Discrete & Continuous Dynamical Systems - B
2003 , Volume 3 , Issue 4
Nonlinear Differential Equations, Mechanics and Bifurcation
A special issue dedicated to David G. Schaeffer's 60th birthday
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This paper treats the quasilinear evolution equations governing the planar motions of incompressible rods. Since incompressibility is here a 2-dimensional phenomenon, a thickness variable enters the governing equations in an essential and novel way. These equations have a mathematical structure strikingly different from that for compressible rods. In contrast to the case for compressible rods, the governing equations admit a priori upper and lower bounds on the stretches without the viscosity becoming singular when these stretches reach their extremes.
This work explores the use of numerical experiments in two specific cases: (1) the discovery of two families of exact solutions to the elastic string equations, and approximately periodic solutions that appear to exist near pseudo-solutions formed from these families; (2) the study of the diffusion-reaction-conduction process in an electrolyte wedge (meniscus corner) of a current-producing porous electrode. This latter work establishes the well-posedness of the electrolyte wedge problem and provides asymptotic expansions for the current density and total current produced by such a wedge. The theme of this paper is the use of computing to discover a result that is difficult or impossible to find without a computer, but which once observed, can then be proven mathematically.
A hypothesis for the prediction of the circumferential wavenumber of buckling of the thin axially-compressed cylindrical shell is presented, based on the addition of a length effect to the classical (Koiter circle) critical load result. Checks against physical and numerical experiments, both by direct comparison of wavenumbers and via a scaling law, provide strong evidence that the hypothesis is correct.
Based on a well-known discrete bifurcation problem (the discretized Euler buckling problem) displaying a highly complex bifurcation diagram, we show how to find fast, global access to the distribution patterns of classical branch-invariants (symmetry groups, nodal properties, stability characteristics), without actually computing the complex diagram. At the core of our method is a symbolic dynamics based labeling system, which can be viewed itself as a (non-classical) global invariant and from which all the classical invariants can be derived. Based on results from the theory of Brownian bridges an approximate sequence of these integer-labels can be obtained very fast, in fair agreement with the measured quantities. Similar labeling systems have been used in other problems, so we argue that our method will be useful for a wider range of boundary value problems displaying spatial complexity characterized by a mixture of regular and random patterns.
The two-fluid equations for two-phase flow, a model derived by averaging, analogy and experimental observation, have the property (in at least some commonly-occurring derivations) of losing hyperbolicity in their principal parts, those representing the chief entries in modeling conservation of mass and transfer of momentum and energy.
Much attention has centered on reformulating details of the model to avoid this awkwardness. This paper takes a different approach: a study of the nonhyperbolic operator itself. The objective is to understand the nature of ill-posedness in nonlinear, as distinct from linearized, models.
We present our initial study of the nonlinear operator that occurs in the two-fluid equations for incompressible two-phase flow. Our research indicates that one can solve Riemann problems for these nonlinear, nonhyperbolic equations. The solutions involve singular shocks, very low regularity solutions of conservation laws (solutions with singular shocks, however, are not restricted to nonhyperbolic equations). We present evidence, based on asymptotic treatment and numerical solution of regularized equations, that these singular solutions occur in the two-fluid model for incompressible two-phase flow. The Riemann solutions found using singular shocks have a reasonable physical interpretation.
Scaling and renormalization group (RG) methods are used to study parabolic equations with a small nonlinear term and find the decay exponents. The determination of decay exponents is viewed as an asymptotically self similar process that facilitates an RG approach. These RG methods are extended to higher order in the small coefficient of the nonlinearity. The RG results are verified in some cases by rigorous proofs and other calculational methods.
We present a framework for modeling a dry geophysical mass of granular material -- a debris or volcanic avalanche or landslide -- flowing over an erodible surface. We also describe a computing environment that incorporates topographical data into a parallel, adaptive mesh computational algorithm that solves the model equations.
A fundamental property of any material is its response to a localized stress applied at a boundary. For granular materials consisting of hard, cohesionless particles, not even the general form of the stress response is known. Directed force chain networks (DFCNs) provide a theoretical framework for addressing this issue, and analysis of simplified DFCN models reveal both rich mathematical structure and surprising properties. We review some basic elements of DFCN models and present a class of homogeneous solutions for cases in which force chains are restricted to lie on a discrete set of directions.
We consider the problem of active feedback control of Rayleigh-Bénard convection via shadowgraphic measurement. Our theoretical studies show, that when the feedback control is positive, i.e. is tuned to advance the onset of convection, there is a critical threshold beyond which the system becomes linearly ill-posed so that short-scale disturbances are greatly amplified. Experimental observation suggests that finite size effects become important and we develop a theory to explain these contributions. As an efficient modelling tool for studying the dynamics of such a controlled pattern forming system, we use a Galerkin approximation to derive a dimension reduced model.
In this paper some new existence results for sub-harmonics are proved for first order Hamiltonian systems with super-quadratic potentials by using two new estimates on $C^0$ bound for the periodic solutions. Applying the uniform estimates on the sub-harmonics, the asymptotic behaviors of sub-harmonics is studied when the systems have globally super-quadratic potentials.
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