The mappings of van der Pol--Dyuffing generator in discrete time

V. V. Zaitsev ; A. N. Shilin

Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, no. 2 (2017), pp. 51-59

Résumé

In the work transition to discrete time in the equation of movement of van der Pol–Dyuffing generator is described. The transition purpose — to create mappings of the generator as subjects of the theory of nonlinear oscillations (nonlinear dynamics) in discrete time. The method of sampling is based on the use of counting of the pulse characteristic of an oscillatory contour as the sampling series for a signal in a self-oscillating ring “active nonlinearity–the resonator–feedback”. The choice of the consecutive scheme of excitement of a contour allows to receive the iterated displays in the form of recurrent formulas. Two equivalent forms of discrete displays of the generator of van der Pol–Dyuffing — complex and valid are presented. In approximation of method of slow-changing amplitudes it is confirmed that the created discrete mappings have dynamic properties of an analog prototype. Also within the numerical experiment it is shown that in case of the high power of generation the effect of changing of frequencies of harmonicas of the generated discrete signal significantly influence dynamics of the self-oscillators. In particular, in the discrete generator of van der Pol–Dyuffing the chaotic self-oscillations are observed.

Détail

Keywords: self-oscillatory system, pulse characteristic, discrete mapping, method of slow-changing amplitudes, chaotic self-oscillations.