Two-dimensional localized structures in harmonically forced oscillatory systems

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.

Spatial asymmetries of resonant oscillations in periodically forced heterogeneous media

“…Notably, for both parametric and additive forcing, Eq. (4) preserves the inversion symmetry (β, ν) → (−β, −ν) [47]. In the following we explore bistability regions of fixed points in the parameter plane spanned by the detuning ν and the forcing amplitude, Γ p or Γ a , for different values of β > 0.…”

“…22 shows for a few β values. For β > β c = √ 3, the range diminishes to zero in a cusp bifurcation at [47]…”

“…Spatially localized resonant oscillations, such as oscillons [1,2,3] and reciprocal oscillons [4,5] are not only inspiring natural phenomena but also mathematically intriguing [6,7,8,9,10,11,12,13,14,15,16,17]. Similarly to most frequency-locking studies of oscillatory media [4,18,19,20,21,22,23,24,25,26,27,28], also localized resonances have been studied within the scope of spatially homogeneous conditions.…”

Spatial Localization in Dissipative Systems

“…Localized solutions need not be periodic, however. Different examples of localized states with chaotic dynamics are known (68,76). The much-studied chimera states in systems of identical globally coupled phase oscillators (77)(78)(79) provide an example in which a set of adjacent oscillators oscillates in phase while the phases of the remaining oscillators remain random.…”

Molding the asymmetry of localized frequency-locking waves by a generalized forcing and implications to the inner ear

“…3(b,d) shows. For a pure additive forcing, the cusp singularity exists for β > √ 3 [59], while for combined forcing it may exist for any |β| > 0. We note that while frequency-locking solutions for either purely parametric or purely additive forcing can be obtained analytically [37], the case of combined forcing requires a numerical approach.…”