Nonlinear science abstracts

NONLINEAR

SCIENCE ABSTRACTS

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Classification Because ofthe

interdisciplinary

side) is simply

the subject

characterntic

(‘ode

nature ofthe

matter:

subject matter

the second category

of

thisjournal.

(on the right-hand

index

articles are classified side). attempts

(‘ode

Subject

listed on the left-hand

nonlinear process or to isolate the

processes or mathematical

techniques

WI

w2 w3 w4 W5 W6 w7 W8 w9

Weakly interacting waves or modes Modulation and averaging Solitary and other strongly nonlinear waves Shock waves Waves in reacting media Random media or random initial conditions Vortices Scattering Inverse problems of wave propagation Regular motion, stability, and asymptotics

Reaction kinetics and inorganic reaction Combustion Phase equilibria Thermal chemistry Electrochemistry Crystallography and liquid crystals Polymers Surface and colloid chemistry Biochemistry Other

mechanisms

RI R2 R3 R4 R5 R6

I2 I3

Ml M2 M3 M4 M5 M6 M7 MX M9 MI0 MII Ml2

Ordinary differential equations Partial differential equations Other equations (difference, delay, integral, Mappings Abstract dynamical systems Numerical analysis and computation Functional analysis Probability and stochastic processes Algebraic and differential geometry Control theory Programming Theory of computation

PI P2 P3 P4 P5 P6 P7 P8 P9 PI0 PII PI2

Physics Classical mechanics Classical field theory Statistical mechanics Fluid mechanics Solid mechanics Plasma physics Solid state physics Quantum mechanics Optics and lasers Quantum fields, elementary Relativity theory Other

Al A2 A3 A4

Galactic dynamics Stellar dynamics Planetary fluid dynamics Other

I4 15

Equilibrium problems El E2 E3

Nonlinear boundary value problems Free boundary problems Nonlinear eigenvalue problems Mathematical

Tl 72 T3

T5 T6 T7 T8 l-9 TIO TII T12 Tl3

and planetary physics

Statistical description of turbulence (chaos) Chaos in conservative systems Chaos in nonconservative systems Transition to turbulence (chaos) Universality. self-similarity, complex time behavior

operator)

T4

particles

Asymptotics in time, space or parameter Periodic orbits, invariant tori, KAM theory Bifurcation Stability theorems Onset of instability Phase transitions, cooperative phenomena, pattern formation Irregular motion

II

Mathematics

Astrophysics

Nonlinear

Wave propagation

Biochemical dynamics Morphological development Active membranes Nerve impulse dynamics Neural dynamics and networks Genetic diffusion Evolutionary mechanisms Population dynamics Other Chemistry

Cl c2 C3 c4 C5 C6 c7 C8 c9 Cl0

The ftrst category(

the dominant

mathematical approach

Biology BI 82 83 B4 B5 B6 87 BX El9

in two categories.

to descnbe

techniques

Exact solution Perturbation methods Phase space geometry (qualitative theory of D.E.‘s) Spectral theory Completely integrable systems, isospectral deformation Dynamical systems theory Hamiltonian mechanics Functional analysis Probability and stochastic processes Complex analysis Algebraic or differential geometry/ topology Computer simulation Algebra (groups, Lie. .)

Nonlinear science abstracts

263

(~4,T8) INITIAL VALUES OF NON-NFGATIVE SOLUTIONS 6F A FILTRATION EQUATION. Maura Ughi, Instituto Mathematico "U. Dini", Universita di Firenze Viale Morgagni, 67/A - 50134 Firenze, Italy. In this paper we are concerned with the existence and uniqueness, in the form of a local measure, for non-negative solutions of the equation of We show that, under filtration of gas through porous medium (A+(u) = u ) . s&lution has a unique locally any non-negative suitable conditions on 4(u) , finite measure as an initial value, its mass being controlled by the value of the solution at any point in any positive time. JOURNAL: Journal of Differential Equations 1

(Ml,T7) A PERTURBATION THEORY NEAR CONVEX HAMILTONIAN SYSTEMS. Ivar Ekland, CEREMADE, Universite Paris-9 Dauphine, 75775 Paris Cedex 16, France. R" x R", depending be a family of functions on Let H_(x,p) E E we seek periodic solutions to on the parameter E . For small Hamilton's equations dp dp. dxi = a HE(x,p) , 2 = 1 H$x,p) 2

dt

api

ax

dt

and we wish to relate them to the periodic soiutions of the unperturbed system The basic assumption is that HO be convex in all variables (x,p) (E = 0). jointly. We then apply the implicit function theorem to the dual action functional, as introduced by F. Clarke and the author. The resulting mathematical statement unifies several classical results - asymptotic expansions near isolated closed orbits, bifurcating trajectories in continuous families of closed orbits, and Weinstein's theorem on the existence of n closed orbits near an equilibrium (the energy level H(x,p)=h being prescribed throughout). Part of this work was done in collaboration with J. Blot. JOURNAL: Journal of Differential Equations 3

(Ml,R2) THE NUMBER OF PERIODIC SOLUTIONS OF 2-DIMENSIONAL PERIODIC SYSTEMS. Fumio Nakajima, Department of Mathematics, Iwate University, Moroika, Japan; George Seifert, Department of Mathematics, Iowa State University, Ames, Iowa, USA. We consider a real 2-dimensional scalar system

.

u

=

tJ(t,u,v),

G = V(t,u,v),

(0 =

& >,

(1)

w-periodic in t, w > 0. where U and V are continuous in (t,u,v) and for each fixed t and Theorem. Suppose U and V are analytic in (u,v) -_I_Suppose there is a compactset satisfy au/au + av/av) ---w -periodic solutions of D for t sufficiently large. Then the number of (1)is finite. This result3is axedoaforced Duffing equation with sal -damping: u + c; + au + f3u = B cos t, where c, a, 8, B are positive constants. Two examples are given; the first is a 2-di&ensionai system which satisfies all the hypotheses of the above the_orem except that the analyticity condition, and which has an condition on U and V is replaced by a C infinite number of w-periodic solutions. 'The second example is a 3-

264

Nonlinearscience abstracts

dimensional system which satisfies all the conditions of our theorem and which has a non-denumerably infinte set of w -periodic solutions. JOURNAL: Journal of Differential Equations (Pl,R2) ON AVERAGING, REDUCTION, AND SYMMETRY IN HAMILTONIAN SYSTEMS. Richard C. Churchill, Department of Mathematics, Hunter College (CUNY), New York; Martin Rummer, Department of Mathematics, University of Toledo, Ohio, USA; David L. Rod, Department of Mathematics and Statistics, University of Calgary, Canada. The existence of periodic orbits for Hamiltonian systems at low positive energies can be deduced from the existence of non-degenerate critical points of an averaged Hamiltonian on an associated "reduced space". Alternatively, in classical (kinetic plus potential energy) Hamiltonians the existence of such orbits can often be established by elementary geometrical arguments. The present paper unifies the two approaches by exploiting discrete symmetries, including reversing diffeomorphisms, that occur in a given system. The symmetries are used to locate the periodic orbits in the averaged Hamiltonian, and thence in the original Hamiltonian when the periodic orbits are continued under perturbations admitting the same symmetries. In applications to the Henon-Heiles Hamiltonian, we illustrate how "higher order" averaging can sometimes be used to overcome degeneracies encountered at first order. JOURNAL: Journal of Differential Equations 4

