Features of parameters space structure for oscillator
with asymmetric soft-spring behavior
B.P. Bezruchko 
, Ye.P. Seleznev 
Physical Department, Saratov State University, Saratov, Russia;
Institute of RadioEngineering and Electronics of
Russian Academy of Sciences,
Saratov Branch, Saratov, Russia
Dissipative oscillator is a basic model of nonlinear dynamics. However some details of its behavior seem to be terra incognita. For instance, one of the poorly studied aspects of behavior for oscillators with a nonlinearity known as asymmetric soft-spring is a range of broad values of an a driving frequency (lager than linear resonant frequency), where the sequence of period adding cycles exists. This work is devoted to numerical and experimental research of the bifurcation sets configuration in this region of parameters. The differential Toda equation being classical oscillator model with given type of nonlinearity [1,2] is numerically studied. Two real oscillatory systems (periodically driven the electrical ``pendulums'' - LR-circuits with various semiconductor diodes [3-9] and with piecewise-linear switched capacity [10]) are experimentally investigated. The mathematical models for the both experimental systems can be differential equations of oscillator with similar potential function.
The regions of existence of various oscillations in the space of parameters of external driving, dissipation and nonlinearity are constructed. It is shown, that the known configuration of regions of existence and evolution of the sequence of period adding cycles in LR-diode circuits with various types of diodes [4-8] takes place in the Toda oscillator. Two typical configurations of existence regions of period adding sequence cycles are recognized. The first one is characterized by the central arrangement of chaos region, in such system bifurcation set structure is formed as in [7,8]. For second configurations, where the region of chaotic behavior is formed at high-frequency boundary of sheet, it is characteristically bifurcation set structure of type as in [9,10].
The submitted results expand the knowledge about the properties of basic oscillator models and demonstrate their place in nature.
The work is executed at financial support of Russian Foundation of Fundamental Research, grant No.96-02-16753.