Showing papers in "Journal of Nonlinear Science in 1996"

Time Integration and Discrete Hamiltonian Systems

Abstract: This paper develops a formalism for the design of conserving time-integration schemes for Hamiltonian systems with symmetry. The main result is that, through the introduction of a discrete directional derivative, implicit second-order conserving schemes can be constructed for general systems which preserve the Hamiltonian along with a certain class of other first integrals arising from affine symmetries. Discrete Hamiltonian systems are introduced as formal abstractions of conserving schemes and are analyzed within the context of discrete dynamical systems; in particular, various symmetry and stability properties are investigated.

The membrane shell model in nonlinear elasticity: A variational asymptotic derivation

Abstract: We consider a shell-like three-dimensional nonlinearly hyperelastic body and we let its thickness go to zero We show, under appropriate hypotheses on the applied loads, that the deformations that minimize the total energy weakly converge in a Sobolev space toward deformations that minimize a nonlinear shell membrane energy The nonlinear shell membrane energy is obtained by computing the Γ-limit of the sequence of three-dimensional energies

Micro-Chaos in Digital Control

Abstract: In this paper we analyze a model for the effect of digital control on one-dimensional, linearly unstable dynamical systems. Our goal is to explain the existence of small, irregular oscillations that are frequently observed near the desired equilibrium. We derive a one-dimensional map that captures exactly the dynamics of the continuous system. Using thismicro-chaos map, we prove the existence of a hyperbolic strange attractor for a large set of parameter values. We also construct an “instability chart” on the parameter plane to describe how the size and structure of the chaotic attractor changes as the parameters are varied. The applications of our results include the stick-and-slip motion of machine tools and other mechanical problems with locally negative dissipation.

Problems and progress in microswimming

Abstract: Stokesian swimming is a geometric exercise, a collective game. In Part I, we review Shapere and Wilczek’s gauge-theoretical approach for a single organism. We estimate the speeds of organisms moving by propagating small amplitude waves, and we make a conjecture regarding a new inequality for the Stokes’ curvature. In Part II, we extend the gauge theory to collective motions. We advocate the influx of nonlinear control theory and subriemannian geometry. Computationally, parallel algorithms are natural, each microorganism representing a separate processor. In the final section, open questions motivated by biology are presented.

Obstruction Results in Quantization Theory

Abstract: Quantization is not a straightforward proposition, as demonstrated by Groenewold’s and Van Hove’s discovery, exactly fifty years ago, of an “obstruction” to quantization. Their “no-go theorems” assert that it is in principle impossible to consistently quantize every classical observable on the phase space R 2n in a physically meaningful way. A similar obstruction was recently found for S 2 , buttressing the common belief that no-go theorems should hold in some generality. Surprisingly, this is not so – it has also just been proven that there is no obstruction to quantizing a torus. In this paper we take first steps towards delineating the circumstances under which such obstructions will appear, and understanding the mechanisms which produce them. Our objectives are to conjecture a generalized Groenewold-Van Hove theorem, and to determine the maximal subalgebras of observables which can be consistently quantized. This requires a

Bifurcation analysis of a metapopulation model with sources and sinks

Abstract: A class of functions describing the Allee effect and local catastrophes in population dynamics is introduced and the behaviour of the resulting one-dimensional discrete dynamical system is investigated in detail. The main topic of the paper is a treatment of the two-dimensional system arising when an Allee function is coupled with a function describing the population decay in a so-called sink. New types of bifurcation phenomena are discovered and explained. The relevance of the results for metapopulation dynamics is discussed.

Rotating n-gon/kn-gon Vortex Configurations

Abstract: We demonstrate the existence of stationary point-vortex configurations consisting ofk vortexn-gons and a vortexkn-gon. These configurations exist only for specific values of the vortex strengths; the relative vortex strengths of such a consiguration can be uniquely expressed as functions of the radii of the polygons. Thekn-gon must be oriented so as to be fixed by any reflection fixing one of then-gons; for sufficiently smallk, we show that then-gons must be oriented in such a way that the entire configuration shares the symmetries of any of then-gons. Necessary conditions for the formal stability of general stationary point-vortex configurations set conditions on the vortex strengths. We apply these conditions to then-gon/kn-gon configurations and carry out a complete linear and formal stability analysis in the casek=n=2, showing that linearly and nonlinearly orbitally stable configurations exist.

A symplectic integrator for riemannian manifolds

Abstract: The configuration spaces of mechanical systems usually support Riemannian metrics which have explicitly solvable geodesic flows and parallel transport operators. While not of primary interest, such metrics can be used to generate integration algorithms by using the known parallel transport to evolve points in velocity phase space.