5

(Ml,T7) N-BODY SPATIAL PARABOLIC ORBITS ASYMPTOTIC TO COLINEAR CENTRAL CONFIGURATIONS. Clark Robinson and D. G. Saari. An orbit for the Newtonian N-body problem is called completely parabolic when all the mutual distances between particles approach infinity while the velocities tend to zero as time goes to infinity. After scaling the size, it is known that the configuration of a parabolic orbit approaches a central configuration. This paper studies the manner in which the scaled position vectors approach the limiting central configuration and the dependence of this approach on the mass ratios. In particular it is shown that for certain mass ratios for three bodies, there exist parabolic orbits with an infinite spin about the limiting axis. It was the possible existence of this infinite spin about the angular momentum axis which kept N. Hulkower from extending his results on parabolic orbits for the planar three body problem to the full three dimensional problem [N. Hulkower, The zero energy three body problem, Indiana Univ. Math. J. 27 (1978), pp. 409-4481. D. Saari and N. Hulkower had previously extended these results to the N-body problem, [D. Saari and N. Hulkower, On the manifolds of total collapse orbits and of orbits for the N-body problem, J. Diff. Equat. 41 (1981), pp. 27-431. They have shown the amount of spin for complete collapse of N-bodies is finite. They also obtained results about the spin for a completely parabolic orbit. The result of this paper concerning specific parabolic orbits for the isosceles three body problem is closely related to those obtained for complete collapse in [R. Devaney, Triple collision in the planar isosceles three body problem, Inventiones Math. 60 (1980), pp. 249-2671. JOURNAL: Journal of Differential Equations

Nonlinear science abstracts

265

6

(Ml, 13) GENERALIZED AIMOST PERIODIC SOLUTIONS AND ERGODIC PROPERTIES OF QUASI-AUTONOMOUS DISSIPATIVE SYSTEMS. Alain Haraux, Laboratoire d' Analyse Numerique, T 55-65, Univ. P et M. Curie 4, Place Jussien, 75005 Paris, France. Let I-Ibe a real Hilbert space, A a maximal monotone operator in H and f : R + H a measurable function which is S1-almost periodic. Assuming that the positive trajectories of the equation du/dt + Au(t)3 f(t) are bounded (in H) for t > 0, we construct a kind of generalized almost periodic "solution" for the-equation, and we show how to deduce information of ergodic type on the asymptotic behavior of the trajectories as t++=. JOURNAL: Journal of Differential Equations (M~,RI) ASYMPTOTIC BEHAVIOR OF TRAJECTORIES FOR SCME NON AUTONOMOUS, ALMOST PERIODIC PROCESSES. Alain Haraux, Laboratoire d' Analyse Numerique, T 55-65, Univ. P. et M. Curie, 4, Place Jussien, 75005 Paris, France. In this paper, asymptotics are studied for some almost periodic processes on a complete metric space (X,d). In section 1, we show that any precompact positive trajectory of a contractive periodic process is asymptotically almost periodic as t++dD. This property does not hold for general almost periodic, contractive processes. In section 2, a compactness result is obtained for weakly almost periodic complete trajectories of some (possibly nonlinear) processes in a uniformly convex Banach space. In section 3, existence of almost periodic trajectories is studied for "affine" processes in a uniformly convex Banach space. These results are applicable to some evolution equations of the form du/dt + A(t)u(t)3f(t), where f(t) is almost periodic: R + V uniformly convex Banach space and A(t) is a periodic, time dependent, m-accretive operator in V. JOURNAL: Journal of Differential Equations 7

(M3,13) EXISTENCE OF CHAOS IN CONTROL SYSTEMS WITH DELAYED FEEDBACK. U. an der Heiden, Universittt Bremen, NW 2, D-28 Bremen 33 and H. 0. Walther, Mathematisches Institut der Universitft, Theresienstr. 39, &8 Miinchen2, F.R. Germany. It is proved that the difference-differentialequation has, for continuous nonlinear functions f and I;(t)- f(x(t-1)) - a x(t) parameters a > 0, a chaotic solution manifold, i.e., there are infinitely many periodic solutions with different periods and there are infinitely many not asymptotically periodic, bounded and mixing solutions. JOURNAL: Journal of Differential Equations 8

9

(M2,E2) A FREE-BOUNDARY PROBLEM FOR A DEGENERATE PARABOLIC SYSTEM. Emmanuele DiBenedetto, Mathematics Research Center, University of Wisconsin, Madison, WI 53706, and R. E. Showalter, Department of Mathematics, RIM 8.100, The University of Texas, Austin, TX 78712, USA. The first initial-boundary value problem is shown to be well-posed which is for the system aa(e)/at + he++)3f,,as(+)/at - 114 + h(+-e)Bf related to a free-boundary problem of Stefan type. The pair a,8 are maximal monotone graphs. The origin of the system as a model for conduction is

266

Nonlinear science abstracts

discussed. Then under appropriate additional assumptions the non-negativity, boundedness and continuity of a solution are established. JOURNAL: Journal of Differential Equations 10

(M2,El) UNIQUENESS OF SOLUTIONS OF NONLINEAR DIRICHLET PROBLEMS. Wei-Ming Ni, School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, USA. In this paper, we consider the uniqueness of radial solutions of the nonlinear giricllet2problem Au + f(u) =O inn with u=O on an, where f satisfies some appropriate conditions and n is ,"bou~%~ ,"d%i domain in Rn which possesses radial symmetry. Our uniqueness results apply to for instance, f(u) - up , p > 1, or more generally Xu + Cisl aiubi, x,0, with appropriate ai> 0 and pi > 1, upper bounds, and Q a ball or an annulus. JOURNAL: Journal of Differential Equations 11

(M2,T8) THE SPACE BV IS NOT ENOUGH FOR HYPERBOLIC CONSERVATION LAWS. Kuo-Shung Cheng, Department of Applied Math, National Chiao Tung University, Hsinchu, Taiwan 300, Republic of China. We give two examples to show that the solution of Ut + f(u)x = 0 is not a BV function when f(O) is not uniformly convex or concave. JOURNAL: Journal of Mathematical Analysis and its Applications. 12

(M2,T8) CONSTRUCTING SOLUTIONS OF A SINGLE CONSERVATION LAW. Kuo-Shung Cheng, Department of Applied Math, National Chiao Tung University, Hsinchu, Taiwan 300, Republic of China. We provide a method to construct the solutions of single hyperbolic conservation law for a bounded and piecewise monotone initial data. Our method employs Lax's explicit formula very naturally and effectively. JOURNAL: Journal of Differential Equations 13

(Ml,R3) FUNCTIONAL DEPENDENCE AND BOUNDARY-VALUE PROBLEMS WITH FAMILIES OF SOLUTIONS. W. S. Loud, School of Mathemtaics, University of Minnesota, 206 Church Street, S. E. Minneapolis, Minnesota 55455, USA. A. Vanderbauwhede, Instituut voor Theoretische Mechanica, Rijksuniversiteit Gent, B-9000 Gent, Belgium. A boundary-value problem (1) x' = f(t,x,X), (2) x(l) = $ (x(O), X) will usually have isolated solutions. However in some cases, the equation determining the initial conditions of solutions of (I), (2) will exhibit for some values of X a "functional dependence", which will result in families of solutions depending on one or several parameters. Conditions are developed under which there is such a family of solutions for all values of X in a neighborhood of X = 0 , and also conditions under which there is a family of solutions for X = 0 , but there are only isolated solutions for small nonzero X . The important condition is the existence of a first integral V(t,x,X) for (1) which is "preserved" by the boundary condition (2), i.e. V(1, +(x,X), X) = V(O,x,X) for a relevant set of values of (x,X).

