Rationalizing Oscillations: Embedding Periodic and Chaotic Dynamics within the Framework of the Response Laws

Oscillatory behavior—whether periodic, quasi-periodic, or chaotic—is often seen as incompatible with the rational-function-based response models constrained by energy, delay, and feedback. In this paper, we argue that oscillations are in fact special manifestations of constrained dynamics and can be embedded within the rational function framework through higher-order approximations, harmonic balance, or nonlinear linearization. By examining Laplace-domain representations of sinusoidal decay, limit cycle approximations of the van der Pol oscillator, and Carleman-linearized models of chaotic trajectories, we show that oscillation is fundamentally rational in nature. This unification demonstrates that the response laws retain descriptive and predictive power even across seemingly non-convergent systems.