On the Liouville Phenomenon in Estimates of Fractal Dimensions of Forced Quasi-Periodic Oscillations

Abstract

The method for studying the fractal dimensions of forced almost periodic oscillations with various differential equations is described in this paper. The method is based on the concept previously introduced of the Diophantine dimension of an almost periodic function, which is closely related to the Diophantine approximations of its frequencies. Diophantine dimensions for some classes of quasi-periodic functions are estimated. The application of this method is demonstrated by the example of a single class of control systems studied by V. A. Yakubovich. As a result, one can observe a number theoretic phenomenon (the Liouville phenomenon), which does not make it possible to control the fractal dimension of forced oscillations with well-approximated frequencies.