Properties of the chaotic oscillations in electric circuit with arc
V.N. Sidorets
Department of Electrical Engineering of the Paton Welding Institute,
Kiev, Ukraine
The autonomous electric circuit with arc governed by three ordinary differential equations was investigated. Many kinds of bifurcations, periodic and chaotic behaviors of this system were observed under variation of two parameters. There were: Hopf bifurcation (supercritical as precursor of period- doubling bifurcations or subcritical with stiff transition to chaotic oscillation); tangent bifurcation with stiff birth stable and unstable cycles; infinite cascade of period-doubling bifurcations with transition to chaos; finite cascade of period-doubling bifurcations with transition to chaos or without one; reverse cascade of period-doubling; intermittency; crisis of attractor; jump bifurcation; overlapping of basins of attractors that leads to metastable chaos and isolated regimes [1,2].
Phenomena above were studied in details by means of constructing the bifurcation diagrams and routes to chaos were classified. Two general physical properties have been lied on the basis of classification: a) softness or stiffness of appearance; b) reversibility or irreversibility of process. We observed three principal patterns of bifurcation diagrams: i) softness and reversibility; ii) stiffness and irreversibility; iii) stiffness and reversibility. Last pattern possesses so called isolated regimes that cannot be observed by a simple physical experiments.
So far as isolated region is difficulty observed on ordinary physical experiment while only one parameter is varied the method of search and realization of these regimes had been developed. It is basing on investigation carried out by us of isolated regimes evolution under vary of second parameter. The isolated regime has appeared may transfer to periodic window in chaotic regimes while second parameter is varied. Thus the method of search and realization of isolated regime is especial physical experiment while two parameters are varied with simultaneous monitoring some phenomenon, for example, the first period-doubling bifurcation.
We have had interesting to periodic window with period 1 while bifurcation diagrams are investigated. It presents on almost all bifurcation diagram. This is considerable distinction of our system from well known ones, for example, logistic equation studied by May and Feigenbaum. In order to investigated this phenomenon the mapping functions for our system were constructed. Mapping function has appeared to have quadratic maximum only on initial stage of chaos development. By increasing of the bifurcation parameter the next minimum, maximum, minimum, maximum, etc. appear successively. If mapping function minima always have quadratic type but maxima are strongly sharpened as cusp. Mapping function has an image of comb with teeth directed upward. The periodic window with period 1 appears when next function maximum crosses a bissectrix of right angle.
Same mapping function had obtained for Lorenz system in Ref. [3]. So we can make a conclusion about some universality of such type of mapping function for physical systems described by ordinary differential equations. Investigation of Poincare section topology of the strange attractor showed that it is possible to generalize the Smale horseshoe to a case of very large tension of phase space in dynamical systems. If arising parts of mapping function may be associated with tension of phase space and falling ones - with folding - Smale horseshoe is transforming to structure which may be called as "roulette" ("Swiss roll").