# Forced synchronization of an oscillator with a line of equilibria

## TL;DR

This paper investigates how a memristor-based oscillator with a line of equilibria synchronizes with external periodic forces, revealing mechanisms of phase and frequency locking and their dependence on system parameters and initial conditions.

## Contribution

It introduces a theoretical and numerical analysis of synchronization phenomena in a non-autonomous oscillator with a line of equilibria, highlighting new synchronization mechanisms.

## Key findings

- Demonstrates phase and frequency synchronization mechanisms.
- Shows continuous dependence of phase locking on parameters and initial conditions.
- Classifies undamped oscillations as a special type of self-sustained oscillations.

## Abstract

The model of a non-autonomous memristor-based oscillator with a line of equilibria is studied. A numerical simulation of the system driven by a periodical force is combined with a theoretical analysis by means of the quasi-harmonic reduction. Both two mechanisms of synchronization are demonstrated: capture of the phase and frequency of oscillations and suppression by an external signal. Classification of undamped oscillations in an autonomous system with a line of equilibria as a special kind of self-sustained oscillations is concluded due to the possibility to observe the effect of frequency-phase locking in the same system in the presence of an external influence. It is established that the occurrence of phase locking in the considered system continuously depends both on parameter values and initial conditions. The simultaneous dependence of synchronization area boundaries on the initial conditions and the parameter values is also shown.