Superexponential Self-Interacting Oscillator

TL;DR

The paper introduces and analyzes the superexponential self-interacting oscillator (SSO), a nonlinear system with unique phase space and transition behaviors, expanding understanding of complex oscillatory dynamics.

Contribution

It presents the first detailed analysis of the SSO, highlighting its unique potential, phase space structure, and transition phenomena, including a double well variant.

Findings

SSO exhibits a transition point with hierarchical singularities.

The oscillator's period shifts from linear decrease to nonlinear increase across the transition.

A kick-like behavior characterizes the crossover dynamics.

Abstract

The superexponential self-interacting oscillator (SSO) is introduced and analyzed. Its power law potential is characterized by the dependence of both the base and the exponent on the dynamical variable of the oscillator. Opposite to standard oscillators such as the (an-)harmonic oscillator the SSO combines both scattering and confined periodic motion with an exponentially varying nonlinearity. The SSO potential exhibits a transition point with a hierarchy of singularities of logarithmic and power law character leaving their fingerprints in the agglomeration of its phase space curves. The period of the SSO consequently undergoes a crossover from decreasing linear to a nonlinearly increasing behaviour when passing the transition energy. We explore its dynamics and show that the crossover involves a kick-like behaviour. A symmetric double well variant of the SSO is briefly discussed.