3.5 Oscillations and Waves
35
Fig. 3.12 An edge dislocation (left) and a screw dislocation (center). Right: Photo of B´ enard’s
original experiment with a penta-hepta defect accentuated by colors
dislocation (Fig. 3.12, left), and perpendicular for a screw dislocation (Fig. 3.12,
center).
The edge dislocation is retained in two-dimensional striped patterns. Just count
the number of ridges crossed when one goes around the frame on the lower left-hand
panel of Fig. 3.12, with the plus sign when moving up and the minus sign when
moving down. Again, we have a unit Burgers vector. A striped pattern is, strictly
speaking, not a crystal: it is patterned only in one direction and, defects excluded,
homogeneous in another one. Hexagonal planar patterns contain penta-hepta defects: adjacent cells with five and seven neighbors. One such defect is accentuated
in B´ enard’s convection pattern in the right-hand panel of Fig. 3.12. Chains of such
defects are seen along the boundary between patches with different orientations in
the central panel of Fig. 3.10.
3.5 Oscillations and Waves
Patterns of another kind, which are possible only in systems sustained in a nonequilibrium state, are dynamic. In fluids, all kinds of dynamic wave patterns are
typical, while stationary patterns of the kind mentioned in the preceding section
(where, of course, the fluid still moves within convection cells) need special arrangements. Oscillatory patterns, like stationary ones, need the combination of an
activator with an inhibitor. The simplest model of this kind was the prey–predator
model devised by Alfred Lotka (1910) and later independently by Vito Volterra. The
prey is an activator: it multiplies by itself, and makes growth of the predator’s population possible. The predator is an inhibitor, depressing the prey’s population and
unable to multiply on its own. The original Lotka–Volterra model is defective, as it
has a continuum of oscillatory solutions with different amplitudes depending on an
analogue of “energy” conserved on each orbit. This non-generic feature is brittle,
and disappears when the equations are modified to take into account any realistic
effect, like saturation of the prey’s growth at higher densities.
35
Fig. 3.12 An edge dislocation (left) and a screw dislocation (center). Right: Photo of B´ enard’s
original experiment with a penta-hepta defect accentuated by colors
dislocation (Fig. 3.12, left), and perpendicular for a screw dislocation (Fig. 3.12,
center).
The edge dislocation is retained in two-dimensional striped patterns. Just count
the number of ridges crossed when one goes around the frame on the lower left-hand
panel of Fig. 3.12, with the plus sign when moving up and the minus sign when
moving down. Again, we have a unit Burgers vector. A striped pattern is, strictly
speaking, not a crystal: it is patterned only in one direction and, defects excluded,
homogeneous in another one. Hexagonal planar patterns contain penta-hepta defects: adjacent cells with five and seven neighbors. One such defect is accentuated
in B´ enard’s convection pattern in the right-hand panel of Fig. 3.12. Chains of such
defects are seen along the boundary between patches with different orientations in
the central panel of Fig. 3.10.
3.5 Oscillations and Waves
Patterns of another kind, which are possible only in systems sustained in a nonequilibrium state, are dynamic. In fluids, all kinds of dynamic wave patterns are
typical, while stationary patterns of the kind mentioned in the preceding section
(where, of course, the fluid still moves within convection cells) need special arrangements. Oscillatory patterns, like stationary ones, need the combination of an
activator with an inhibitor. The simplest model of this kind was the prey–predator
model devised by Alfred Lotka (1910) and later independently by Vito Volterra. The
prey is an activator: it multiplies by itself, and makes growth of the predator’s population possible. The predator is an inhibitor, depressing the prey’s population and
unable to multiply on its own. The original Lotka–Volterra model is defective, as it
has a continuum of oscillatory solutions with different amplitudes depending on an
analogue of “energy” conserved on each orbit. This non-generic feature is brittle,
and disappears when the equations are modified to take into account any realistic
effect, like saturation of the prey’s growth at higher densities.