267

Nonlinear science abstracts

An abstract theorem is given which puts the idea of functional dependence in a Banach space setting, with consideration given to existence of families of solutions for equations in a Banach space. The abstract results are then applied to the boundary-value problem (l),(2). The question of linear boundary-value problems is studied with respect to allowable ranks for the matrices in the boundary conditions which will permit a family of solutions of a prescribed dimension. The results of families of solutions of (l), (2) are applied to the question of families of periodic solutions when (l), (2) is a periodic system. Examples are given of a two-dimensional boundary-value problem with families of solutions. JOURNAL: Journal of Differential Equations 14

(M3,T8) SOLVABILITY OF SOME OPERATOR EQUATIONS AND PERIODIC SOLUTIONS OF NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS. A. Canada and P. Martinez-Amores, Departamento de Ecuaciones Funcionales, Universidad de Granada, Granada, Spain. In this paper we give sufficient conditions for the solvability of operator equations Lx =INx in normed spaces, with L a linear noninvertible operator but with a finite-dimensional kernel and N a nonlinear operator. The proofs of our theorems use a Leray-Schauder type existence theorem in the frame of the coincidence degree theory. Our theorems generalize some results of J. Mawhin (The solvability of soms operator equations with a quasibounded nonlinearity in normed spaces. J. Math. Anal. Appl. 45, (1974), 455-467). The aim of our abstract theorems is to apply them to the problem of the existence of periodic solutions for periodic vector functional differential equations of arbitrary order of the form x(~)+A x(~-')+...+A x' = g(t , x t’ 1 m-l

x'tw*Pt

(m-1)). m> 1.

The main assumptions on the nonlinearity g are a growth condition which generalize to the vector case those considered by J. R. Ward (Asymptotic conditions for periodic solutions of ordinary differential equations. Proc. Amer. Math. Sot. 81, 3, (1981), 415-420); in particular, we include cases where N is quasibounded or it is of exponential type, and an asymptotic condition related with the conditions of Landesman-Lazer type. some applications and examples are given. JOURNAL: Journal of Differential Equations 15

(M2,W3) STABLE AND UNSTABLE MANIFOLDS FOR THE NONLINEAR WAVE EQUATION WITH DISSIPATION. Clayton Reller, Department of Mathematics, College of the

U

Assume that

tt

+

aut -Au + f(u) = 0 ,

(*)

f'(O) > 0, and Assume further that this condition amounts to the degeneycy of u. as a critical point f8r the energy functional J(u) = /(1/2/Vul + F(u))dx where F(u) = jIf(s>ds. Then, in a neighborhood of = 0,

268

Nonlinear science abstracts

1 2 there exists a non-empty finite-dimensional invariant (u_,O)EH=H XL , manyfold (resp., an infinite-dimensional invariant manifold S) such that any solution of (*> with initial data (u(0),ut(O)) in U (resp., T) approaches (uo,O) in H as t + - 0~ (resp., t + + =) . Because is unstable as a solution of (*). u*+,u JOtiNAL: Journal of Differential Equations 16

(M7,TlO) ON A GENERALIZED HILBERT PROBLEM. Roger G. Newton, Physics Department, Indiana University, Bloomington, IN 47405, USA. The problem analyzed is to find functions f, , meromorphic in Cf , respectively, with values that are linear operators on a Banach space, and such that their boundary values on R satisfy the equation f = wf, , where the operator-valued function w as well as the positions of tFe poles of f* and the ranges of their residues are given. Uniqueness results are obtained, under certain conditions an index is proved to exist, and the determination of f* is reduced to the solution of a generalization of Marchenko's fundamental equation. The results are applied to inverse scattering and inverse spectral problems. JOURNAL: none given 17

(~3,T8) MEAN-FIELD BOUNDS AND CORRELATION INEQUALITIES. Alan D. Sokal, Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USA. I prove a new correlation inequality for class of N-component classical ferromagnets (l< N < 4). This inequality implies that the correlation functions decay ex';onentiallyand the spontaneous magentization is zero, above the mean-field critical temperature. JOURNAL: Journal of Statistical Physics 18

(P4,T7) SINGULAR POISSON TENSORS. Robert G. Littlejohn, University of California, Los Angeles, CA 90024, USA. The Hamiltonian structures discovered by Morrison and Greene for various fluid equations were obtained by guessing a Hamiltonian and a suitable Poisson bracket formula, expressed in terms of noncanonical (but physical) coordinates. In general, such a procedure for obtaining a Hamiltonian system does not produce a Hamiltonian phase space in the usual sense (a symplectic manifold), but rather a family of symplectic manifolds. To state the matter in terms of a system with a finite number of degrees of freedom, the family of symplectic manifolds is parametrized by a set of Casimir functions, which are characterized by having vanishing Poisson brackets with all other functions. The number of independent Casimir functions is the corank of the Poisson tensor Jij the components of which are the coordinates among themselves. Thus, thesi Casimir function exist only when the Poisson tensor is singular. JOURNAL: none given 19

(M5,Tll) THE MORSE INDEX OF AN ISOLATED INVARIANT SET IS A CONNECTED SIMPLE SYSTEM. Henry L. Kurkland, Department of Mathematics, Boston University, Boston, Mass. 02215, USA. C. Conley in his monograph Isolated Invariant Sets and the Morse Index, CBMS regional conference series 1138,defines an index for isolated invariant sets of a flow. For given isolated invariant set, this index

Nonlinearscience abstracts

269

consists of a collection of pointed topological spaces, which are called index spaces and which are formed from quotients of subsets of phase space, all of the same homotopy type (the common homotopy type is the homotopy index of the isolated invariant set) and a collection of homotopy classes of maps between the index spaces where each homotopy class is represented by an admissible finite composition of flow and inclusion induced maps between index spaces. The two collections form the objects and morphisms respectively of a small category which is called by Conley, the Morse index. The Morse index of an isolated invariant set generalizes the classical Morse index of a nondegenerate critical point of a vectorfield in that the classical index is a non-negative integer n where n is the dimension of the unstable nmnifold to the critical point, and considered as an isolated invariant set, the homotopy index of the critical point is the homotopy type of a pointed n-sphere. A complete development of the Morse index is given in the title paper of this abstract which is independent of Conley's monograph, except for the proof of existence of index pairs used to form index spaces. The main object of the paper is to prove that between any two objects in the Morse index there exists one and only one morphism between them in the Morse index. This fact embodies the defining property of "connected simple system." This unique morphism is always a homotopy equivalence, and, in particular, the only selfmorphism of an index space is the homotopy class of the identity. The result is important for the further development of the index to allow application to existence proofs in differential equations, and in subsequent papers by the author will be applied to non-linear reaction-diffusionequations. JOURNAL: Journal Differential Equations 42 (1981), 234-259. 20

(P3,T5) TIME CORRELATION FUNCTIONS FRM LINEAR AND NONLINEAR KINETIC EQUATIONS. James W. Dufty, Department of Physics, University of Florida, Gainesville, Florida 32611, USA; Rosalio F. Rodriguez, Department0 de Fisica, Facultad de Ciencias, UNAM, 09850 Mexido, D.F., Mexico. A wide class of time correlation functions may be represented in terms of the effective dynamics in the single particle phase space by performing a partial average over all degrees of freedom except one. It is observed that the associated kinetic equation for this reduced description is necessarily linear for the class of correlation functions considered, Nevertheless, the nonlinear Boltzmann equation has been applied in this context and shown to accurately predict mode coupling effects at low density. This apparent paradox is resolved by showing that the correlation functions may be calculated instead from an equation for the microscopic phase space density, before averaging over any degrees of freedom. The microscopic phase space density satisfies a nonlinear equation (the Klimontovich equation) which for hard spheres is closely related to the non-linear Boltzmann equation. The relationship between these two descriptions is discussed and some of the advantages of calculating correlation functions from approximate solutions to the Klimontovich equation for the phase space density are described. As an example, the dense fluid ring kinetic equation is obtained simply and directly. JOURNAL: Journal of Statistical Physics (submitted)

