Biomedical subjects
Publications and source records attributed to Robert Gilmore.
Strange attractors can exhibit bifurcations just as periodic orbits in these attractors can exhibit bifurcations. We describe two classes of large-scale bifurcations that strange attractors can undergo. For each we provide a mechanism. These bifurcations are illustrated in a simple class of three-dimensional dynamical systems that contains the Lorenz system.
We establish conditions for distinguishing between two topologically identical strange attractors that are enclosed by identical bounding tori, one of which is generated by a flow restricted to that torus, the other of which is generated by a flow in a different bounding torus and either imaged or lifted into the first bounding torus.
The algorithm for determining a global Poincaré section is applied to a previously studied dynamical system on R2 x S1 and a one-parameter family of embeddings of the strange attractor it generates into R3. We find that the topological properties of the attractor are embedding dependent to a limited extent. These embeddings rigidly preserve mechanism, which is a simple stretch and fold. The embeddings studied show three discrete topological degrees of freedom: parity, global torsion, and braid type of the genus-one torus bounding the embedded attractor.
Strange attractors with Lyapunov dimension d(L) <3 can be classified by branched manifolds. They can also be classified by the bounding tori that enclose them. Bounding tori organize branched manifolds (classes of strange attractors) in the same way as the branched manifolds organize the periodic orbits in a strange attractor. We describe how bounding tori are constructed and expressed in a useful canonical form. We present the properties of these canonical forms and show that they can be uniquely coded by analogs of periodic orbits of period g-1, where g is the genus. We describe the structure of the global Poincaré surface of section for an attractor enclosed by a genus- g torus and determine the transition matrix for flows between the g-1 components of the Poincaré surface of section. Finally, we show how information about a bounding torus can be extracted from scalar time series.
There is at present a doubly discrete classification for strange attractors of low dimension, d(L)<3. A branched manifold describes the stretching and squeezing processes that generate the strange attractor, and a basis set of orbits describes the complete set of unstable periodic orbits in the attractor. To this we add a third discrete classification level. Strange attractors are organized by the boundary of an open set surrounding their branched manifold. The boundary is a torus with g holes that is dressed by a surface flow with 2(g-1) singular points. All known strange attractors in R3 are classified by genus, g, and flow type.
A strange attractor (SA) with symmetry group G can be mapped down to an image strange attractor SA without symmetry by a smooth mapping with singularities. The image SA can be lifted to many distinct structurally stable strange attractors, each equivariant under G, all with the same image SA. If the symbolic dynamics of the image SA requires s symbols sigma(1),sigma(2), em leader,sigma(s), then |G|s symbols are required for symbolic dynamics in the covers, and there are |G|(s) distinct equivariant covers. The covers are distinguished by an index. The index is an assignment of a group operator to each symbol sigma(i):sigma(i)-->g(alpha(i)). The subgroup H subset G generated by the group operators g(alpha(i)) in the index determines how many disconnected components (|G|/|H|) each equivariant cover has. The components are labeled by coset representatives from G/H. The structure of each connected component is determined by H. A simple algorithm is presented for determining the number and the period of orbits in an equivariant attractor that cover an orbit of period p in the image attractor. Modifications of these results for structurally unstable covers are summarized by an adjacency diagram.
This article details a new method that has been developed to transmit auditory and visual information to people who are deaf or hard of hearing. In this method, ordinary fluorescent lighting is modulated to carry an assistive data signal throughout a room while causing no flicker or other distracting visual problems. In limited trials with participants who are deaf or hard of hearing, this assistive system, combined with commercial voice recognition software, showed statistically significant improvement in sentence recognition compared to recognition of audio-only or audio-plus-speech-reading stimuli.
Type-I intermittencies are common phenomena that are often observed in the neighborhood of periodic windows when a control parameter is varied. These intermittencies usually have a single reinjection channel, that is, a single type of laminar phase was observed. Recently, type-I intermittencies with two reinjection channels were reported in several systems. In this paper, it will be shown that type-I intermittencies with n channels of reinjection are associated with the coexistence of n stable periodic orbits that are mapped into each other under a symmetry. A procedure to build type-I intermittency with n reinjection channels using the n-fold cover of an image system is presented. Cases up to n=3 are explicitly given with the covers of the centered Rössler system.