Pattern subharmonic

R. Baumann, K. Kassner, C. Misbah, D. E. Temkin. Spatial subharmonics, irrational patterns, and disorder in eutectic growth. Phys Rev Lett 74 1597, 1995. 

At interesting phenomenon occurs in the case of other resonance horns we have studied it for the case of the 3/1 resonance. The torus pattern breaks when the subharmonic periodic trajectories locked on it for small FA decollate from the torus as FA increases. We are left then with two attractors a stable period 3 and a stable quasi-periodic trajectory. This is a spectacular case of multistability (co-existence of periodic and quasi-periodic oscillations). The initial conditions will determine the attractor to which the system will eventually converge. This decollation of the subharmonics from the torus was predicted by Greenspan and Holmes (1984). They also predicted chaotic trajectories close to the parameter values where the subharmonic decollation occurs. 

So far we have discussed EC instabilities driven by a sinusoidal AC voltage. When the AC driving voltage U t) with period T is asymmetric , i.e. U(t -h T/2) —U t), besides the conductive and the dielectric symmetries there is a subharmonic pattern where the director dynamics is 2T-periodic in time. The effect of flexopolarization on standard and non-standard EC for an asymmetric driving voltage has been analysed. One recovers in principle the scenarios with symmetric driving described... 

The experimental investigation of three-dimensional instabilities of film flows is presented in the paper by Liu, Schneider and Gollub (1995) and several distinct transverse instabilities are found to deform the travelling waves a synchronous mode (in which the deformations of adjacent wave front are in phase) and a subharmonic mode (in which the modulations of adjacent wave front are out of phase - in this case the herringbone patterns result). 

Figure 2.2 Patterns in vertically vibrated granular materials, (a) Overhead view of a square pattern produced by vibrating 60,000 0.55 mm lead spheres in a 55 mm cell at / = 22 Hz and maximum acceleration amplitude of 3.000 times gravity. The pattern is illuminated with low angle lighting to accentuate the peaks and valleys, (b) Oblique view of the square pattern from (a). The pattern is //2 subharmonic, repeating every other drive cycle. On the other cycle, the peaks become valleys and the valleys become peaks, (c) Collection of patterns at various driving frequencies and amplitudes, (top row) //2 Stripes, //2 Hexagons //4 Stripes, (bottom row) //4 Hexagons, //4 Phase bubbles, //4 Phase domain pattern. (Adapted from Bizon, C. et al., Phys. Rev. Lett., 80, 57, 1998 Moon, S.J. et al., Phys. Rev. E, 65, 011301, 2002.)...

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