Volume 3, Issue 1, January 1993
Index of content:
3(1993); http://dx.doi.org/10.1063/1.165974View Description Hide Description
In this note we show how to find patterned solutions in linear arrays of coupled cells. The solutions are found by embedding the system in a circular array with twice the number of cells. The individual cells have a unique steady state, so that the patterned solutions represent a discrete analog of Turing structures in continuous media. We then use the symmetry of the circular array (and bifurcation from an invariant equilibrium) to identify symmetric solutions of the circular array that restrict to solutions of the original linear array. We apply these abstract results to a system of coupled Brusselators to prove that patterned solutions exist. In addition, we show, in certain instances, that these patterned solutions can be found by numerical integration and hence are presumably asymptotically stable.
3(1993); http://dx.doi.org/10.1063/1.165967View Description Hide Description
Reactive lattice gas automata simulations show that Turing structure can form on a mesoscopic scale and are stable to molecular fluctuations in this domain. Calculations on the Sel’kov model suggest that Turing instabilities can give rise to global spatial symmetry breaking in ATP concentration within the cell cytoplasm with a mesoscopic Turing scale well within typical cell dimensions. This leads to a new mechanism for the global breaking of energy distribution in the cell. It also leads to reappraisal of the importance of the Turing effect on extended biochemical spatialstructures and energy transport available to cellmorphogenesis.
3(1993); http://dx.doi.org/10.1063/1.165973View Description Hide Description
In closed systems of the Belousov–Zhabotinsky reaction a large number of dynamic states found in open systems is sampled as they evolve in time. During such slow aging processes of thin solution layers, prepared under appropriately chosen chemical conditions, an unexpectedly rich variety of spiral tip behavior was observed experimentally. Within a (concentration, time) parameter plane, the movement of free ends of waves was classified as follows: (a) in a stable domain–periodic rigid rotation with cores of small (200 μm) or very large (2 mm) diameter; quasiperiodic compound motion along a hypocycle, a straight loopy line or an epicycle; complex meandering composed of possibly more than two components; (b) rectilinear tip motion indicating the boundary of spiral wave stability; and (c) in an unstable domain—shrinking of open ends of wave fronts during propagation. The main properties of these parameters are compared with recently published computer calculations.
3(1993); http://dx.doi.org/10.1063/1.165975View Description Hide Description
The dynamic evolution of a chemical reaction–diffusion pattern and its interaction with hydrodynamicflow is investigated by two‐dimensional velocimetry and spectrophotometry based on microscope video imaging techniques. Oscillatory deformation and turbulent decomposition of chemical wave fronts are observed which are induced by a pronounced oscillatoryflow excited spontaneously in a Belousov–Zhabotinsky solution layer with a free surface.
3(1993); http://dx.doi.org/10.1063/1.165976View Description Hide Description
Fluctuations in resting depth of breathing (tidal volume) at constant breathing rate in the anesthetized adult rat exhibit fractal properties when analyzed by a rescaled range method characterized by a mean (±SD) exponent H=0.83±0.02 and 0.92±0.03 with and without sighs, respectively, for up to 400 breaths. Values of H determined from shuffled tidal volumes and simulated tidal volumes taken randomly from a Gaussian distribution of mean and variance approximating that of the actual data are consistent with the expected value of H=0.5 for an independent random process with finite variances. An empirical description is proposed to predict the change in H with length of time record.
3(1993); http://dx.doi.org/10.1063/1.165977View Description Hide Description
One‐dimensional maps with complete grammar are investigated in both permanent and transient chaotic cases. The discussion focuses on statistical characteristics such as Lyapunov exponent, generalized entropies and dimensions, free energies, and their finite size corrections. Our approach is based on the eigenvalue problem of generalized Frobenius–Perron operators, which are treated numerically as well as by perturbative and other analytical methods. The examples include the universal chaos function relevant near the period doubling threshold. Special emphasis is put on the entropies and their decay rates because of their invariance under the most general class of coordinate changes. Phase‐transition‐like phenomena at the border state of chaos due to intermittency and super instability are presented.
3(1993); http://dx.doi.org/10.1063/1.165978View Description Hide Description
A model of a hard oscillator with analytic solution is presented. Its behavior under periodic kicking, for which a closed form stroboscopic map can be obtained, is studied. It is shown that the general structure of such an oscillator includes four distinct regions; the outer two regions correspond to very small or very large amplitude of the external force and match the corresponding regions in soft oscillators (invertible degree one and degree zero circle maps, respectively). There are two new regions for intermediate amplitude of the forcing. Region 3 corresponds to moderate high forcing, and is intrinsic to hard oscillators; it is characterized by discontinuous circle maps with a flat segment. Region 2 (low moderate forcing) has a certain resemblance to a similar region in soft oscillators (noninvertible degree one circle maps); however, the limit set of the dynamics in this region is not a circle, but a branched manifold, obtained as the tangent union of a circle and an interval; the topological structure of this object is generated by the finite size of the repelling set, and is therefore also intrinsic to hard oscillators.
