Spiral states are commonly observed in pattern formation on a sphere. Li et al.(Phys. Rev. E, Vol.71, 016301 [2005]) previously proposed a method to generate the spiral states by adopting mixed modes with adjacent wavenumbers like l = l0, l0 + 1, as the initial condition of a time-developing numerical simulation. Herein, using the Swift-Hohenberg equation as an example, we show that Li's empirical method can be successful only in particular regions of parameter space. Tracing the state by a continuation method generated from the static state via three symmetry breakings, the spiral state may be regarded as a resonant state generated by the combination of a few degenerate unstable modes with different azimuthal wave numbers.
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This work has been supported in part by KAKENHI (20K04294).
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Science and technology reports of Kansai University = 関西大学理工学研究報告