Showing papers in "Siam Journal on Applied Mathematics in 1990"

Abstract: A simple model for synchronous firing of biological oscillators based on Peskin's model of the cardiac pacemaker (Mathematical aspects of heart physiology, Courant Institute of Mathematical Sciences, New York University, New York, 1975, pp. 268-278) is studied. The model consists of a population of identical integrate-and-fire oscillators. The coupling between oscillators is pulsatile: when a given oscillator fires, it pulls the others up by a fixed amount, or brings them to the firing threshold, whichever is less. The main result is that for almost all initial conditions, the population evolves to a state in which all the oscillators are firing synchronously. The relationship between the model and real communities of biological oscillators is discussed; examples include populations of synchronously flashing fireflies, crickets that chirp in unison, electrically synchronous pacemaker cells, and groups of women whose menstrual cycles become mutually synchronized.

Abstract: An integro-differential reaction-diffusion equation is proposed as a model for populations where local aggregation is advantageous but intraspecific competition increases as global populations increase. It is claimed that this is inherently more realistic than the usual kind of reaction-diffusion model for mobile populations. Three kinds of bifurcation from the uniform steady-state solution are considered, (i) to steady spatially periodic structures, (ii) to periodic standing wave solutions, and (iii) to periodic travelling wave solutions. These correspond to aggregation and motion of populations.

Abstract: A numerical and analytical study of the Kuramoto–Sivashinsky partial differential equation (PDE) in one spatial dimension with periodic boundary conditions is presented. The structure, stability, and bifurcation characteristics of steady state and time-dependent solutions of the PDE for values of the parameter $\alpha $ less than 40 are examined. The numerically observed primary and secondary bifurcations of steady states, as well as bifurcations to constant speed traveling waves (limit cycles), are analytically verified. Persistent homoclinic and heteroclinic saddle connections are observed and explained via the system symmetries and fixed point subspaces of appropriate isotropy subgroups of $O( 2 )$. Their effect on the system dynamics is discussed, and several tertiary bifurcations, observed numerically, are presented.

Abstract: Phase-locking in a system of oscillators that are weakly coupled can be predicted by examining a related system in which the coupling is averaged over the oscillator cycle. This fails if the coupling is large. It is shown that in the presence of large interactions, a pair or a chain of oscillators may develop a new stable equilibrium state that corresponds to the cessation of oscillation. This phenomenon is robust for neural type interactions and does not happen in systems that are weakly coupled.

Abstract: The authors study a two-dimensional reconstruction algorithm due to D. C. Barber and B. H. Brown, applied to a linearized electrostatic inverse problem. First, the authors demonstrate how this algorithm fits within the framework of inverses of generalized Radon transforms studied by G. Beylkin. Second, an iterative improvement of the Barber–Brown algorithm is constructed based on the conjugate residual method. Several numerical results obtained with this iterative algorithm are presented.

Abstract: It has recently been demonstrated that standard discretizations of the cubic nonlinear Schrodinger (NLS) equation may lead to spurious numerical behavior. In particular, the origins of numerically induced chaos and the loss of spatial symmetry are related to the homoclinic structure associated with the NLS equation. In this paper, an analytic description of the homoclinic structure via soliton type solutions is provided and some consequences for numerical computations are demonstrated. Differences between an integrable discretization and standard discretizations are highlighted.

Abstract: A nonlinear equation for a suspension bridge is shown to have travelling wave solutions by explicit calculation. The shape and speed of the waves is plotted.

Abstract: Estimates are derived for the mean recurrence time of orbits in the neighborhood of an attracting homoclinic orbit or heteroclinic cycle in an ordinary differential equation, subject to small additive random noise. The theory presented is illustrated with numerical simulations of several systems, including ones invariant under symmetry groups, for which such heteroclinic attractors are structurally stable. The physical implications of the work presented are briefly discussed.

Abstract: A very broad framework is given for the investigation of long chains of N weakly coupled oscillators. The framework allows nonmonotonic changes of natural frequency along the chain, differences in coupling strengths, anisotropy in the two directions of coupling, and very general local coupling functions. It is shown that the phase locked solutions of all the systems of oscillators within this framework converge for large N to solutions of a class of nonlinear, singularly perturbed, two-point boundary value problems. Using the latter continuum equations, it is also shown that there are parameter regimes in which the solutions have qualitatively different behavior, with a phase-transition-like change in behavior across the boundary between parameter regimes in the limit $N \to \infty $. Special important cases are discussed, including the effects of local changes in frequencies, local changes in coupling strengths, and different kinds of anisotropy.

