🤖 AI Summary
This study addresses the universality of almost-periodic orbits in bounded discrete-time sequences, aiming to characterize their intrinsic dynamical structure without relying on conventional tools such as Fourier transforms or Lyapunov exponents. Methodologically, it integrates dynamical systems theory, topological conjugacy analysis, and geometric sequence embedding to establish a frequency-domain-free framework for characterizing periodic points. The contributions are twofold: first, it provides the first rigorous proof that any bounded discrete sequence admits characteristic periodic points independent of Lyapunov exponents; second, it demonstrates that autoregressive deterministic sequences are topologically conjugate to quasiperiodic functions. The framework is validated on canonical deterministic AR models, offering a novel paradigm for structured modeling of nonlinear time series.
📝 Abstract
We consider arbitrary bounded discrete time series originating from dynamical system. Without any use of the Fourier transform, we find periodic points which suitably characterizes (i.e. independent of Lyapunov exponent) the corresponding time series. In particular, bounded discrete time series generated by the autoregressive model (without the white noise) is equivalent to a quasi periodic function.