Dmitry Savin

This article deals with the dynamics of a pulse-driven self-oscillating 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 high-period periodic attractors.

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.