270

Nonlinear science abstracts

21

(M3,TB) A CLASS OF QUASILINEAR PARABOLIC EQUATIONS WITH INFINITE DELAY AND APPLICATION TO A PROBLEM OF VISCOELASTICITY. Michael Renardy, Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, Minnesota 55455, USA. A semigroup approach to differential-delayequations is developed which reduces such equations to ordinary differential equations on a Banach space of histories and seems more suitable for certain partial integrodifferential equations than the standard theory. The method is applied to prove a local-time existence theorem for equations of the form utt = On a formal level, it is demonstrated that g(uXt,uk)X,, where ag/au > 0 . that the stretching of fil&nts of viscoelastic liquids can be described by an equation of this form. JOURNAL: Journal of Differential Equations 22

(PB) QUANTUM MECHANICS AND NONLINEAR WAVES. Philip Barnes Burt, Clemson University, Clemson, South Carolina, 29631 USA. A distinctive and fundamental property of interacting quantum systems is that through virtual processes the interaction is always present. This persistent interaction and its concomitant intrinsic nonlinearity is discussed in this book by first examining canonical quantization in order to isolate the physical principles from their customary mathematical framework. Irrespective of the form of field equations quantum theory as an independent, linear superposition principle for probability amplitudes. Effects of persistent interactions are described in these amplitudes by using operator solutions of the nonlinear field equations along with this principle. The inclusion of persistent interactions eliminates divergences in Boson amplitudes through proper normalization. Important application of these ideas include the description of spin zero meson resonances, pion-pion scattering, quasi particles in Josephson junctions and spontaneous mass generation through self interactions. An extensive discussion of intrinsic nonlinearity in classical systems, especially plasma, provides a basis for generalization to quantum theories. Classical nonlinear excitations such as solitary waves, solitons and their recent generalizations are studied in detail Published by Halwood Academic Publishers 23

(PlO,T2) FAILURE OF PERTURBATION THEORY IN NONLINEAR FIELD THEORIES. Philip B. Burt, Department of Physics and Astronomy, Clemson University, Clemson, South Carolina 29631, USA. Perturbative solutions of the field equations of a class of nonlinear field theories do not exist. The cases studied include SU(2) gauge theories with specific gauge assumptions. JOURNAL: none given (M5,R4) ON SOME STABILITY NOTIONS IN TOPOLOGICAL DYNAMICS. Saber Elaydi, Department of Mathematics, Kuwait University, P. 0. Box 5069, Kuwait. For dynamical systems with R (the real numbers) as a phase, a number of stability concepts are available in the literature. We address the problem of extending some of these notions to more general transformation groups, over a topological group. Unilateral stability (positive or negative) is extened to P-stability, where P is a replete semi-group in the phase 24

Nonlinear science abstracts

271

group. The results in the paper confirm that this proposed mode is quite reasonable. Dp- and JP-stabilities are introduced and relations with PLiapunov stability and P-characteristic 0 are established, where Dp and JP refers, respectively, to the prolongation and the prolongation limit relations relative to P. JOURNAL: Journal of Differential Equations 25

(P5,El) MULTIPLE EQUILIBRIA OF ELASTIC STRINGS UNDER CENTRAL FORCES; HIGHLY SINGULAR NONLINEAR BOUNDARY VALUE PROBLEMS OF THE BERNOULLIS. S. Antman and Peter Wolfe, Department of Mathematics, University of Maryland, College Park, Maryland 20742, USA. In this paper we study the multiplicity of equilibrium states of non-linearly elastic strings under central forces which may be infinite at their center. We find that these problems have a multiplicity of both regular and singular solutions with the analysis of the latter requiring a careful extension of the governing laws of mechanics to handle infinite forces. The regular solutions are of two types; purely radial and non-radial. We show how the knowledge of the radial solutions leads to multiplicity results for non-radial solutions via the Leray-Schauder degree theory. JOURNAL: Journal of Differential Equations 26

(Ml,R2) APPROXIMATE CONSTANTS OF MOTION FOR CLASSICALLY CHAOTIC VIBRATIONAL DYNAMICS; VAGUE TORI, SEMICLASICAL QUANTIZATION AND INTRAMOLECULAR ENERGY FLOW. Randall B. Shirts and William P. .Reinhardt,Department of Chemistry, University of Colorado and Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colorado 80309, USA. Coupled oscillator systems are known to exhibit regular and chaotic clasical motions. In this and the preceding paper by Jaff& and Reinhardt, we find substantial short time regularity even in the chaotic regions of phase space for what appears to be a large class of systems. This regularity is demonstrated by the behavior of approximate constants of motion calculated by Pad& summation of the Birkhoff-Gustavson normal form expansion and is attributed to remnants of destroyed invariant tori in phase space. The remnant torus-like manifold structures are used to justify both EinsteinBrillouin-Keller semiclassical quantization procedures for obtaining quantum energy levels even in the absence of complete tori and to form a theoretical basis for the calculation of rate constants for intramolecular mode-mode energy transfer. These results are illustrated in a thorough analysis of the HbnonHeiles oscillator problem, and the possible generality of the conclusions is demonstrated by the Barbanis Hamiltonian as well as recent results of Wolf and Hase. JOURNAL: J. Chem. Phys. 27

(Ml,R2) CHAOTIC DYNAMICS, SEMICLASSICAL QUANTIZATION, AND MODE-MODE ENERGY TRANSFER; THE BOULDER VIEW. William P. Reinhardt, Department of Chemistry, University of Colorado and Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colorado 80309, USA. Some consequences of classical chaos for semiclassical quantization, the correspondence principle, and molecular energy randomization

Nonlinear science abstracts

212

are outlined. It is suggested that for many of the classically chaotic systems of current interest that the short to intermediate time motion is not actually very much different than for quasiperiodic orbits. This allows introduction of the concept of approximate constants of the motion (or vague tori). The existence of such approximate constants isused to quantize the classical motion, and to set up a framework for definition and calculation of mode-mode energy transfer constants. Preliminary results of calculation of such a rate constant are given. The possible relation of this calculation to quantum dynamics is briefly outlined in the two limits of "large" and "small" values of Planck's constant (an essential consideration for understanding such results). JOURNAL: J. Phys. Chem. 28

(Ml,R2) UNIFORM SEMICLASSICAL QUANTIZATION OF REGULAR AN CHAOTIC CLASSICAL DYNAMICS ON THE HENON-EILES SURFACE. Charles Jaff& and William P. Reinhardt, Department of Chemistry, University of Colorado and Joint Institute for Laboratory Astrophysics, University of Colorado and National Bureau of Standards, Boulder, Colorado 80309, USA. Qualitative arguments (made quantitative in the accompanying article by R. Shirts and W. P. Reinhardt) are put forward which indicate that the apparently chaotic dynamics on the H&non-Heiles surface display sufficient regularity on a short to intermediate (but not long) time scale to allow use of standard EBK quantization techniques, taking advantage of the partial manifold structure that these remarks imply. A complete uniform semiclassical quantization is carried out using the time independent technique of the Birkhoff-Gustavson normal form, recently introduced in the context of semiclassical quantization by Swimm and Delos. JOURNAL: J. Chem. Phys. A GENERALIZATION OF THE ONSAGER RECIPROCITY THEOREM. James Hurley (P3) and Claude Garrod, Department of Physics, University of California, Davis, CA 95616, USA. The Onsager reciprocity theorem has been since its discovery in 1931 one of the most prominent successes of nonequilibrium statistical mechanics. Its interpretation and application suffer from several limiting assumptions made in its derivation. The most notable of these is the linear relation between the currents and the affinities. A generalization of the Onsager theorem has been obtained, which is compratively free of ad hoc assumptions. JOURNAL: Phys. Rev. Lett. (submitted) 29

SYMMETRY PROPERTIES OF NONLINEAR BARRIER COEFFICIENTS. Claude (P3) Garrod and James P. Hurley, Department of Physics, University of California, Davis, CA 95616, USA. This paper concerns the properties of a symmetric barrier between two reservoirs. The barrier can pass K conserved quantities. Tbe current of the ith quantity is assumed to satsify the nonlinear relation

