provided strong experimental and numerical evidence of elastostatical chaos. For
example, solving the 3D-elastica problem of a very long twisted steel tape as an initial
value problem, the numerical solution qualitatively agrees with the random soliton
loops actually observed in experiments.
An experiment you can try setting up for yourself requires only a long flat rubber
band. Stretch it, twist it, and release it very slowly. Loops may form at random
locations along the band, in agreement with the chaotic solutions of (6.65) which
holds for this case too. Considering only the experimental observations, you might
suspect the loops to occur at locations having some kind of material or geometrical
imperfection. However, as a consequence of the statical-dynamical analogy, loops
will form at random even if the structure was ideal with no imperfections at all.
6.8 Spatial and Spatiotemporal Chaos
For some systems the spatial variables may describe disordered patterns, fixed in
time or slowly varying. The hallmark of this phenomenon is a rapid decay of spatial
correlations, so that one cannot predict the formation of more distant patterns from
local observations. This is called spatial chaos, to distinguish it from temporal
chaos associated with disordered behavior in time.
Elastostatical chaos (Sect. 6.7) is one example of spatial chaos. Another type is
related to granular materials or fluids. For example, spatial chaos has been observed
in the form of disordered patterns of standing waves on the surface of a fluid
enclosed in a vertically oscillating container (e.g., Tufillaro et al. 1989).
Certain coupled dynamical systems may display spatial as well as temporal
disorder. Then disordered patterns move around unpredictably in time. This is
called spatiotemporal chaos. See Cross and Hohenberg (1994) for an introduction
to this new and rather unsettled area of research.
6.9 Controlling Chaos
What good is chaos? Research shows that it can be used to stabilize lasers, electronic circuits and animal hearts. Also, one may utilize chaotic dynamics in the
design of systems that can rapidly switch between a large number of states in
response to very small control forces.
Clever uses of chaos may be a promising area for future research, though, at
present it is beyond the scope of this introduction to chaos. We briefly mention
some basic ideas, and refer interested readers to, e.g., Bishop and Xu (1996), Chen
(2000), Ditto and Pecora (1993), Kapitaniak (1993,1996), Lenci S, Rega G (2003),
Myneni et al. (1999), Ott et al. (1990), Pecora and Caroll (1990), Shinbrot et al.
(1990), and Thompson and Bishop (1994).
6.7 Elastostatical Chaos
381
example, solving the 3D-elastica problem of a very long twisted steel tape as an initial
value problem, the numerical solution qualitatively agrees with the random soliton
loops actually observed in experiments.
An experiment you can try setting up for yourself requires only a long flat rubber
band. Stretch it, twist it, and release it very slowly. Loops may form at random
locations along the band, in agreement with the chaotic solutions of (6.65) which
holds for this case too. Considering only the experimental observations, you might
suspect the loops to occur at locations having some kind of material or geometrical
imperfection. However, as a consequence of the statical-dynamical analogy, loops
will form at random even if the structure was ideal with no imperfections at all.
6.8 Spatial and Spatiotemporal Chaos
For some systems the spatial variables may describe disordered patterns, fixed in
time or slowly varying. The hallmark of this phenomenon is a rapid decay of spatial
correlations, so that one cannot predict the formation of more distant patterns from
local observations. This is called spatial chaos, to distinguish it from temporal
chaos associated with disordered behavior in time.
Elastostatical chaos (Sect. 6.7) is one example of spatial chaos. Another type is
related to granular materials or fluids. For example, spatial chaos has been observed
in the form of disordered patterns of standing waves on the surface of a fluid
enclosed in a vertically oscillating container (e.g., Tufillaro et al. 1989).
Certain coupled dynamical systems may display spatial as well as temporal
disorder. Then disordered patterns move around unpredictably in time. This is
called spatiotemporal chaos. See Cross and Hohenberg (1994) for an introduction
to this new and rather unsettled area of research.
6.9 Controlling Chaos
What good is chaos? Research shows that it can be used to stabilize lasers, electronic circuits and animal hearts. Also, one may utilize chaotic dynamics in the
design of systems that can rapidly switch between a large number of states in
response to very small control forces.
Clever uses of chaos may be a promising area for future research, though, at
present it is beyond the scope of this introduction to chaos. We briefly mention
some basic ideas, and refer interested readers to, e.g., Bishop and Xu (1996), Chen
(2000), Ditto and Pecora (1993), Kapitaniak (1993,1996), Lenci S, Rega G (2003),
Myneni et al. (1999), Ott et al. (1990), Pecora and Caroll (1990), Shinbrot et al.
(1990), and Thompson and Bishop (1994).
6.7 Elastostatical Chaos
381