# Networks of planar Hamiltonian systems

## TL;DR

This paper studies networks of diffusively coupled planar Hamiltonian systems, exploring synchronization and Turing-like instabilities, revealing unique behaviors due to their Hamiltonian structure compared to other oscillator networks.

## Contribution

It introduces a new class of coupled networks with Hamiltonian dynamics and analyzes their synchronization and instability phenomena.

## Key findings

- Existence of synchronization phenomena in Hamiltonian networks
- Identification of Turing-like instabilities in these systems
- Unusual behaviors compared to other oscillator networks

## Abstract

We introduce diffusively coupled networks where the dynamical system at each vertex is planar Hamiltonian. The problems we address are synchronisation and an analogue of diffusion-driven Turing instability for time-dependent homogeneous states. As a consequence of the underlying Hamiltonian structure there exist unusual behaviours compared with networks of coupled limit cycle oscillators or activator-inhibitor systems.