30

Ji

= Aij

ABj + BijkgABj

A8k ABQ

Nonlinearscience abstracts

273

are the affinity differences across the barrier and A AtQ's are functions of the average affinities of the reservoirs. $1 is is symmetric in all indices. l+;RN&jk~Jo . u r. of Stat. Phys. (Submitted)

where the zzwn 31

(M8,Il) DIFFERENTIAL FRACTAL DIMENSION OF RANDOM WALK AND ITS APPLICATIONS TO PHYSICAL SYSTEMS. Hideki Takayasu, Deparment of Physics, Nagoya University, Nagoya, 464 Japan. In order to characterize the complexity of the path of any randomly walking test particle, we introduce the new conception, the differential fractal dimension (in short d.f.d.). This conception seems to have a special importance for the analysis of turbulence, because it clearly represents the complexity of the path observed by a scale r (r is any given length), while the scales of the observation are particularly important for the theory of turbulence. It is shown that, as far as the diffusion and the kinetic energy are concerned, we can treat inclusively both microscopic (or thermal) motions and macroscopic (or turbulent) motions by means of the d.f.d.. As an example, we obtain an analytical expression of the d.f.d. for an one-dimensional random walk with finite mean-free-path. JOURNAL: none given 32

(M5,Tll) THE MORSE-INDEX, REPELLER-ATTRACTOR PAIRS AND THE CONNECTION INDEX FOR SEMIFLOWS ON NONCOMPACT SPACES. Kmysztof P. Rybakowski, Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. This paper extends the Morse index theory of C. C. Conley to semiflows n on a noncompact metric space X. x is assumed to satisfy a hypothesis related to conditional a-contraction. In Section 1 we collect background material. In Section 2 we define quasi index pairs and the Morse index of a compact, isolated invariant set K. We prove that the Morse index is a connected simple system. In Section 3 we study repeller-attractor pairs in K. We define index triples, prove their existence and several properties leading to the concepts of the connection index, the connection map and the splitting class. Finally, in Section 4, we consider paths (continuous families) of pairs (n,K) and study continuations of the Morse and the connection indices along such paths. The present paper is a sequel to the author's previous work: On the homotopp index for infinite-dimensionalsemiflows, Trans. Amer. Math. Sot., to appear. JOURNAL: Journal of Differential Equations 33

(M4,T9) DIFFUSION IN DISCRETE NON-LINEAR DYNAMICAL SYSTEMS. S. Grossmsnn and H. Fujisaka, Fachbereich Physik, Phillips-UniversitIit, D-3550 Marburg, FRG. Diffusive processes in one-dimensional discrete chaotic systems are considered. Drift and diffusion coefficient are calculated, showing critical behavior including logarithmic corrections. A Kubo-formula for the diffusion coefficient in terms of the time correlation function is given. Nondiffusive states may exist in certain parameter windows showing drift

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with broken symmetry or strict localization. Period doubling bifurcations and states of periodic chaos describing chaotic but non-diffusive drift occur. JOURNAL: none given 34

(P6,TlZ) RESISTIVE MHD STUDIES OF HIGH-BETA TOKAMAK PLASMAS. V. E. Lynch, H R. Hicks, J. A. Holmes, Computer Sciences; B. A. Carreras, L. Garcia, Fusion Energy Division; Oak Ridge National Laboratory, Oakridge, TN 37830, USA. Numerical calculations have been performed to study the magnetohydrodynamic (MHD) activity in high-beta tokamaks such as ISX-B. These initial value calculations have been built on earlier low-beta techniques, but the beta effects create several new numerical issues. These issues are discussed and resolved. In addition to time-stepping modules, our system of computer codes includes equilibrium solvers (used to provide an initial condition) and output modules, such as a magnetic field line follower and an xray diagnostic code. The transition from current-driven modes at low beta to predominantly pressure-driven modes at high beta is described. The nonlinear studies yield x-ray emissivity plots which are compared with experiment. JOURNAL: none given 35

(B4,Rl) THRESHOLD PHENOMENA FOR A REACTION-DIFFUSION SYSTEM. David Terman, Mathematics Research Center, University of Wisconsin-Madison, Madison, WI 53706, USA We consider the pure initial value problem for the system of equations + f(v) - w Et+ - VW), Wt =

Vt

=v

E,Y ) 0 ,

the initial data being (v(x,O),w(x,O)) = (+(x),0). Here f(v) = -v + H(v - a) where H is the Heaviside step function and a E (0,1/2) . This system is of the Fitzhugh-Nagumo type, and has several applications including nerve conduction and distributed chemical/biochemicalsystems. It is demonstrated that this system exhibits a threshold phenomenon. This is done by considering the curve s(t) defined by s(t) = sup{ x:v(x,t) = a 1. The initial datum, It is proven that the 9 s(t) = m . +(x), is said to be superthreshold if initial datum is superthreshold if Cp(x)F a on a sufficiently long interval, $(x> is sufficiently smooth, and +(x) decays sufficiently fast to zero as 1x1 + 0. JOURNAL: Journal of Differential Equations 36

(~5,B8) SOME QUESTIONS AND OPEN PROBLEMS IN CONTINUUM MECHANICS AND POPUW\TION DYNAMICS. Morton E. Gurtin, Department of Mathematics, Carnegie-Mellon University, Pittsburgh, PA 15213, USA. This paper presents some unsolved problems in continuum mechanics and population dynamics. The problems concern: buckling of a viscoelastic beam; minimizing the residual stress that occurs during the cooling of a constrained visco-elastic body; dispersal of age-dependent biological populations;,populationmodels that are perturbations of chaotic systems. JOURNAL: Journal of Differential Equations

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275

37

(P3,T9) SMALL DEVIATIONS FROM LOCAL EQUILIBRIUM FOR A PROCESS WHICH EXHIBITS HYDRODYNAMICAL BEHAVIOR. A. De Masi, Instituto Matematico Universita dell'Aquila, L'Aquila, Italy; N. Ianiro, Institute Meccanica, Ingegneria, Universita dell'Aquila, L'Aquila, Italy; E. Presutti, Instituto Matematico, Universita di Roma, P. le A. Moro, 00185 Roma, Italy. The symmetric simple exclusion process where infinitely many particles move randomly on Z, jump with probability on nearest neighbor sites and interact by simple exclusion is considered. It is known that the only extremal invariant measures are Bernoulli, that each measure, in a suitable class, after a "macroscopic" time is locally described, at a zero order approximation, by a Bernoulli measure with parameter depending on macroscopic space and time, and that the so defined equilibrium profile satisfies the heat equation. Small deviations from local equilibrium in the hydrodynamical limit are investigated. It is proven, under suitable assumptions, that at first order the state is Gibbs with one and two body potentials whose strength depends only on macroscopic space and time and on the equilibrium profile. More precisely the one body potential is linear (on the microscopic positions of the particles) and proportional to the macroscopic space gradient of the equilibrium parameter at that time, so that Fourier law holds. The two body potential varies on a macroscopic scale and does not depend on the microscopic positions of the particles; it is given by the value of the covariance of the Gaussian "macroscopic density fluctuation field". JOURNAL: J. Statistical Physics (submitted) 38

(P3,T9) SMALL DEVIATIONS FROM LOCAL EQUILIBRIUM FOR A PROCESS WHICH EXHIBITS HYDRODYNAMICAL BEHAVIOR. A. De Masi, Instituto MatematicoUniversita dell'Aquila, L'Aquila, Italy; P. Ferrari, Instituto de Matemtice e Estatistica, Universidade de Sao Paulo, Sao Paulo, S. P. Brasil; N. Ianiro, Instituto mecanica, Ingegneria, Universita dell'Aquila, L'Aquila, Italy; E. Presutti, Instituto Matematico, Universita di Roma, P. le A. Moro, 00185 Roma, Italy. The symmetric simple exclusion process on Z with sources at f L, L E N is considered. The stationary measure pL is studied in the limit as L diverges. The first order correction to its limit is proven to be of order l/L and it is explicitely computed. The result is in agreement with the analysis of the model from a hydrodynamical point of view initiated in [l]. JOURNAL: J. Statistical Physics (submitted) 39

(M4,IZ) DECAY OF STATISTICAL DEPENDENCE IN CHAOTIC ORBITS OF DETERMINISTIC MAPPINGS. Celso Grebogi and Allan N. Kaufman, Lawrence Berkeley Laboratory, University of California, Berkley, CA 94720, USA. A numerical study is made of the decay of statistical dependence in chaotic orbits of the iterated mappings of Chirkov and of Rannou. The decay appears to be exponential; the decay exponent is proportional to, but smaller than, the Liapunov exponent. JOURNAL: Physical Review A, -24 (1981) 2829.