The Limits of Hamiltonian Structures in Three-Dimensional Elasticity, Shells, and Rods

Abstract: This paper uses Hamiltonian structures to study the problem of the limit of three-dimensional (3D) elastic models to shell and rod models. In the case of shells, we show that the Hamiltonian structure for a three-dimensional elastic body converges, in a sense made precise, to that for a shell model described by a one-director Cosserat surface as the thickness goes to zero. We study limiting procedures that give rise to unconstrained as well as constrained Cosserat director models. The case of a rod is also considered and similar convergence results are established, with the limiting model being a geometrically exact director rod model (in the framework developed by Antman, Simo, and coworkers). The resulting model may or may not have constraints, depending on the nature of the constitutive relations and their behavior under the limiting procedure.

A Nonlinear Extensible 4-Node Shell Element Based on Continuum Theory and Assumed Strain Interpolations

Abstract: A quadrilateral continuum-based C 0 shell element is presented, which relies on extensible director kinematics and incorporates unmodified three-dimensional constitutive models. The shell element is developed from the nonlinear enhanced assumed strain (EAS) method advocated by Simo & Armero [1] and formulated in curvilinear coordinates. Here, the EAS-expansion of the material displacement gradient leads to the local interpretation of enhanced covariant base vectors that are superposed on the compatible covariant base vectors. Two expansions of the enhanced covariant base vectors are given: first an extension of the underlying single extensible shell kinematic and second an improvement of the membrane part of the bilinear element. Furthermore, two assumed strain modifications of the compatible covariant strains are introduced such that the element performs well even in the case of very thin shells.

Nonexistence of travelling wave solutions to nonelliptic nonlinear schrödinger equations

Abstract: By deriving Pohojaev-type identities we prove that nonelliptic nonlinear Schrodinger equations do not admit localized travelling wave solutions. Similary, we prove that the Davey-Stewartson hyperbolic-elliptic systems do not support travelling wave solutions except for a specific range of the parameters that comprises the DS II focusing case (where the existence of lumps is well known).

An Impetus-Striction Simulation of the Dynamics of an Elastica

Abstract: This article concerns the three-dimensional, large deformation dynamics of an inextensible, unshearable rod To enforce the conditions of inextensibility and unshearability, a technique we call the impetus-striction method is exploited to reformulate the constrained Lagrangian dynamics as an unconstrained Hamiltonian system in which the constraints appear as integrals of the evolution We show here that this impetus-striction formulation naturally leads to a numerical scheme which respects the constraints and conservation laws of the continuous system We present simulations of the dynamics of a rod that is fixed at one end and free at the other

Heteroclinic cycles in bifurcation problems with O (3) symmetry and the spherical Bénard problem

Abstract: It has been known since a paper of Armbruster and Chossat ([AC91]) that robust heteroclinic cycles between equilibria can bifurcate in differential systems which are invariant under the action of the groupO(3) defined as the sum of its “natural” irreducible representations of degrees 1 and 2 (i.e., of dimensions 3 and 5). Moreover, these cycles can be seen numerically in the simulation of the amplitude equations resulting from a center manifold reduction of the Benard problem in a nonrotating spherical shell with suitable aspect ratio ([FH86]). In the present work we first generalize the results of [AC91] to the interactions of irreducible representations of degrees l and l+1 for any l>0. Heteroclinic cycles of various types are shown to exist under certain “generic” conditions and are classified. We show in particular that these conditions are satisfied in most cases when the differential system proceeds from a l, l+1 mode interaction bifurcation in the spherical Benard problem.

Dynamical Problems for Geometrically Exact Theories of Nonlinearly Viscoelastic Rods

Abstract: This paper surveys recent results and open problems for the equations of motion for geometrically exact theories of nonlinearly viscoelastic and elastic rods. These rods can deform in space by undergoing not only flexure and torsion, but also extension and shear. The paper begins with a derivation of the governing equations, which for viscoelastic rods form a quasilinear system of hyperbolic—parabolic partial differential equations of high order. It then derives the energy equation and discusses difficulties that can arise in getting useful energy estimates. The paper next treats constitutive assumptions precluding total compression. The paper then discusses the curious asymptotic problems that arise when the inertia of the rod is small relative to that of a rigid body attached to its end. The paper concludes with discussions of traveling waves and shock structure, Hopf bifurcation problems, and problems of control.

Gravity Waves on the Surface of the Sphere

Abstract: We propose a Hamiltonian model for gravity waves on the surface of a fluid layer surrounding a gravitating sphere. The general equations of motion are nonlocal and can be used as a starting point for simpler models, which can be derived systematically by expanding the Hamiltonian in dimensionless parameters. In this paper, we focus on the small wave amplitude regime. The first-order nonlinear terms can be eliminated by a formal canonical transformation. Similarly, many of the second order terms can be eliminated. The resulting model has the feature that it leaves invariant several finite-dimensional subspaces on which the motion is integrable.

