Delay-induced depinning of localized structures in a spatially inhomogeneous Swift-Hohenberg model

We report on the dynamics of localized structures in an inhomogeneous Swift-Hohenberg model describing pattern formation in the transverse plane of an optical cavity. This real order parameter equation is valid close to the second-order critical point associated with bistability. The optical cavity is illuminated by an inhomogeneous spatial Gaussian pumping beam and subjected to time-delayed feedback. The Gaussian injection beam breaks the translational symmetry of the system by exerting an attracting force on the localized structure. We show that the localized structure can be pinned to the center of the inhomogeneity, suppressing the delay-induced drift bifurcation that has been reported in the particular case where the injection is homogeneous, assuming a continuous wave operation. Under an inhomogeneous spatial pumping beam, we perform the stability analysis of localized solutions to identify different instability regimes induced by time-delayed feedback. In particular, we predict the formation of two-arm spirals, as well as oscillating and depinning dynamics caused by the interplay of an attracting inhomogeneity and destabilizing time-delayed feedback. The transition from oscillating to depinning solutions is investigated by means of numerical continuation techniques. Analytically, we use an order parameter approach to derive a normal form of the delay-induced Hopf bifurcation leading to an oscillating solution. Additionally we model the interplay of an attracting inhomogeneity and destabilizing time delay by describing the localized solution as an overdamped particle in a potential well generated by the inhomogeneity. In this case, the time-delayed feedback acts as a driving force. Comparing results from the later approach with the full Swift-Hohenberg model, we show that the approach not only provides an instructive description of the depinning dynamics, but also is numerically accurate throughout most of the parameter regime.

Figure 1

(a) Stable localized solution $q_0\left(\mathbf{x}\right)$ obtained by 2D numerical integration of Eq. (1), as well as three localized eigenfunctions ${\phi }_k\left(\mathbf{x}\right)$ as solutions of the linear eigenvalue problem for $\alpha =0$: (b) translational mode; (c) growth mode; (d) deformation mode. Other parameters are $L_x=L_y=19.6,a_1=2.0,a_2=\frac{4}{3},Y_0=-0.4,C=1.0,A=0,B=0$.

Figure 2

Eigenvalues ${\mu }_k$ of Eq. (1) for $\alpha =0$ corresponding to the drift-inducing modes (blue solid line), the growth-inducing mode (red dash-dotted line), and the deformation-inducing modes (green dotted line) for different amplitudes $A$ of the inhomogeneity $Y$.

Figure 3

Frequency $\omega$ of the oscillations of the localized structure around the inhomogeneity obtained from direct numerics in one dimension (dotted line) and the imaginary part of the eigenvalue corresponding to the unstable translational mode (solid line). The dotted vertical line marks the onset of the oscillations, i.e., the first bifurcation point. The error bars are given by $\mathrm{\Delta }\omega =\frac{2\pi }{\mathrm{\Delta }t}$, where $\mathrm{\Delta }t$ is the sampling time of the simulation.

Figure 4

Direct numerical simulations in one dimension with a fixed delay time $\tau =1$. (Left) A localized solution oscillates around the inhomogeneity for $\alpha =1.035$. (Right) A localized structure gets depinned from the inhomogeneity and starts to drift freely for $\alpha =1.073$. The amplitude of the inhomogeneity is fixed to $A=0.2$. Other parameters are $a_1=2.0,a_2=\frac{4}{3},Y_0=-0.4,C=1.0,B=4.0$.

Figure 5

Period time $T$ of the oscillatory solution depending on the delay strength $\alpha$ obtained by numerical continuation using dde-biftool. The magnified version shows the parameter region in the vicinity of the bifurcation point ${\alpha }_{\text{crit2}}$, where the saddle-node bifurcation of limit cycles sets in. The small box at the top shows the Floquet multipliers ${\mu }_F$ of the stable (black solid line, *) and the unstable (red dotted line, +) periodic branch at the marked positions.

Figure 6

Formation of a spreading spiral observed in direct numerical simulations in two dimensions. Parameters are $\alpha =0.31,\tau =5.7,L_x=L_y=64.0,a_1=2.0,a_2=\frac{4}{3},Y_0=-0.4,C=1.0,A=2.0,B=4.0$.

Figure 7

Maximum amplitude of the oscillations ${\mathbf{R}}_{\text{max}}$ in dependence of the delay strength $\alpha$ obtained by direct numerical simulations (crosses) and by the order parameter model (7) (black solid line). The dotted line marks the first Hopf bifurcation point ${\alpha }_{\text{crit}}=1.0219$ found both in the direct numerical simulations and in the order parameter model. However, Eq. (7) is only valid in the direct vicinity of ${\alpha }_{\text{crit}}$ and loses its validity for increasing delay strengths.

Figure 8

Potential well $V\left(\mathbf{R}\right)$ and attracting force $F\left(\mathbf{R}\right)$ of an inhomogeneity $Y$ calculated numerically for $A=0.2,B=4.0$.

Figure 9

Maximum amplitude of the oscillations ${\mathbf{R}}_{\text{max}}$ in dependence of the delay strength $\alpha$ obtained by direct numerical simulations (crosses) and by the potential well model (black line). The dotted lines mark the two bifurcation points in the direct numerical simulations. The first bifurcation leading to an oscillation of the structure appears at the same value ${\alpha }_{\text{crit}}=1.0219$ in the direct numerical simulations and in the potential well model, respectively. The depinning instability occurs in the potential well model at $\alpha >{\alpha }_{\text{crit2}}$, i.e., for a larger delay strength than in the direct numerical simulations.

Figure 10

Eigenvalues ${\lambda }_{k,0}$ of Eq. (1) for $\alpha =1.1,\tau =1$ (the same values as used in Figs. 12 and 13) corresponding to the drift-inducing modes (blue solid line), the growth-inducing mode (red dash-dotted line), and the deformation-inducing modes (green dotted line) for different amplitudes $A$ of the inhomogeneity $Y$. One can clearly see that for a wide range of different amplitudes $A$, the translational eigenfunctions are unstable, whereas the other localized eigenfunctions are stable.

Figure 11

Eigenvalues ${\lambda }_{k,0}$ of Eq. (1) for $\alpha =0.31,\tau =5.7$ (the same values as used in Fig. 6) corresponding to the drift-inducing modes (blue solid line), the growth-inducing mode (red dash-dotted line), and the deformation-inducing modes (green dotted line) for different amplitudes $A$ of the inhomogeneity $Y$. Depending on the choice of the amplitude $A$, either an instability of the translational eigenfunctions, both the translational and deformation modes, or only the deformation modes is induced.

Figure 12

Pinned oscillatory solution in two dimensions. The black cross marks the center of the inhomogeneity. Parameters are $\alpha =1.1,\tau =1,L_x=L_y=32.0,a_1=2.0,a_2=\frac{4}{3},Y_0=-0.4,C=1.0,A=1.0,B=4.0$.

Figure 13

Depinning solution in two dimensions. The black cross marks the center of the inhomogeneity Parameters are $\alpha =1.1,\tau =1,L_x=L_y=64.0,a_1=2.0,a_2=\frac{4}{3},Y_0=-0.4,C=1.0,A=0.2,B=4.0$.