# Phase and frequency linear response theory for hyperbolic chaotic oscillators

## TL;DR

This paper develops a linear response theory for hyperbolic chaotic oscillators, extending phase response concepts from limit cycle oscillators to chaotic systems, with practical methods for measuring phase shifts.

## Contribution

It introduces a shadowing-based phase and frequency response framework for hyperbolic chaotic flows, enabling analysis of phase shifts under perturbations.

## Key findings

- Defines phase sensitivity function via an adjoint equation
- Provides a method to construct phase and shadowing trajectories explicitly
- Identifies the limits of linear response regime in chaotic oscillators

## Abstract

We formulate a linear phase and frequency response theory for hyperbolic flows, which generalizes phase response theory for autonomous limit cycle oscillators to hyperbolic chaotic dynamics. The theory is based on a shadowing conjecture, stating the existence of a perturbed trajectory shadowing every unperturbed trajectory on the system attractor for any small enough perturbation of arbitrary duration and a corresponding unique time isomorphism, which we identify as phase, such that phase shifts between the unperturbed trajectory and its perturbed shadow are well defined. The phase sensitivity function is the solution of an adjoint linear equation and can be used to estimate the average change of phase velocity to small time dependent or independent perturbations. These changes of frequency are experimentally accessible giving a convenient way to define and measure phase response curves for chaotic oscillators. The shadowing trajectory and the phase can be constructed explicitly in the tangent space of an unperturbed trajectory using co-variant Lyapunov vectors. It can also be used to identify the limits of the regime of linear response.