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40

(M2,T5) INVERSE SCATTERING FOR THE ONE-DIMENSIONAL STARK EFFECT AND APPLICATION TO THE CYLINDRICAL KdV EQUATION. S. Graffi, on leave from Instituto de Matematica di Bologna, 40126 Bologna, Italy, and E. Harrell, Department of Mathematics, the Johns Hopkins University, Baltimore, MD 21218, USA. We develop the inverse spectral and scattering theory for onedimensional Stark operators, i.e. --d2 + fx + u(x) on L2 ( R) . dx2 The potential u is determined in a special case; this allows existence and approximate solitary-wave behavior to be proved for solutions of a non-linear evolution equation corresponding to the Stark Hamiltonian. Connection is made to the solitary solution discovered by Calogero and Degasperis. This behavior is associated with resonances in much the same way as solitons are associated with bound states in the theory without the linear term fx. JOURNAL: Ann. Inst. Henri Poincard Vol. XXXVI, no 1, 1982, p. 41-58. 41

(P8,T2) ESTIMATING TUNNELING PHENOMENA. Evans M. Harrell II, Department of Mathematics, The Johns Hopkins University, Baltimore, MD 21218, USA. Several tunneling phenomena are surveyed. A unified, rigorous treatment of them can be based on a simple technique of integration by parts coupled with growth estimates of eigenfunctions, using, for example, WKB approximations. In particular, the analysis of Harrell and Simon of the resonance widths in the hydrogen Stark effect can be simplified. JOURNAL: International Journal of Quantum Chemistry Vol. XXI, 199-207 (1982) 42

(M4,15) RENORMALIZATION FOR STOCHASTIC LAYERS. D. F. Escande, Laboratoire de Physique des Mileux Ionis&, Ecole Polytechnique, 91128 Palaisseau, France. Chirikov's standard mapping for stochastic layers is derived in a simpler and more general way. A one-parameter renormalization transformation yields, analytically, many universal quantities defined in a previous renormalization theory. Expressions are given for the width of stochastic layers that correct several previous estimates. JOURNAL: none given 43

(~l,T6) QUADRUPLE COLLISION IN THE TRAPEZOIDAL ~-BODY PROBLEM. E. A. Lacomba, Math Department, Universidad Autbnoma Metropolitana, Izt., P. 0. Box 55-534, 09340 Mexico City, Mexico. In this paper the author began to study the topology and nature of equilibrum points in the total collision manifold of the trapezoidal 4-body problem, by following McGehee's method of blowing up the total collision singularity. One is given four particles with masses equal in pairs, and initial conditions are given so that they always form the vertices of a symmetric trapezoid. We have three degrees of freedom: the two semibases and the height. The topology of the collision manifold is described, both before and after regularization of most of the binary collisions. It is then shown that

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277

there are exactly 2 convex and 2 collinear central configurations, giving rise to 8 equilibrium points of the fictitious flow in the total collision manifold. JOURNAL: In Classical Mechnaics and Dynamical Systems, Marcel Dekker, New York, 1981. (~l,T6) ~ouv~~wrs VOISINS DE COLLISION QUADRUPLE DANS IE PROBLEME TRAPEZOIDAL DES 4 CORPS. E. A. Lacomba, Math Department, Universidad Autonoma Metropolitana, Izt., P. 0. Box 55-534, 09340 Mexico City, Mexico We continue here the study of the trapezoidal 4 body problem, started by the author in a previous work. The dimensions and inter-connections of the invariant submanifolds of the equilibrium points in the total collision manifold, are studied. We conclude with the description of some important motions in the problem. The principal features of this problem is the non existence of triple collisions, and the embedding of two simpler problems with one less degree of freedom: the rectilinear trapezoidal, and the rectangular problems (the last one only if u = m). As usual, there are some bifurcations for some values of the mass ratio. They change some of the interconnections. The set of initial conditions of orbits on a given energy surface going to quadruple collision, is a union of 4 submanifolds: two of them have dimension 2, while the others have dimension 3. Similarly for ejection orbits from quadruple collision. JOURNAL: not given 44

45

(Pl,T6) ANALYSIS OF SOME DEGENERATE QUADRUPLE COLLISIONS. C. Simo, Facultat de Matematiques, Universitat de Barcelona, Gran Via 585, Barcelona 7, Spain, and E. A. Lacomba, Math Department, Universidad Autonoms Metropolitana, Izt., P. 0. Box 55-534, 09340 Mexico City, Mexico. We consider the trapezoidal problem of four bodies. This is a special problem where only three degrees of freedom are involved. The blow up method of McGehee can be used to deal with the quadruple collision. Two degenerate cares are studied in this paper; the rectangular and the collinear problems. They have only two degrees of freedom and the analysis of total collapse can be done in a way similar to the one used for the collinear and isosceles problems of three bodies. We fully analyze the flow on the total collision manifold, reducing the problem of finding heteroclinic connections to the study of a single ordinary differential equation. For the collinear case from which arises a one parameter family of equations analysis is done for extreme values of the parameter and numerical computations fill up the gap for intermediate values. Dynamical consequences for possible motions near total collision as well as for regularization are obtained. JOURNAL: To appear in Celestial Mechanics Congress, Oberwolfach 1981.

278

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46

(~l,T6) BOUNDARY MANIFOLDS FOR ENERGY SURFACES IN CELESTIAL MECHANICS. E. A. Lacomba, Math Department, Universidad Autonoma Metropolitana, Izt., P. 0. BOX 55-534, 09340 Mexico City, Mexico; and C. Simo, Facultat de Matematiques, Universitat de Barcelona, Gran Via 585, Barcelona 7, Spain. We complete McGehee's picture of introducing a boundary (total collision) manifold to each energy surface. This is done by constructing the missing components of its boundary as other submanifolds, representing now the asymptotic behaviour at infinity. It is necessary to do separately each case h = 0, h > 0 or h < 0. In the first case, we recover the known result that the behavior at total escape is the same as in total collision. In particular, we explain why the situation is.radically different in the h > 0 case, than in the zero energy case. In the case of h < 0 we have many infinitely many manifold components, and the general situation is not quite well understood. Finally, our results for h > 0 are shown to be valid for general homogeneous potentials. JOURNAL: To appear in Celestial Mechanics Congress, Oberwolfach 1981. 47

(M5,T6) UNSTABLE WEAR ATTRACTORS. Ronald A. Knight, Mathematics Division, Northeast Missouri State University. Our objective in this paper is to continue the process of classification and characterization of weak attractors initiated by the author in an earlier paper. In particular we obtain additional characterizations of those weak attractors which are saddle sets and bilateral weak attractors. JOURNAL: Proceedings of the American Mathematical Society 48

