# Devil's Staircases in Continuous Systems with Modulated Forcing

## TL;DR

This paper demonstrates that harmonically modulated forcing in continuous systems near a Hopf bifurcation can produce devil's staircase phenomena, expanding the understanding of resonance structures beyond traditional discrete models.

## Contribution

It reveals that modulated forcing induces devil's staircase behavior in continuous systems, supported by theoretical analysis and experimental validation with a semiconductor laser.

## Key findings

- Harmonic modulation leads to multiple frequency locking regions.
- Experimental results confirm theoretical predictions.
- Modulated forcing introduces complex resonance structures in continuous systems.

## Abstract

The discrete circle map is the archetypical example of a driven periodic system, showing a complex resonance structure under a change of the forcing frequency known as the devil's staircase. Adler's equation can be seen as the direct continuous equivalent of the circle map, describing locking effects in periodic systems with continuous forcing. This type of locking produces a single fundamental resonance tongue without higher order resonances, and a devil's staircase is not observed. We show that, with harmonically modulated forcing, nonlinear oscillations close to a Hopf bifurcation generically reproduce the devil's staircase even in the continuous case. Experimental results on a semiconductor laser driven by a modulated optical signal show excellent agreement with our theoretical predictions. The locking appears as a modulation of the oscillation amplitude as well as the angular oscillation frequency. Our results show that by proper implementation of an external drive, additional regions of stable frequency locking can be introduced in systems which originally show only a single Adler-type resonance tongue.