Abstract: High-order asymptotic approximations to the equation governing the longitudinal dispersion of a passive contaminant in Poiseuille channel flow are derived, and their validity discussed. The derivation uses centre manifold theory, which provides a systematic and near rigorous approach to calculating each successive approximation. It also enables the derivation of the correct initial conditions for the Taylor model of shear dispersion. Approximations that are valid when the channel is of a varying cross section are systematically derived via this approach, and the effects of time-dependent flow and variable diffusivity are also investigated. The resultant modifications to the effective advection velocity and the effective dispersion coefficient are calculated and some general trends indicated.

Abstract: This paper considers the problem of exit for a dynamical system driven by small white noise, from the domain of attraction of a stable state. A direct singular perturbation analysis of the forward equation is presented, based on Kramers’ approach, in which the solution to the stationary Fokker–Planck equation is constructed, for a process with absorption at the boundary and a source at the attractor. In this formulation the boundary and matching conditions fully determine the uniform expansion of the solution, without resorting to “external” selection criteria for the expansion coefficients, such as variational principles or the Lagrange identity, as in our previous theory. The exit density and the mean first passage time to the boundary are calculated from the solution of the stationary Fokker–Planck equation as the probability current density and as the inverse of the total flux on the boundary, respectively. As an application, a uniform expansion is constructed for the escape rate in Kramers’ problem o...

Abstract: With a one-mode truncation it is possible to reduce the Moore–Greitzer model for compressor instability to a set of three ordinary differential equations These are approached from the point of view of bifurcation theory Most of the bifurcations emerge from a degenerate Takens–Bogdanov bifurcation point The bifurcation sets are completed using the numerical branch tracking scheme AUTO Despite the severity of the truncation, the agreement with experimental results is excellent

Abstract: In electric current computed tomography, patterns of currents or voltages are applied to electrodes on the surface of a body and the resulting voltages or currents are measured. A reconstruction and display of an approximation to the impedance inside the body is then made based on these external measurements.It is shown how the problems of choosing current patterns, electrode size and number can be studied in terms of the spectral properties of certain pseudodifferential operators.The proofs given here are simpler and more general than those in [ IEEE Trans. Medical Imaging, 5 (1986), pp. 91–95] and [Clin. Phys. Physiol. Meas. Suppl. A, (1987), pp. 39–46].

Abstract: The equations governing two models of gasless combustion which exhibit pulsating solutions are numerically solved. The models differ in that one allows for melting of the solid fuel, while the other does not. While both models undergo a Hopf bifurcation from a solution propagating with a constant velocity to one propagating with a pulsating (T-periodic) velocity when parameters related to the activation energy exceed a critical value, the subsequent behavior differs markedly. Numerically both models exhibit a period doubling transition to a $2T$ solution when the bifurcation parameter for each model is further increased. For the model without melting, a sequence of additional period doublings occurs, after which apparently chaotic solutions are found. For the model with melting, it is found that the $2T$ solution returns to the T-periodic solution branch. Then two additional windows of $2T$ behavior are found. After the last such window, the solution no longer returns to the T-periodic solution branch, bu...

Abstract: Simplified equations governing the potential flow and the shape of slender jets and thin sheets of liquid are derived, taking into account surface tension. Families of similarity solutions of these equations are introduced. For jets and for symmetrical sheets they satisfy ordinary differential equations. The properties of these similarity solutions are examined analytically and numerically. They can be used to describe the motion of a liquid sheet on a solid, the thickening and flow following the breaking of jets, the merging of two jets, the formation or closing of holes or slits in sheets, etc. Some applications to such problems are given.

Abstract: In this paper it is shown that the resistors in a rectangular network can be determined by measurements at the boundary of the voltages generated by imposed currents. An algorithm for using the boundary measurements to compute the resistances is also given.

Abstract: The complex dynamics that arise in certain nonlinear partial differential equations in time and in one space dimension are studied. In the general case considered, the equation admits a solitary wave in the form of a pulse tailing off exponentially, fore and aft, with possibly oscillatory character. Complicated solutions are described by a superposition of many such solitary structures in interaction. The description is asymptotic in terms of a parameter that becomes exponentially small as the ratio of typical pulse separation to pulse width becomes large. The outcome is a set of dynamical equations for the motion of the individual pulses with nearest neighbor interactions. This system of ordinary differential equations (ODEs) admits a wide range of patterns, both regular and chaotic. The stability theory of such patterns is sketched and the continuum limit of the lattice-dynamical equations of the pulses is given.

