Abstract
A dynamical model of the Swift-Hohenberg type is proposed to describe the formation of twelvefold quasipattern as observed, for instance, in optical systems. The model incorporates the general mechanisms leading to quasipattern formation and does not need external forcing to generate them. Besides quadratic nonlinearities, the model takes into account an angular dependence of the nonlinear couplings between spatial modes with different orientations. Furthermore, the marginal stability curve presents other local minima than the one corresponding to critical modes, as usual in optical systems. Quasipatterns form when one of these secondary minima may be associated with harmonics built on pairs of critical modes. The model is analyzed numerically and in the framework of amplitude equations. The results confirm the importance of harmonics to stabilize quasipatterns and assess the applicability of the model to other systems with similar generic properties.
Highlights
The existence of planar quasipatterns with N -fold rotational symmetry and N 5 has been predicted and studied in a variety of nonlinear dynamical systems
The observation of twelvefold quasipatterns has been reported in an autonomous optical system for a wide range of parameters [18,19,20]
The medium is driven by a laser beam, and the main observation is that, on increasing the input power, bifurcations lead from a homogeneous state to hexagonal structures, and to twelvefold quasiperiodic patterns
Summary
The existence of planar quasipatterns with N -fold rotational symmetry and N 5 has been predicted and studied in a variety of nonlinear dynamical systems. We identify the minimum ingredients for the robust observation of these spatial structures, namely, a marginal stability curve with at least two local minima, quadratic nonlinearities which favor hexagonal triads of spatial modes, and cubic nonlinearities which favor multimode patterns. Due to the quadratic nonlinearities of the dynamics, hexagonal patterns built on equilateral triads of unstable modes may be a solution.
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