Symmetry Methods in Collisionless Many-Body Problems

Abstract: We formulate an appropriate symmetry context for studying periodic solutions to equal-mass many-body problems in the plane and 3-space. In a technically tractable but unphysical case (attractive force a smooth function of squared distance, bodies permitted to coincide) we apply the equivariant Moser-Weinstein Theorem of Montaldiet al. to prove the existence of various symmetry classes of solutions. In so doing we expoit the direct product structure of the symmetry group and use recent results of Dionneet al. on ‘C-axial’ isotropy subgroups. Along the way we obtain a classification of C-axial subgroups of the symmetric group. The paper concludes with a speculative analysis of a three-dimensional solution to the 2n-body problem found by Davieset al. and some suggestion for further work.

Multilayer Beams: A Geometrically Exact Formulation

Abstract: We review and extend our recent work on a new theory of multilayer structures, with particular emphasis on sandwich beams/1-D plates. Both the formulation of the equations of motion in the general dynamic case and the computational formulation of the resulting nonlinear equations of equilibrium in the static case based on a Galerkin projection are presented. Finite rotations of the layer cross sections are allowed, with shear deformation accounted for in each layer. There is no restriction on the layer thickness; the number of layers can vary between one and three. The deformed profile of a beam cross section is continuous, piecewise linear, with a motion in 2-D space identical to that of a planar multibody system that consists of three rigid links connected by hinges. With the dynamics of this multi (rigid/flexible) body being referred directly to an inertial frame, the equations of motion are derived via the balance of (1) the rate of kinetic energy and the power of resultant contact (internal) forces/couples, and (2) the power of assigned (external) forces/couples. The present formulation offers a general method for analyzing the dynamic response of flexible multilayer structures undergoing large deformation and large overall motion. With the layersnot required to have equal length, the formulation permits the analysis of an important class of multilayer structures with ply drop-off. For sandwich structures, an approximated theory with infinitesimal relative outer-layer rotations superimposed onto finite core-layer rotation is deduced from the general nonlinear equations in a consistent manner. The classical linear theory of sandwich beams/1-D plates is recovered upon a consistent linearization. Using finite element basis functions in the Galerkin projection, we provide extensive numerical examples to verify the theoretical formulation and to illustrate its versatility.

Semirotors in the Josephson junction equations

Abstract: We consider the dynamics of arrays ofN-series coupled Josephson junctions, under pure resistive and capacitive loads. In the limit of the junction capacitance becoming large, we prove the existence of semirotor solutions. These are periodic solutions in which the phase difference across the gap ink of the junctions oscillates with small amplitude while the remainingN—k phase differences increase by 2π radians per period. We investigate the stability of these solutions and report observations of chaotic behavior associated with these solutions.

A generalization of the theory of normal forms

Abstract: Normal forms allow the use of a restricted class of coordinate transformations (typically homogeneous polynomials) to put the bifurcations found in nonlinear dynamical systems into a few standard forms. We investigate here the consequences of relaxing the restrictions of the form of the coordinate transformations. In the Duffing equation, a logarithmic transformation can remove the nonlinearity: in one interpretation, the nonlinearity is replaced by a branch cut leading to a Poincare section. When the linearized problem is autonomous with diagonal Jordan form, we can remove all nonlinearities order by order using these singular coordinate transformations.

Reduced spaces for coupled rigid bodies

Abstract: The motion of two identical, axially symmetric coupled rigid bodies with constant linear momentum gives rise to a Hamiltonian system with a fairly large symmetry group, namely,SO(3)×S 1 ×S 1 , which in turn leads to Hamiltonian flows on reduced spaces. In this paper, we illustrate the use of equivariant symplectomorphisms and the reduction in stages procedure in determining the topology of these reduced spaces. It is shown that the reduced spaces corresponding to regular momenta are either two- or four-dimensional and, in the four-dimensional case, the reduced space gets blown up (or blown down) as the momentum value crosses the singular boundary.

Reduced spaces and their relation to relative equilibria

Abstract: It is known that the Hamiltonian motion of a mechanical system with symmetry induces Hamiltonian flows on reduced phase spaces. In this paper we apply Morse theory to study the relationship between the topology of the reduced space and the number of relative equilibria in the corresponding momentum level set. Our attention is restricted to simple mechanical systems with compact configuration space and compact symmetry group. We begin by showing that the set of relative equilibria in a level set of the momentum map is compact. We then employ techniques from Morse theory to prove that the number of orbits of relative equilibria with momentum in the coadjoint orbit of a given regular momentum value is bounded below by the the sum of Betti numbers of the corresponding reduced space when the Hamiltonian is fibre quadratic and the reduced Hamiltonian is nondegenerate. In addition, for a certain class of group actions on the configuration manifold, it is shown that the above result extends to Hamiltonians of the form potential plus kinetic.

Local characterization of invariant sets of an autonomous differential inclusion. Boundary of unstable manifolds

Abstract: An application in robotics motivates us to characterize the evolution of a subset in state space due to a compact neighborhood of an arbitrary dynamical system—an instance of a differential inclusion. Earlier results of Blagodat·skikh and Filippov (1986) and Butkovskii (1982) characterize the boundary of theattainable set and theforward projection operator of a state. Our first result is a local characterization of the boundary of the forward projection ofa compact regular subset of the state space.