(M3,T8) A NOTE ON DEGREE THEORY FOR GRADIENT MAPPINGS. Herbert Amann, Math. Institut, Universitat Ziirich,Freiestr. 36, CH-8032 Ziirich, Switzerland. In this note we give a simple proof for the essentially known fact that the Leray-Schauder degree of the gradient of a coercive functional on a large ball of a Hilbert space is one. As a sim le application we show that the local index of an isolated local minimum of a CP-functional on a Hilbert space equals one. JOURNAL: Proceedings of the American Mathematical Society 49

(P5,T12) FREQUENCY TUNING BY MAGNETIC FIELD IN JOSEPHSON JUNCTION FLUXON OSCILLATORS. S. N. Erne and A. Ferrigno, Physikalisch-Technische Bundesanstalt, Institut Berlin, D-1000 Berlin 10, West Germany; and S. Di Genova and R. D. Parmentier, Istituto di Fisica, Universita di Salema, I-84100 Salerno, Italy. The effect of applied magnetic field on the frequency of oscillation of long Josephson tunnel junctions current-biased on zero-field steps is studied by numerical simulation and by a mechanical analog. The frequency increases with field for larger values of the bias and decreases for smaller values, corresponding to a rotation about some mid-point of the zerofield step in the sense of higher differential resistance. JOURNAL: not given

Nonlinearscience abstracts 50

279

(M4,13)

ANALYSIS OF FLOW HYSTERESIS BY A ONE-DIMENSIONAL MAP. Simon Fraser and Raymond Rapral, Department of Chemistry, University of Toronto, Toronto, Ontario M5S lA1, Canada. The structure of a regi.oRof stable period three for a non-linear dissipative system described by the Rossler equations and a two-parameter cubic map is studied. The intricate configuration of this region, which is bordered by intermittent type chaos on one side, subharmonic cascades on the other and possesses several sharp features, is shown to be associated with bistability and hysteresis of the orbits of the flow or map. Locally, the two-parameter cubic map successfully models many features of the differential flow. The mechanism which gives rise to the hysteresis is quite general and corresponds to a cusp catastrophe. This process is described in detail for the map nd related to the same phenomenon in the flow. JOURNAL: not given 51

(M1,14) RENORMALIZATION APPROACH TO NON-INTEGRABLE HAMILTONIANS. D. F. Escande, Laboratoire de Physique des Milieux Ionises Groupe de Recherche No 29 du Centre National de la Recherche Scientifique Ecole Polytechnique, 91128 Palaiseau Cedex, France. A review is given of th@ results obtained with F. Doveil and A. Mehr by an pproximate renormalization method applied to the hamiltonian H (u,x,t) = v4 /Z - M cos x - P cox k (x - t). This method is in the spirit of RAM theory and allows to compute fairly accurately universal quantlties, the threshold s(v) of destabilization of KAM tori and of cycles with average velocity v, and the threshold of large schale stochasticity. The link between tori and nearby cycles is demonstrated, the mean residue of cycles is analytically computed and Greene's main assertions on the standard mapping are proven. The graph of s(v) is shown to be a fractal. Expressions are given for the width of stochastic layers that correct previous estimates. It is shown how to apply the results for H to a large class of two degrees (one degree time-dependent) hamiltonian systems. JOURNAL: not given 52

(M4,15) EXISTENCE OF A FIXED POINT OF THE DOUBLING TRANSFORMATION FOR AREA-PRESERVING MAPS OF THE PLANE. J. P. Eckmann, H. Koch (permanent address: Harvard University, Department of Physics), and P. Wittwer, Departement de Physique Theorique, Universite de Geneve, 1211 Geneve 4, Switzerland. We describe a computer assisted proof of the existence of a fixed point b of the doubling transformation (functional composition) for areapreserving maps of the plane: 0 is obtained from the solution S to a fixed point equation RX = S for generating functions of area-preserving maps. This establishes, with mathematical rigour, an important part of Feigenbaum universality for conservative, two-dimensional systems. JOURNAL: not given 53

(A2,R2) OSCILLATIONS OF AN EXTENDED IONIZATION REGION. J. Robert Buhler and Oded Regev, Physics Department, University of Florida, Gainesville, Florida 32611, USA. We describe a mechanism by which an extended ionization region, although vibrationally stable, can undergo oscillations of an unexpectedly

280

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science abstracts

complex nature. The oscillations can be quasi-steady periodic oscillations in either of two possible states (hysteresis), they can be of the relaxation type, periodic or aperiodic, or of a limit cycle type with possible bumps, depending sensitively on the model parameters. The values and range of our one-zonemodel parameters would suggest an identification with the long-period variables, which are observed to vary from regular, to semi-regular to irregular. JOURNAL: not given 54

(M2,T8) INTERPOLATION OF NONLINEAR PARTIAL DIFFERENTIAL OPERATORS AND GENERATION OF DIFFERENTIABLE EVOLUTIONS. Eric Schechter, Department of Mathematics, Box 21, Station B, Vanderbilt University, Nashville, TN 37235, USA. By interpolating between Sobolev spaces we find that many partial differential operators become continuous when restricted to a sufficiently small domain. Hence soma techniques from the theory of ordinary differential equations can be applied to some p.d.e.'s. Using these ideas, in this paper we study a class of nonlinear evolutions in a Banach space. We obtain some very simple existence and continuous dependence results. The theory is applicable to reaction-diffusionequations, dispersion equations, and hyperbolic equations before shocks develop. JOURNAL: Journal of Differential Equations 55

(P4,W4) THE RIEMANN PROBLEM FOR A CLASS OF CONSERVATION LAWS OF MIXED TYPE. Michael Shearer, Department of Mathematics, Duke University, Durham, N C 27706, USA. The Riemann problem for a 2 x 2 system of conservation laws of mixed hyperbolic and elliptic type is solved uniquely for arbitrary initial step data. For certain initial data, the solution includes stationary shock waves that satisfy neither the Maxwell equal area rule for phase transitions nor the admissibility criteria often associated with shock wave solutions o hyperbolic systems. JOURNAL: Journal of Differential Equations 56

(M4,T6) REPELLERS FOR REAL ANALYTIC MAPS. David Ruelle, Institut des Hautes Etudes Scientifiques,35, Route de Chartree, 91440 Bures-surYvette, France. The purpose of this note is to prove a conjecture of D. Sullivan that when the Julia set J of a rational function f is hyperbolic, the Hausdorff dimension of J depends real analytically on f. We shall obtain this as corollary of a general result on repellers of real analytic maps (see Corollary 5). JOURNAL: not given (P8,12) "QUANTUM CHAOS" IN THE IRREGULAR SPECTRUM. Philip Tech&as, Department of Chemistry,ColumbiaUniversity, New York, New York 10027, USA. We select an initial state Y(0) "at random" rom an appropriate 5 to be still in ensemble and evaluate the probability P(t) = I<$(O)($(t)>l that state at time t. If the energy spectrum of the system is characteristic 57

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281

of the regular regime, we find that the long-time behavior of P(t) is nonstatistical; in the irregular spectrum we find statistical behavior. JOURNAL: not given 58

(P3,T12) MONTE CARLO TECHNIQUE FOR VERY LARGE ISING MODELS. C. Ralle and V. Winkelmann, University of Cologne, Computing Center, Robert-RochStrasse 10, D-5000 Cologne 41, FRG. Rebbi's multi-spin coding technique is improved and applied to the kinetic Ising model with size 600 * 600 * 600. We give the central part of our computer program (for a CDC Cyber 76), which will be helpful also in a simulation of smaller systems, and describe the other tricks necesary to go to large lattices. The magnetization M at t = 1.4*Tc is found to decay asymptotically as exp(-t/2.90) if t is measured in Monte Carlo steps per ’ spin, and M(t=O) = 1 initially. JOURNAL: To appear in J. Statist. Phys. Vol. 28 - #4 - Aug. 82. 59