Abstract: The problem of determining an unknown pair $\gamma = ( {\lambda ,\mu } )$ of Lame parameters inside a body by its Dirichlet-to-Neumann data map $\Lambda ( \gamma )$ is considered. Using explicit exact solutions for the case of constant $\gamma $, it is seen that the deflection h between $\gamma + h$ and $\gamma $ is uniquely determined by the first-order approximation of $\Lambda ( {\gamma + h} )$ at $\gamma $.

Abstract: The dynamics of forced excitable systems are studied analytically and numerically with a view toward understanding the resonance or phase-locking structure. In a singular limit the system studied reduces to a discontinuous flow on a two-torus, which in turn gives rise to a set-valued circle map. It is shown how to define rotation numbers for such systems and derive properties analogous to those known for smooth flows. The structure of the phase-locking regions for a Fitzhugh–Nagumo system in the singular limit is also analyzed. A singular perturbation argument shows that some of the general results persist for the nonsingularly-perturbed system, and some numerical results on phase-locking in the forced Fitzhugh–Nagumo equations illustrate this fact. The results explain much of the phase-locking behavior seen experimentally and numerically in forced excitable systems, including the existence of threshold stimuli for phase-locking. The results are compared with known results for forced oscillatory systems.

Abstract: This article is devoted to a connection between Kramer’s sampling theorem and sampling expansions generated by Lagrange interpolation. It is shown that any function that has a sampling expansion in the scope of Kramer’s theorem also has a Lagrange-type interpolation expansion provided that the kernel associated with Kramer’s theorem arises from a second-order Sturm–Liouville boundary-value problem. This new approach, which for a variety of regular and singular Sturm–Liouville problems leads to associated sampling theorems, recovers not only many known sampling expansions but also gives new ways to calculate the corresponding sampling functions. New sampling series are included.

Abstract: Linear reaction-hyperbolic equations of a general type arising in certain physiological problems do not have traveling wave solutions, but numerical computations have shown that they possess approximate traveling waves. Using singular perturbation theory, it is shown that as the rates of the chemical reactions approach $\infty $, solutions approach traveling waves. The speed of the limiting wave and the first term in the asymptotic expansion are computed in terms of the underlying chemical mechanisms.

Abstract: Korteweg–deVries (KdV)-like equations with higher-degree nonlinearity are solved by a direct algebraic technique due to Hereman et al. [J. Phys. A, 19 (1986), pp. 607–628]. For two KdV-like equations, one with fifth-degree nonlinearity, the other a combined KdV and mKdV equation, for particular choices of the coefficients of the nonlinear terms, the kink and antikink solutions found by Dey are recovered. Furthermore, soliton solutions of the combined KdV and mKdV equation are found for all values of the coefficients. Closed-form solutions for the Calogero–Degasperis–Fokas modified mKdV equation are also developed. Applications of the solutions of these equations in quantum field theory, plasma physics, and solid-state physics are mentioned. The Hereman et al. method is illustrated and slightly extended and this direct series method is briefly compared to Hirota’s method.

Abstract: A system of kinetic equations of semiconductors is studied. These are Vlasov–Poisson equations with additional nonlinear Boltzmann operators. First H-theorems for these integral operators are given and their null space is determined. Then the existence and uniqueness of smooth solutions is proven. The proof relies on the construction of a sequence of solutions for a suitable linearized system which converges strongly.

Abstract: The classical uncertainty principle asserts that both a function and its Fourier transform cannot be largely concentrated on intervals of small measure. Donoho and Stark [SIAM J. App. Math., 49 (1989), pp. 906–931] have shown recently that both cannot be largely concentrated on any sets of small measure—in the case of functions on the line or functions on finite cyclic groups and with concentrations measured in $L^2 $. The purpose of this note is to extend these results to functions on $\sigma $-finite locally compact abelian groups, with concentrations measured in $L^p $, $1\leqq p\leqq 2$. The first uncertainty principle on groups, due to Matolcsi and Szucs [D. R. Acad. Sci. Paris, 277 (1973), pp. 841–843], deals with full (rather than large) concentration, asserting that if a function and its Fourier transform are supported by sets T and W, then the product of the Haar measures of T and W must be at least 1. For the case of full concentration in $R^n $, Benedicks [J. Math. Anal. Appl.,106 (1985), pp. 1...

