THE EXACT ALMOST SURE STABILITY FOR A SPECIFIC CLASS OF NONLINEAR ITO DIFFERENTIAL-EQUATIONS

The classic stability studies for linear Ito differential equations were developed by Khas'minskii in the 1960s. The main concept is to norm the solution and study the properties of the normed vector on the surface of the unit sphere. In the 1970's many ordinary second-order dynamical systems were generated to their exact stability regions by Kozin and his students. The recent methods due to Wedig are the most efficient ways to determine the stability regions and Lyapunov exponents for the Ito one degree of freedom equations. There has not been in the past an exact study for non-linear Ito equations. In this paper we shall show that there is a class of homogeneous non-linear oscillators that can be transformed on the unit sphere and the exact stability regions can be determined. Two simple examples will be presented. © 1990.