of activation of individual oscillators. Illustrations are given for successive birth of tori of increasing dimension via quasi-periodic Hopf bifurcations.

VARIETY OF TYPES OF CRITICAL BEHAVIOR AND MULTISTABILITY IN PERIOD-DOUBLING SYSTEMS WITH UNIDIRECTIONAL COUPLING NEAR THE ONSET OF CHAOS

The dynamics of two unidirectionally coupled period-doubling systems is investigated depending on three relevant parameters (control parameters of subsystems and coupling). There is a hierarchy of critical behavior types. Feigenbaum’s critical surfaces existing in the parameter space are bounded by tricritical lines and intersect along the bicritical line. These lines, in turn, intersect at a new multicritical point BT. Universality and scaling properties for all the critical situations are discussed, and the table of critical indices is given.

CATASTROPHE THEORETIC CLASSIFICATION OF NONLINEAR OSCILLATORS

Catastrophe theory is employed to classify different types of nonlinear oscillators, basing on the complication of their potentials. By using Thoms catastrophe unfoldings as oscillator potentials, we have introduced more general models to describe the dynamics of nonlinear oscillators, differing from each other by the form of their potential wells and by the possibility of escape. Spreading the investigation in the space of the parameters of the potential function, we have revealed that our examples defined via Thoms catastrophe unfoldings have some type of universal properties in the context of forced oscillations. For oscillators with nonescaping solutions, we have detected such typical bifurcation structures as crossroad areas and spring areas, and have described the universal scenario of their evolution under the forcing amplitude variation. On increasing the potential function degree, the complexity of the charts of the dynamical regimes results from the repetition of the described bifurcation scenario. For oscillators with escaping solutions, such general properties were investigated, as dependence of the charts of the dynamical regimes and the basins on the parameters of the potential function. We have observed that these properties are typical in a broad range of the control parameters.

MULTI-PARAMETER CRITICALITY IN CHUA'S CIRCUIT AT PERIOD-DOUBLING TRANSITION TO CHAOS

Investigation of non-Feigenbaum types of period-doubling universality is undertaken for a single Chuas circuit and for two systems with a unidirectional coupling. Some codimension-2 critical situations are found numerically that were known earlier for bimodal 1D maps. However, the simplest of them (tricritical) does not survive in a strict sense when the exact dynamical equations are used instead of the 1D map approximation. In coupled systems double Feigenbaums point and bicritical behavior are found and studied. Scaling properties that are the same as in two logistic maps with a unidirectional coupling are illustrated.

BICRITICAL DYNAMICS OF PERIOD-DOUBLING SYSTEMS WITH UNIDIRECTIONAL COUPLING

The simplest case of bicritical behavior arises in a system of two logistic maps with unidirectional coupling in the point of a parameter plane where lines of transition to chaos in both subsystems meet. We develop a renormalization group analysis of the bicriticality and find the corresponding fixed point universal function and constants featuring the scaling properties of the second system while the first one is in the Feigenbaum critical state. Fractal properties of the bicritical attractor and its quantitative characteristics (σ-functions, f(α)-spectra, generalized dimensions) are considered. It is shown that the bicriticality may be observed as well in lattice models of flow systems consisting of more than two coupled elements.

Multistability and transition to chaos in the degenerate Hamiltonian system with weak nonlinear dissipative perturbation

Abstract The effect of small nonlinear dissipation on the dynamics of a system with the stochastic web which is linear oscillator driven by pulses is studied. The scenario of coexisting attractors evolution with the increase of nonlinear dissipation is revealed. It is shown that the period-doubling transition to chaos is possible only for the third-order resonance and only hard transitions can be seen for all other resonances.