International Journal of Bifurcation and Chaos

Traveling waves of excitation organize physical, chemical, and biological systems in space and time. In the biological context they serve to communicate information rapidly over long distances and to coordinate the activity of tissues and organs. An example of particular beauty, complexity and importance is the three-dimensional rotating scroll wave observed in the Belousov–Zhabotinskii reaction and in the ventricle of the heart. A scroll wave rotates around a filamentous phase singularity that weaves through the three-dimensional medium. At any instant of time the geometry of the scroll wave can be reduced to the spatial arrangement of a ribbon whose edges are the singular filament and the tip of the scroll wave. This ribbon, when it closes on itself, must satisfy the topological constraint L = Tw + Wr, where L is the linking number of the two edges of the ribbon, Tw is the total twist of the ribbon, and Wr is the writhing number of the singular filament. We discuss the origin of this equation and its implications for scroll wave statics and dynamics.

A number of basic algorithms for the numerical analysis and control of bifurcation phenomena are described. The emphasis is on algorithms based on pseudoarclength continuation for ordinary differential equations. Several illustrative examples computed with the AUTO software package are included. This is Part II of the paper that appeared in the preceding issue [Doedel et al., 1991] and that mainly dealt with algebraic problems.

The structure of the bifurcation set for non-symmetric Lorenz model is studied. Regions of existence of simple and complex dynamics are pointed out. It is shown that there exist two qualitatively distinct scenarios of transition to chaos.

Data measurements from a dynamical system may be used to build triangulations and tesselations which can — at least when the system has relatively low-dimensional attractors or invariant manifolds — give topological, geometric and dynamical information about the system. The data may consist of a time series, possibly reconstructed by embedding, or of several such series; transients can be put to good use. The topological information which can be found includes dimension and genus of a manifold containing the state space. Geometric information includes information about folds, branches and other chaos generators. Dynamical information is obtained by using the tesselation to construct a map with stated smoothness properties and having the same dynamics as the data; the resulting dynamical model may be tested in the way that any scientific theory may be tested, by making falsifiable predictions.

This paper is concerned with nodal points, isolated points and cusp points (PD) appearing on a locus of period-doubling bifurcation points and the accurate location of them. Such points can be observed in periodic differential systems (e.g., Duffing's equation) in nonlinear oscillations. In a suitable two-dimensional parameter-plane, such points can be formulated as double (singular) points of plane curves defined by a bifurcation equation, and a method for computing them is proposed.

This paper deals with the foundations of a theory of iterated algebraic functions. The results concern a modification of the shift space of all forward orbits, which one can get as a projective limit of Riemann surfaces. Our interest in this topic stems from computer experiments. The connection between the theory and these experiments is only sketched in the introduction.

The considered endomorphism is

The simplest case of bicritical behavior arises in a system of two logistic maps with unidirectional coupling in the point of a parameter plane where lines of transition to chaos in both subsystems meet. We develop a renormalization group analysis of the bicriticality and find the corresponding fixed point universal function and constants featuring the scaling properties of the second system while the first one is in the Feigenbaum critical state. Fractal properties of the bicritical attractor and its quantitative characteristics (σ-functions, f(α)-spectra, generalized dimensions) are considered. It is shown that the bicriticality may be observed as well in lattice models of flow systems consisting of more than two coupled elements.

The problem of dynamic loss of stability during line-of-sight pursuit navigation of a vehicle under automatic control of a heading sliding mode autopilot is considered. The linear and nonlinear controller gains and the lookahead distance of the guidance law are the main bifurcation parameters. Hopf bifurcation theory techniques are used in order to establish stability of the resulting periodic solutions after the initial loss of stability of straightline motion. The analytical results are confirmed by numerical integrations. It is shown that both linear and nonlinear methods are required in order to evaluate the dynamic response of the system.

