Two-dimensional localized structures in harmonically forced oscillatory systems

TL;DR

This paper investigates two-dimensional localized structures in the complex Ginzburg-Landau equation near bifurcation points, revealing stable oscillons, growth mechanisms, and novel depinning dynamics through numerical simulations.

Contribution

It introduces a detailed bifurcation analysis of localized structures in 2D Ginzburg-Landau systems near Turing bifurcations, highlighting stability and dynamic behaviors.

Findings

Stable oscillons exist over wide parameter ranges.

Localized target patterns grow via ring insertion, similar to defect-mediated snaking.

Depinning dynamics of patterns are observed in simulations.

Abstract

Two-dimensional spatially localized structures in the complex Ginzburg-Landau equation with 1:1 resonance are studied near the simultaneous occurrence of a steady front between two spatially homogeneous equilibria and a supercritical Turing bifurcation on one of them. The bifurcation structures of steady circular fronts and localized target patterns are computed in the Turing-stable and Turing-unstable regimes. In particular, localized target patterns grow along the solution branch via ring insertion at the core in a process reminiscent of defect-mediated snaking in one spatial dimension. Axisymmetric oscillons on these solution branches are found to be stable over a wide parameter interval, and subject to various types of instability otherwise. Direct numerical simulations reveal novel depinning dynamics of localized target patterns in the radial direction, and of circular and planar localized hexagonal patterns in the fully two-dimensional system.