Excitable media, which range from autocatalytic chemical systems to biological cells and tissues, can maintain organized structures in the form of rotating spiral waves of excitation. The dynamics of spiral waves in two-dimensional systems have been shown to be susceptible to control by external fields (such as electric, thermal and optical). In three dimensions, the analogues of spiral waves are scroll waves. Here we show that an external field--a temperature gradient--can be used to control a particular class of scroll waves called scroll rings. The gradient allows scroll rings to be precisely oriented in space, and their spontaneous shrinkage to be accelerated, decelerated or even reversed (so that the ring expands). The temperature gradient also influences the lifetimes of the scroll rings. We suggest that these dynamics are likely to be generic to other types of field gradients and other excitable media.

Scroll waves in an excitable medium rotate about tubelike filaments, whose ends, when they exist, can lie on the external boundary of the medium or be pinned to an inclusion. We derive a topological rule that governs such pinning. It implies that some configurations cannot occur although they might otherwise have been expected. Heart tissue provides an application of these concepts. Computational illustrations based on a FitzHugh-Nagumo model are given.

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The mechanism of destabilization of rotating vortices of chemical activity (scroll waves) induced by a parameter gradient is studied in a gel-immobilized Belousov-Zhabotinsky (BZ) reaction. Destabilization occurs when a temperature gradient above a certain critical value is applied parallel to the filament of an initially simple scroll wave. Direct observation of the vortex filament using video imaging provides evidence that the mechanism of destabilization involves the spontaneous creation of a helical perturbation of the filament which grows and eventually collides with the boundaries of the medium, fragmenting into multiple filaments. Evidence is presented that is phenomenon is the result of an instability caused by a gradient-induced filament drift.

The chaotic dynamics from a nonlinear electronic circuit is shown to exhibit the universal topological structure of maps on an annulus. This suggests that the corresponding universality class is large enough to include physical systems. We suggest that low-dimensional strange attractors fall into a few classes, each characterized by distinct universal topological features.PACS numbers: 05.45.+b Several routes to chaos in dynamical systems have been proposed and analyzed in detail, the best known being the period doubling 1 and the quasiperiodic routes. 2 At the onset of chaos, such dynamical systems exhibit scaling, and certain qualitative and quantitative features of the transition are universal. 3 " 5 Many experimental systems are believed to make similar transitions to chaos, with the path determined by evaluating universal features of the transition that are unique to that path. 6,7 Some examples of universal quantities that can be used for this purpose are the generalized dimensions 4 or singularity spectra 5 of the attracting set and the trajectory scaling function 3 of the orbit at the onset.Much less is known about dynamical systems beyond the onset of chaos. Chaotic experimental systems relax to regions of phase space with very complex structure and zero measure. Generally such objects are fractal, and have structure on ever smaller length scales. They are called strange attractors, and are characterized by metric invariants (which do not change under smooth transformations) such as fractal dimension and Lyapunov exponent. However, they are not universal, and hence do not yield much detailed information about the experimental system: e.g., how to model the dynamics.An alternative class of invariants that can be used to characterize a strange attractor is the set of all periodic orbits. 8 " 10 Periodic orbits are topological invariants; i.e., any change of coordinates will not change the periodicity of an orbit. Thus changing the point of observation or the variable that is being observed in an experiment will not change the cycle structure. This is important since the characterization of the attractor should be robust.The significance of the cycles is seen from the following observations. First, since the strange attractor is the set of points of the phase space visited by the orbit after the transients have settled down, motion on it is ergodic. Thus the orbit of any point P on the strange attractor will make arbitrarily close returns to P. Because of the smoothness and nonlinearity of the dynamics, one should in general be able to move P by a small amount so that the close return becomes exact: i.e., there is a periodic point arbitrarily close to P. This means that the periodic points are dense on the strange attractor. Since the motion on the attractor is chaotic, these cycles have to be unstable.The second important observation is that the structure of the strange attractor in the neighborhood of a periodic point and the motion of points in this neighborhood are determined by the tangent space of the...

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The conjectured universality of the Hausdorff dimension of the fractal set formed by the set of the irrational winding parameter values for critical circle maps is shown to follow from the universal scalings for quadratic irrational winding numbers.