4. Dimensional Aspects of Almost Periodic Dynamics

The first part (Sects. 4.2, 4.3, 4.5 and 4.6) of the present chapter contains several approaches to the investigation of the Fourier spectrum of almost periodic solutions to various differential equations. The core element here is the Cartwright theorem [6] that links the topological dimension of the orbit closure of an almost periodic flow and the algebraic dimension of its frequency module (Theorem 4.8). The next step is an extension of this theorem to non-autonomous differential equations (Theorem 4.11) originally presented in [7]. Applications of Cartwright’s theorems are given for almost periodic ODEs based on the approach due to R. A. Smith (Theorem 4.12) and for DDEs based on results of Mallet-Paret from [16] (Theorem 4.14). In Sect. 4.7 we develop a method for studying fractal dimensions of forced almost periodic oscillations using some kind of recurrence properties. This approach differs from the one due to Douady and Oesterlé and highly relies on almost periodicity. Some fundamental ideas firstly appeared in the works of Naito (see [17, 18]) and then were developed in [1, 2]. In Sect. 4.8 we study forced almost periodic oscillations in Chua’s circuit and compare the analytical upper estimates of the fractal dimension of their trajectory closures with numerical simulations given by the standard box-counting algorithm.