Driven transitions between megastable quantized orbits

TL;DR

This paper analyzes a nonlinear oscillator with state-dependent delay that exhibits multiple stable quantized orbits, demonstrating controlled transitions between these orbits driven by external harmonic forcing, akin to quantum jumps.

Contribution

It introduces a model of megastability with nested limit cycles and analytically describes driven transitions between quantized orbits, including resonance and beating effects.

Findings

Analytical description of increasing amplitude spectrum of quantized orbits

Demonstration of driven transitions between energy levels via external forcing

Identification of amplitude locking and basin effects in phase space

Abstract

We consider a nonlinear oscillator with state-dependent time-delay that displays a countably infinite number of nested limit cycle attractors, \emph{i.e.} megastability. In the low-memory regime, the equation reduces to a self-excited nonlinear oscillator and we use averaging methods to analytically show quasilinear increasing amplitude of the megastable spectrum of quantized quasicircular orbits. We further assign a mechanical energy to each orbit using the Lyapunov energy function and obtain a quadratically increasing energy spectrum and (almost) constant frequency spectrum. We demonstrate transitions between different quantized orbits, i.e. different energy levels, by subjecting the system to an external finite-time harmonic driving. For large driving amplitude with frequency close to the limit cycle frequency, resonance drives transitions to higher energy levels. Alternatively, for large driving amplitude with frequency slightly detuned from limit-cycle frequency, beating effects can lead to transitions to lower energy levels. Such driven transitions between quantized orbits form a classical analog of quantum jumps. For excitations to higher energy levels, we show amplitude locking where nearby values of driving amplitudes result in the same response amplitude, i.e. the same final higher energy level. We rationalize this effect based on the basins of different limit cycles in phase space. From a practical viewpoint, our work might find applications in physical and engineering system where controlled transitions between several limit cycles of a multistable dynamical system is desired.