The speed of an inclined ruck.

Steady rucks in an elastic beam can roll at constant speed down an inclined plane. We examine the dynamics of these travelling-wave structures and argue that their speed can be dictated by a combination of the physical conditions arising in the vicinity of the 'contact points' where the beam is peeled off the underlying plane and stuck back down. We provide three detailed models for the contact dynamics: viscoelastic fracture, a thermodynamic model for bond formation and detachment and adhesion mediated by a thin liquid film. The results are compared with experiments.

Pattern dynamics in a checkerboard map.

Differential equations often have solutions in the forms of trains of coherent structures such as pulses and antipulses. For such systems, the methods of singular perturbation theory permit the derivation of pattern maps that predict the sequence of spacings between successive pulses. Here we apply such a procedure to cases where two distinct kinds of pulse (or antipulse) may coexist in the system. In that case, direct application of the method leads to multivalued maps that make for complicated descriptions, especially when the succession of pulse types becomes chaotic. We show how this description may be simplified by using maps arrayed in checkerboard style to provide causal descriptions of both the successions of pulse spacings and the order in which the different kinds of pulse go by.

Synchronized family dynamics in globally coupled maps.

The dynamics of a globally coupled, logistic map lattice is explored over a parameter plane consisting of the coupling strength, varepsilon, and the map parameter, a. By considering simple periodic orbits of relatively small lattices, and then an extensive set of initial-value calculations, the phenomenology of solutions over the parameter plane is broadly classified. The lattice possesses many stable solutions, except for sufficiently large coupling strengths, where the lattice elements always synchronize, and for small map parameter, where only simple fixed points are found. For smaller varepsilon and larger a, there is a portion of the parameter plane in which chaotic, asynchronous lattices are found. Over much of the parameter plane, lattices converge to states in which the maps are partitioned into a number of synchronized families. The dynamics and stability of two-family states (solutions partitioned into two families) are explored in detail. (c) 1999 American Institute of Physics.

Synchronizing Moore and Spiegel.

This paper presents a study of bifurcations and synchronization {in the sense of Pecora and Carroll [Phys. Rev. Lett. 64, 821-824 (1990)]} in the Moore-Spiegel oscillator equations. Complicated patterns of period-doubling, saddle-node, and homoclinic bifurcations are found and analyzed. Synchronization is demonstrated by numerical experiment, periodic orbit expansion, and by using coordinate transformations. Synchronization via the resetting of a coordinate after a fixed interval is also successful in some cases. The Moore-Spiegel system is one of a general class of dynamical systems and synchronization is considered in this more general context. (c) 1997 American Institute of Physics.

Checkerboard maps.

When a map has one positive Lyapunov exponent, its attractors often look like multidimensional, Cantorial plates of spaghetti. What saves the situation is that there is a deterministic jumping from strand to strand. We propose to approximate such attractors as finite sets of K suitably prescribed curves, each parametrized by an interval. The action of the map on each attractor is then approximated by a map that takes a set of curves into itself, and we graph it on a KxK checkerboard as a discontinuous one-dimensional map that captures the quantitative dynamics of the original system when K is sufficiently large. (c) 1995 American Institute of Physics.