Phase-space structure of two-dimensional excitable localized structures

Abstract

In this work we characterize in detail the bifurcation leading to an excitable regime mediated by localized structures in a dissipative nonlinear Kerr cavity with a homogeneous pump. Here we show how the route can be understood through a planar dynamical system in which a limit cycle becomes the homoclinic orbit of a saddle point ͑saddle-loop bifurcation͒. The whole picture is unveiled, and the mechanism by which this reduction occurs from the full infinite-dimensional dynamical system is studied. Finally, it is shown that the bifurcation leads to an excitability regime, under the application of suitable perturbations. Excitability is an emergent property for this system, as it emerges from the spatial dependence since the system does not exhibit any excitable behavior locally.

Dynamics of localized and patterned structures in the Lugiato-Lefever equation determine the stability and shape of optical frequency combs

“…Later studies have then demonstrated the existence of CSs in the LLE [11][12][13][14][15]. The LLE was originally obtained through a mean-field approximation, describing the dynamics of the slowly varying amplitude of the electromagnetic field u(ϕ, t) in the paraxial limit, where ϕ = x is the spatial coordinate transverse to the propagation direction.…”

Bifurcation analysis to the Lugiato–Lefever equation in one space dimension

“…However, a cavity soliton has a different root from the so-called ''soliton'' in a conservative system, because it is a result from a balance among input driving force, detuning, and dissipation for light in a nonlinear medium. We can see the brief history and the underlying nonlinear optics of cavity and feedback solitons in the review article written by Professors, Ackemann and Firth [3] and the references therein ( [4][5][6] for example).…”

Effects of a localized beam on the dynamics of excitable cavity solitons

“…1 past the saddle-node bifurcation at I s = 0.6857 (SN point), a pair of stationary (stable, upper branch, and unstable, middle branch) localized solutions in the form of CS are found. In this parameter region, these structures are not essentially different to the solutions found in the homogeneous case [24]. Increasing I s the stable high-amplitude CS undergoes a Hopf bifurcation.…”

“…The localized pump helps to spatially fixed the CS and to control some of its dynamical properties. After the saddle-node bifurcation that creates the CS, it starts oscillating and overall exhibit a plethora of bifurcations that are shown to be organized by three codimension-2 points: a TakensBogdanov point, (which is also present for homogeneous pump as discussed already in [23,24]), a Cusp, and a Saddle-Node Separatrix Loop (SNSL) points. In this scenario a saddle-loop bifurcation connects the TakensBogdanov and the SNSL and the Cusp is connected to the other codimension-2 bifurcations by two saddle-node lines.…”