Random walk of oscillator on the plane

with non-homogeneous friction

V.б.Dobrynsky, н.б.IvБnПv

G.V.Kurdyumov Institute of Metal Physics of NASU

Kyiv, 03142, Ukraine

ivanov@imp.kiev.uaљљљљљ dobry@imp.kiev.ua

With help of computer simulation it was studied a stochastic wandering of nonlinear oscillator. There takes place the external periodic force acting on the oscillator along y-axe, and the impulse noise in xy-plane. At this we suppose, that dumping friction for the oscillator movement, which is caused by surrounded medium, has nonlinear behavior, growing monotonously along x-axe. It is found, that there exists a statistic stochastic dependence of displacement direction of Brownian movement of the oscillator. This dependence consists in, that there takes place the Brownian drift of the oscillator in the direction, where the medium friction is lowing. It is stated also, that systems dynamic is more sensitive to the noise action, when the frequency of external force coincides with the own frequency of the oscillator, then when this frequencies are outside the resonance.

A question of how determinism and stochasticity interact in order that some physical phenomenon takes place is till now of one of the most important and fundamental problems of the modern physics. A purposeful study of this problem as well as of how chaos and order interconnects each other was started relatively recently. In doing so it was found that there are such conditions that nonlinear systems (even of low dimensionality relatively) demonstrate a behavior of sufficient high regularity and stochastic stabilizes the behavior and brings often to the one more order rather than disorder.

The present article is devoted to the behavior feature study of the nonlinear oscillator (material point) dynamics on the plane with no homogeneous friction which oscillations arises under an action of one-dimensional periodic force and the whole dynamics is a result of the joint action both of the harmonic force and sustained two-dimensional impulse noise. Simulating with aid of computer this dynamics we are of interest at first to study its friction that is a projection onto the direction transversal to that of the sinusoidal force action. In doing so we suppose that, due to sole the impulse noise action, the latter must be the Brownian walks. For a simplification of mathematical model a coordinate system is chosen in such a way that the periodic force acts along the y-axis and its amplitude is sinusoidal (i.e. depends on time as a sine). As for the impulse noise,

the impulse amplitudes all are identical but directions from which the impulses act on the point all are different and vary at random and are determined by a random number generator. A corollary of such the coordinate system choice is that we should expect to observe the Brownian walk dynamics along the x-axis.