Dmitry Savin
Publications:
Golokolenov A. V., Savin D. V.
Attractors of a Weakly Dissipative System Allowing Transition to the Stochastic Web in the Conservative Limit
2023, Vol. 19, no. 1, pp. 111124
Abstract
This article deals with the dynamics of a pulsedriven selfoscillating system — the Van
der Pol oscillator — with the pulse amplitude depending on the oscillator coordinate. In the
conservative limit the “stochastic web” can be obtained in the phase space when the function
defining this dependence is a harmonic one. The paper focuses on the case where the frequency of
external pulses is four times greater than the frequency of the autonomous system. The results of
a numerical study of the structure of both parameter and phase planes are presented for systems
with different forms of external pulses: the harmonic amplitude function and its power series
expansions. Complication of the pulse amplitude function results in the complication of the
parameter plane structure, while typical scenarios of transition to chaos visible in the parameter
plane remain the same in different cases. In all cases the structure of bifurcation lines near the
border of chaos is typical of the existence of the Hamiltonian type critical point. Changes in
the number and the relative position of coexisting attractors are investigated while the system
approaches the conservative limit. A typical scenario of destruction of attractors with a decrease
in nonlinear dissipation is revealed, and it is shown to be in good agreement with the theory
of 1:4 resonance. The number of attractors of period 4 seems to grow infinitely with the decrease
of dissipation when the pulse amplitude function is harmonic, while in other cases all attractors
undergo destruction at certain values of dissipation parameters after the birth of highperiod
periodic attractors.

Isaeva O. B., Obychev M. A., Savin D. V.
Dynamics of a discrete system with the operator of evolution given by an implicit function: from the Mandelbrot map to a unitary map
2017, Vol. 13, No. 3, pp. 331348
Abstract
An abstract discrete time dynamical system, given by an implicit function of the values of
a variable at successive moments of time, is presented. The dynamics of this system is defined
ambiguously both in reverse and forward time. An example of a system of such type is described in
the works of Bullett, Osbaldestin and Percival [Physica D, 1986, vol. 19, pp. 290–300; Nonlinearity,
1988, vol. 1, pp. 27–50]; it demonstrates some features of the behavior of Hamiltonian systems.
The map under study allows a smooth transition from the case of the explicitly defined evolution
operator to an implicit one and, further, to the “conservative” limit, corresponding to the
symmetric evolution operator satisfying the unitarity condition. Being created on the basis of
the complex Mandelbrot map, it demonstrates the transformation of the phenomena of complex
analytical dynamics to “conservative” phenomena and allows us to identify the relationship
between them.