(Al,T12) ON THE STABILITY OF SCHWARZSCHILD'S TRIAXIAL GALAXY MODEL. B. F. Smith, Theoretical and Planetary Studies Branch, NASA Ames Research Center and R. H. Miller, University of Chicago, Chicago, IL, USA. The numerical model for triaxial galaxy constructed by Schwarzschild (1979) has been tested for stability by means of fully selfconsistent three-dimensional n-body numerical experiments that use a representation of this model to start the integration. The model is rugged and robust. No growing disturbances with growth rate in excess of 0.5 per crossing time could be detected, This limit is strong enough to indicate that the model is free of rapidly growing dynamical instablities that could limit its value in dynamical studies. JOURNAL: not given 60

(M5,T6) QUADRATIC MORSE-SMALE VECTOR FIELDS WHICH ARE NOT STRUCTURALLY STABLE. Carmen Chicone, Department of Mathematics, University of Missouri, Columbia, MO 65211, USA, and Douglas Shafer, Department of Mathematics, University of Missouri, Columbia MO 65211, USA, and Department of Mathematics, University of North Carolina, Charlotte, NC 28223, USA. An example is given of a quadratic system in the plane which is Morse-Smale but not structurally stable. Also, it is proved that no such example exists for a quadratic system which is a gradient. JOURNAL: Proceedings Amer. Math. Society THE SIMULATION OF RANDOM PROCESSES ON DIGITAL COMPUTER: (M12) INELUCTABLE ORDER. W. F. Darsow and M. J. Frank, Department of Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA; T. Erber and T. M. Rynne, Department of Physics, Illinois Institute of Technology, Chicago, IL 60616, USA. Non-random features of pseudo-random number generators are usually regarded as defects which may be minimized by improving the algorithms or utilizing larger computers. There are however certain elements of order which cannot be avoided even on digital devices of arbitrarily large capacity. For instance on an N-state machine pseudo-random number generators will terminate on fixed points or fall into loops after approximately m steps. 61

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Combinatorial arguments then may be used to show that for any finite device it is highly improbably that there are more than three or four distinct terminal loops. All pseudo-random sequences merging into these loops may be traced backwards to their initial numbers --- the resulting pattern of 'ancestornumbers' can be charted in detail for any computer even for non-invertible algorithms. The conflicting requirements of randomness and finite numerical precision lead to an ordered distribution of the set of initial numbers. In this sense neither the initial nor the final states of a simulation of chaotic behavior can ever be random. The 'few-loop' constraint could generate patterns of self-organization in non-equilibrium systems. Experimental evidence from the hysteresis of Ewing arrays supports this conjecture. JOURNAL: not given 62

(P4,T9) THE LANGEVIN-EQUATION APPROACH TO DYNAMICS OF DENSE FLUIDS. Hiroshi Ueyema, Department of Physics, College of General Education, Osaka University, Toyonaka 560. The stochastic-differential-equationapproach to dynamical problems of dense fluids is presented. The stochastic Boltzmann-Enskog equation is derived from the Liouville equation of a classical hard sphere system. By the method of the Chapman-Enskog expansion, it is shown that the equation reduces near the local equilibrium to the equations of fluctuations in fluids of Landau and Lifshitz. Discussions are given on the possiblity of using the Landau-Lifshitz formula as a microscopic expression of transport coefficients. JOURNAL: Progress of Theoretical Physics, Vol. 66, No. 6, 1981. 63

(P7,T4) ELECTRON EIGENSTATES IN LATTICES WITH INCOMMENSURATE POTENTIALS. T. Hogg, Department of Physics, Stanford University, Stanford, CA 94305, USA; B. A. Huberman, Xerox Palo Alto Research Center, Palo Alto, CA 94304, USA. We show that all the eigenatates of an electron in a quasiperiodic potential, with an arbitrarily large number of incommensurate frequencies in any dimension, are extended, i.e., the wavefunction does not go to zero at infinity regardless of the strength of the potential. This implies that localization requires a more extreme type of disorder than that described by quasiperiodic functions. JOURNAL: submitted to Physical Review B 64

(Al,T12) A NUMERICAL EXPERIMENT ON THE EQUILIBRIUM AND STABILITY OF A ROTATING GALACTIC BAR. Richard H. Miller, Peter 0. Vandervoort, Daniel E. Welty, Department of Astronomy and Astrophysics, University of Chicago; Bruce F. Smith, Theoretical and Planetary Studies Branch, NASA Ames Research Center. A self-consistent, three-dimensionalnumerical experiment is performed on an N-body system whose initial state is a realization of a certain theoretical model of a rotating triaxial galaxy. The model is a stellar-dynamical counterpart of a uniformly rotating polytrope of index equal to 0.5. The aim of the experiment is to study the equilibrium of the system and, in particular, to test its stability. The experimental system behaves in the mean like a realization of the theoretical model for at least seven

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283

crossing times. The principal departure of the system from equilibrium is an oscillation which we identify as a radia: pulsation. There is no indication in its behavior that the system is unstable with respect to any mode with an efolding time shorter than or of the order of two crossing times. Certain changes that occur in the state of the system are interpreted, with the aid of the theoretical model, as secular changes which result from a slight failure of our numerical methods to conserve the mass, energy, and angular momentum of the system; these effects are small enough that they do not vitiate the experiment on a dynamical time scale. The distribution of stars in the phase space of a single star appears to relax on a time scale comparable with the time of relaxation derived from the theory of two-body encounters. A special feature of this work is that the behavior of the experimental system can be compared with expectations derived from a theoretical model and the dynamical results rest on the mutual consistency of the model and the system. JOURNAL: not given 65

(Al,T12) THE DYNAMICAL INSTABILITY OF A ROTATING, AXISYMMETRIC GALAXY WITH RESPECT TO A DEFORMATION INTO A BAR. Peter 0. Vandervoort, Department of Astronomy and Astrophysics, The University of Chicago, Chicago, IL, USA. On the basis of the tensor virial equations of the second order, a criterion has been derived for the dynamical instability of any rotating, axisymmetric galaxy with respect to a bar mode. Although the new criterion for instability differs in important ways from the well-known conjecture by Ostriker and Peebles, it does not vitiate the constraints envisaged by those authors that considerations of stability would impose on galactic dynamics and galactic structure. JOURNAL: not given 66

(Al,R5) THE EFFECT OF GRAVITATIONAL RADIATION ON THE SECULAR STABILITY OF A ROTATING, AXISYMMETRIC GALAXY. Peter 0. Vandervoort and James R. Isper, Department of Astronomy and Astrophysics, The University of Chicago, Chicago, IL, USA. In a post-Newtonian approximation, it is shown that the emission of gravitational radiation makes a Dedekind-like mode of second-harmonic oscillation secularly unstable in all of the rotating members of a certain Maclaurin-like sequence of axisymmetric stellar systems. The limiting, nonrotating member of the sequence is a point of neutral stability, and it is also the point of bifurcation at which a Dedekind-like sequence of nonaxisymmetric stellar systems branches off from the Maclaurin-like sequence. The Maclaurin-like and Dedekind-like systems that are considered here belong to the family of stellar systems, of the form of elliptical disks, that has been constructed by Freeman and by Hunter. JOURNAL: not given 67

(M3,T5) NONLINEAR DIFFERENTIAL-DIFFERENCEMATRIX EQUATIONS WITH nDEPENDENT COEFFICIENTS. M. Bruschi, Istituto di Fisica dell'Universita, Roma, Italy; 0. Ragnisco, Istituto Nazionale di Fisica Nucleare, Sezione di Roma, Roma, Italy. We derive a class of nonlinear differential-differencematrix equations with linearly n-dependent coefficients solvable via the Inverse Spectral Transform. JOURNAL: not given