q-damped Oscillator and degenerate roots of constant coefficients q-difference ODE

TL;DR

This paper introduces and solves the classical q-damped oscillator model using Jackson q-exponentials, explores its micro-structure, and extends solutions to q-difference equations with degenerate roots, revealing oscillatory, unbounded behaviors.

Contribution

It provides explicit solutions for q-damped oscillators in all damping cases and extends to general q-difference equations with degenerate roots, highlighting new solution structures.

Findings

Solutions oscillate but are unbounded and non-periodic.

Micro-structure of solutions exhibits self-similarity.

Extended solutions for equations with degenerate roots are constructed.

Abstract

The classical model of q-damped oscillator is introduced and solved in terms of Jackson q-exponential function for three different cases, under-damped, over-damped and the critical one. It is shown that in all three cases solution is oscillating in time but is unbounded and non-periodic. By q-periodic function modulation, the self-similar micro-structure of the solution for small time intervals is derived. In the critical case with degenerate roots, the second linearly independent solution is obtained as a limiting case of two infinitesimally close roots. It appears as standard derivative of q-exponential and is rewritten in terms of the q-logarithmic function. We extend our result by constructing n linearly independent set of solutions to a generic constant coefficient q-difference equation degree N with n degenerate roots.