3(1993); http://dx.doi.org/10.1063/1.165965View Description Hide Description
This paper reports experimental observations of codimension‐two heteroclinic bifurcations in an autonomous third‐order electrical circuit. The paper also reports confirmations by computer simulations. In the laboratory experiments, a pair of programmable resistors are used in order to adjust two bifurcation parameters. In the associated two‐parameter space, several codimension‐one bifurcation sets are experimentally measured to capture codimension‐two bifurcation structures. All of these bifurcation sets are numerically confirmed by exact bifurcation equations which are derived from piecewise‐linear circuit dynamics.
3(1993); http://dx.doi.org/10.1063/1.165966View Description Hide Description
In the description of bifurcations in a family of maps of an n‐torus it is natural to consider phase‐locked regions in the parameter space that correspond approximately to the sets of parameter values for which the maps have invariant tori. The extreme case of phase‐locking is resonance, where the torus map has a periodic orbit. We study a family of maps of an n‐torus that only differ from a family of torus translations by a small nonlinear perturbation. The widths of the phase‐locked regions for this family generally increase linearly with the perturbation amplitude. However, this growth varies to a higher power law for families of maps that are given by trigonometric polynomials (the so‐called Mathieu‐type maps). The exponent of the asymptotic power law can be found by simple arithmetic calculations that relate the spectrum of the trigonometric polynomial to the unperturbed translation. Perturbation theory and these calculations predict that typical resonance regions for the family of Mathieu‐type maps are narrow elliptical annuli. All these results are illustrated in a number of numerical examples.
3(1993); http://dx.doi.org/10.1063/1.165968View Description Hide Description
In this paper is shown how to interpret the nonlinear dynamics of a class of one‐dimensional physical systems exhibiting soliton behavior in terms of Killing fields for the associated dynamical laws acting as generators of torus knots. Soliton equations are related to dynamical laws associated with the intrinsic kinematics of space curves and torus knots are obtained as traveling wave solutions to the soliton equations. For the sake of illustration a full calculation is carried out by considering the Killing field that is associated with the nonlinear Schrödinger equation. Torus knot solutions are obtained explicitly in cylindrical polar coordinates via perturbation techniques from the circular solution. Using the Hasimoto map, the soliton conserved quantities are interpreted in terms of global geometric quantities and it is shown how to express these quantities as polynomial invariants for torus knots. The techniques here employed are of general interest and lead us to make some conjectures on natural links between the nonlinear dynamics of one‐dimensional extended objects and the topological classification of knots.
3(1993); http://dx.doi.org/10.1063/1.165969View Description Hide Description
The existence of spatially chaotic deformations in an elastica and the analogous motions of a free spinning rigid body, an extension of the problem originally examined by Kirchhoff are investigated. It is shown that a spatially periodic variation in cross sectional area of the elastica results in spatially complex deformation patterns. The governing equations for the elastica were numerically integrated and Poincaré maps were created for a number of different initial conditions. In addition, three dimensional computer images of the twisted elastica were generated to illustrate periodic, quasiperiodic, and stochastic deformation patterns in space. These pictures clearly show the existence of spatially chaotic deformations with stunning complexity. This finding is relevant to a wide variety of fields in which coiled structures are important, from the modeling of DNA chains to video and audio tape dynamics to the design of deployable space structures.
3(1993); http://dx.doi.org/10.1063/1.165971View Description Hide Description
This paper presents a new ray theory for the propagation of sound waves in nonuniformly moving media. It is found that the ray equations in weakly inhomogeneous and slowly moving media are analogous to the equations of motion of charged particles in nonuniform electric and magnetic fields. The adiabatic approximation is used to study the problem of the propagation of sound rays in a model of near‐ocean‐bottom waveguide with horizontal flow and slowly varying parameters along the direction of propagation of the wave. A general formula is derived that describes the transverse displacement of the trajectory of the ray relative to the direction of propagation of the wave.
Waveguide propagation of intense electromagnetic radiation in slightly inhomogeneous nonlinear media3(1993); http://dx.doi.org/10.1063/1.165972View Description Hide Description
The propagation of self‐localizing beams of electromagnetic waves in the form of nonlinear waveguides in a slightly inhomogeneous medium is studied analytically and numerically. The trajectories of the axial ray are studied as a function of its direction and the field strength at the initial point on the basis of a nonlinear scalar Helmholtz equation. Analytic expressions are derived. The longitudinal refractive index, the field intensity, and the waveguide radius are plotted as functions of the instantaneous position of the point on the axial ray. Deep penetration of the beam into the opaque region and the position of the screening surface are studied as functions of the parameters of the beam and the medium. A steady‐state 3D problem is analyzed for a power‐law nonlinearity with an arbitrary power. A 2D problem is analyzed for the case of a ponderomotive nonlinearity with saturation.