# Noise-Induced Chaos and Signal Detection by the Nonisochronous Hopf Oscillator

## TL;DR

This paper analyzes how a nonisochronous Hopf oscillator responds to stimuli, revealing that noise-induced chaos enhances sensitivity and temporal resolution, with exact solutions for responses to stimuli and chaotic dynamics.

## Contribution

It provides the first analytic response functions for a nonisochronous Hopf oscillator and links noise-induced chaos to improved signal detection capabilities.

## Key findings

- Noise induces chaotic dynamics near the system's sensitivity peak.
- Chaotic regime enables detection of brief, low-amplitude signals.
- Period-doubling cascade to chaos observed with increasing forcing strength.

## Abstract

The Hopf oscillator has been shown to capture many phenomena of the auditory and vestibular systems. These systems exhibit remarkable temporal resolution and sensitivity to weak signals, as they are able to detect sounds that induce motion in the sub-nanometer regime. In the present work, we find the analytic response function of a nonisochronous Hopf oscillator to a step stimulus and show that the system is most sensitive in the regime where noise induces chaotic dynamics. We show that this regime also provides a faster response and enhanced temporal resolution. Thus, the system can detect a very brief, low-amplitude pulse. Finally, we subject the oscillator to periodic delta-function forcing, mimicking a spike train, and find the exact analytic expressions for the stroboscopic maps. Using these maps, we find a period-doubling cascade to chaos with increasing force strength.