The self-oscillating system with compensated dissipation – the dynamics of the approximate discrete map

Abstract

We study the van der Pol oscillator subject to an external forcing given by δ-pulses depending on the dynamical variable. In order to simplify the computations, an approximate map is derived by solving the equations between subsequent pulses. It is shown that the period-doubling cascade in this map with nonlinear dissipation converges to a critical point of Hamiltonian type with scaling properties which are known for conservative systems.

Keywords:

• self-oscillating system
• dissipative systems
• period-doubling
• scaling law
• conservative systems
• quasiperiodicity

AMS Subject Classifications::

• 37F25
• 37E20
• 37E30
• 65P30
• 37M05

Acknowledgements

First of all, the authors want to thank Dr. I.R. Sataev from the Kotel'nikov Institute of Radioengineering and Electronics of RAS, Saratov branch for his very useful advice and discussions. This work was in part supported by the Russian Ministry for Science and High Education (analytical program “Development of science potential of High School” No. 2.1.1/1738) and by the grant of the President of Russian Federation for young scientists MK-905.2010.2. D.S. would also like to thank the Carl von Ossietzky University Oldenburg for the financial support of his visit and the group of Complex Systems for their hospitality.