In his paper [Takens, 1981] on strange attractors and turbulence, Floris Takens proves a theorem giving conditions under which a discrete-time dynamical system can be reconstructed from scalar-valued partial measurements of internal states. We discuss Takens' theorem in terms suitable for a general audience, and give an alternative and more detailed proof of this important result, making use of two basic facts. The first is the Whitney embedding theorem, which we use in an alternative to Takens' original argument away from periodic points of small periods. Near the periodic points we adapt a proof that typical scalar-output linear time-invariant control systems are completely observable.

This paper discusses synaptic inhibition of one pacemaker neuron by another, using data from living synapses. Spike discharges were assimilated to point processes.

Inhibitory rate scale and behavior form. (i) Forms (p:q locked and others) with similar prevalent spectral components assembled monotonically with p:q. Between different lockings, intermittent, messy and other intermittent forms staggered characteristically; hoppings were interspersed. (ii) Locked, intermittent and messy forms occupied about 1/3 each of the rate scale. Individually, the 1:1, 2:1 and 1:2 locked domains were the widest, and seemed continuous; individual intermittent and messy domains were very narrow.

Step-like inhibitory transients induced abrupt postsynaptic changes opposing them, which over- or under-shot and slowly returned in either orderly or complicated (chaotic?) ways to steady states. Input-output relations around inhibitory trains resembled those of first-order lead-lag systems distorted by asymmetric sensitivity to change and saturation. Postsynaptic natural discharges separated into "slow" less variable, and "fast" more variable categories with somewhat different inhibited behaviors.

Formal modeling is introduced by summarizing comparable models, the data-assumption discrepancies, and reasonable conjectures as to eventual models.

Numerical studies of an apparently stable knotted singularity are used here to seek out regularities of vortex dynamics in three-dimensional excitable media. Initial conditions were contrived to produce a vortex filament in the form of a trefoil (3:2 torus) knot, using the piecewise linear 'B-kinetics', with excitability parameter 'g' set to 0.9. The simulation was pursued for 3000 time units, or about 125 wave rotations, during which time the knot was observed to rigidly 'precess' about its symmetry axis, completing one turn every 96 wave rotations, and also translate along that same axis, at about 1 percent of wavespeed. These motions, described with respect to the local Frenet frame, were shown to be highly correlated with local geometry and twist, yielding possible 'laws of filament motion', which might prove viable in other contexts. The trefoil knot is the first organizing center to be studied in detail which exhibits nonuniform twist, and its dynamical behavior supports the notion that the arclength derivative of twist is an important determinant of filament motion, first postulated by Keener in 1988. The knot also exhibited changes in twist distribution, and, to a lesser extent, local geometry, within one wave rotation period, adding an unforeseen complication to efforts at understanding vortex dynamics. The knot is compact, only about a wavelength in radius, so filament interactions may play a prominent role in determining the stability and behavior of the object. The locus of wave collisions was determined for the knot; in places this surface apparently adjoins the filament itself. In the course of this study, two different criteria for locating the filament were compared, and many analytic and graphical utilities were developed, which may prove useful in further studies of three-dimensional excitable media.

In this letter we report the numerical observation of the co-existence of three distinct chaotic attractors for at least one choice of parameters (α = 15.60, β = 28.58, m₀ = −1/7, m₁ = 2/7) in Chua's circuit. This new phenomenon may have escaped earlier detection because the three attractors are located very close to each other and requires the "zooming" ability of our confinor approach to uncover them.

In a syncytial tissue, the intracellular and extracellular spaces may have different degrees of anisotropy. This property allows generation of a re-entrant wave front in a two-dimensional sheet of tissue after two successive stimuli are applied through the same electrode, if the second stimulus is delivered during the vulnerable phase of the first action potential. The re-entrant wave front contains four phase singularities placed symmetrically about the position of the electrode.

The two-dimensional stationary concentration pattern (Turing structure) recently observed by Castets, Dulos, Boissonade and De Kepper in the –Malonic Acid (CIM) reaction has been qualitatively simulated on the basis of a realistic chemical mechanism.