Two-dimensional localized chaotic patterns in parametrically driven systems

We study two-dimensional localized patterns in weakly dissipative systems that are driven parametrically. As a generic model for many different physical situations we use a generalized nonlinear Schrödinger equation that contains parametric forcing, damping, and spatial coupling. The latter allows for the existence of localized pattern states, where a finite-amplitude uniform state coexists with an inhomogeneous one. In particular, we study numerically two-dimensional patterns. Increasing the driving forces, first the localized pattern dynamics is regular, becomes chaotic for stronger driving, and finally extends in area to cover almost the whole system. In parallel, the spatial structure of the localized states becomes more and more irregular, ending up as a full spatiotemporal chaotic structure.

Figure 1

Evolution of $\left|A\right(x,y,t\left)\right|$, color-coded on the right, for a chaotic localized pattern at $\gamma =0.9$ after an initial time interval $\mathrm{\Delta }t=5.6\times {10}^3$ has elapsed. Note the blue, nonzero background.

Figure 2

Energy function $Q$ of a chaotic localized two-dimensional pattern, as a function of (a) time and (b) its corresponding Fourier power spectrum at $\gamma =0.9$, after a time interval $\mathrm{\Delta }t=5.6\times {10}^3$ has elapsed.

Figure 3

Largest Lyapunov exponent as a function of $\gamma$ at $\nu =-0.05$.

Figure 4

Bifurcation diagram of $Q_{max}$ as a function of $\gamma$. The inset represents the number of different values of $Q_{max}$.

Figure 5

Time average of the width, ${\langle \mathrm{\Delta }\rangle }_t$, as a function of $\gamma$.

Figure 6

A snapshot at $t=5.6\times {10}^3$ of $\left|A\right|$ for localized and extended patterns at different values of $\gamma$: for $\gamma =0.6$ a localized stationary one with ${\lambda }_{max}=-0.007$ (upper left), for $\gamma =0.7$ a localized quasiperiodic one with ${\lambda }_{max}=0.000$ (upper right), for $\gamma =0.9$ a localized chaotic one with ${\lambda }_{max}=0.049$ (lower left), and for $\gamma =1.1$ an extended chaotic one with ${\lambda }_{max}=0.134$ (lower right). Videos can be found in the Supplemental Material [59] for $\gamma =0.7$ and 0.9.

Figure 7

Three-dimensional representation of the localized chaotic structure at $\gamma =0.9$, shown in the lower left part of Fig. 6. Appropriate videos can be found in the Supplemental Material [59].