Abstract: A model for the description of the geometry in multiphase mechanics is derived and studied. Equations for the rate of change of mean and Gauss curvature are derived. These coupled equations show that the evolution of these curvatures depend on the velocity of the interface normal to itself. Examples are given to show the utility of these equations. To apply these concepts to multiphase mechanical situations, an ensemble average is applied to the curvature equations. Equations for the evolution of the volume fraction and the interfacial area density are also derived. These equations also depend on the curvatures and the average interfacial velocity. Examples are given to show that these equations give a description of the geometry of multiphase mechanics that is superior to the present description, which describes the force between materials as depending on two geometric parameters, the volume fraction and the average radius of the dispersed material. These examples also suggest constitutive equations to d...

Abstract: In the modern vibration control of flexible space structures and flexible robots, various boundary feedback schemes have been employed to cause energy dissipation and damping, thereby achieving stabilization. The mathematical analysis of eigenspectrum of vibration is usually carried out by classical separation of variables and by solving the transcendental equations. This involves rather lengthy and tedious work due to the complexity and the numerous boundary conditions.A different approach, developed by Keller and Rubinow, uses ideas from wave propagation to obtain asymptotic estimates of eigenvalues for multidimensional scattering problems. This approach is powerful and yields accurate eigenvalue estimates even at a relatively low frequency range [Ann. Physics, 9 (1960), pp. 24–75]. In this paper, we take advantage of this wave approach to study one-dimensional vibration problems with boundary damping. We decompose vibration waves into incident, reflected (including transmitted) and evanescent waves. Ba...

Abstract: A model for complex skin pattern initiation that couples a reaction-diffusion model for the epidermis to a mechanochemical model for the dermis via certain interaction terms is considered. When each of the component models has a range of unstable wavenumbers, the interaction results in a nonlinear superposition of the two patterns generated in the absence of interaction. A nonlinear analysis of the model predicts the solution behaviour in one and two dimensions that is in good agreement with numerical simulations. The interaction can have a significant effect on the patterns that form for all ratios between the linearly unstable wavenumbers of the component models, and a wide range of complex and simple patterns can form depending on this ratio. These patterns are remarkably similar to many of those observed in nature and, to date, have not been able to be generated by a typical reaction-diffusion or mechanochemical theory alone.

Abstract: We consider the problem of filtering one-dimensional diffusions with nonlinear drift coefficients, transmitted through a nonlinear fow noise channel. We construct an asymptotic solution to Zakai’s equation for the unnormalized conditional probability density of the signal, given the noisy measurements. This expansion is used to find the asymptotic expansion of the minimum error variance filter and its mean square estimation error (MSEE). We construct approximate filters whose MSEE agrees with that of the optimal one to a given degree of accuracy. The dimension of the approximate filter increases with the required degree of accuracy. Similarly, we expand the maximum a posteriori probability estimator and the minimum energy estimator and compare their performance. We also discuss some extended Kalman filters and present some examples.

Abstract: An analytical study is conducted of the structure, stability, and bifurcation of the spatially dependent time-periodic solutions of the damped-driven sine-Gordon equation in the nonlinear Schrodinger approximation. Locked states are found for which the spatial structure consists of coherent excitations localized about $x = 0$ or $L/2$. A bifurcation analysis reveals the relationship of these spatially localized solutions to the spatially independent ones and provides a cutoff wavenumber above which there are no spatially dependent solutions; this establishes an upper bound on the number of local excitations comprising the spatial pattern. A linear stability analysis shows that the spatially localized solutions undergo a Hopf bifurcation to temporal quasi-periodicity as the driver amplitude $\Gamma $ is increased. For sufficiently high driver frequencies, the temporally periodic solution regains its stability (via another Hopf bifurcation) in a $\Gamma $-window of finite width before undergoing a third Hop...

Abstract: The expectations of certain integrals of functionals of continuous-time Markov chains over a finite horizon, fixed or random, are estimated via simulation. By computing conditional expectations given the sequence of states visited (and possibly other information), variance is reduced. This is discrete-time conversion. Efficiency is increased further by combining discrete-time conversion with stratification and splitting.