# Julian I. Palmore's research while affiliated with University of Illinois, Urbana-Champaign and other places

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## Publications (26)

The study of highly unstable nonlinear dynamical systems—chaotic systems—has emerged recently as an area of major interest and applicability across the mathematical, physical and social sciences. This interest has been triggered by advances in the past decade, particularly in the mathematical understanding of complex systems. An important insight t...

The central theme of this Institute is scaling in disordered systems, especially the use of both regular (periodic) and ‘random’ fractals to describe the scaling properties. In a related set of lectures, Wolfram1 has shown how both regular and pseudo-random states can be generated systematically via simple integer maps called cellular automata. The...

In this paper we explore aspects of computer arithmetic from the viewpoint of dynamical systems. We demonstrate the effects of finite precision arithmetic in three uniformly hyperbolic chaotic dynamical systems: Bernoulli shifts, cat maps, and pseudorandom number generators. We show that elementary floating-point operations in binary computer arith...

We observe that pseudo-random number generators, familiar to all programmers, are examples of deterministic chaotic dynamical systems. We discuss the implications of this finding and compare computer generation of pseudo-random numbers to the theoretical ideal of a (noncomputable) random sequence.

We generate phase portraits of autonomous differential equations by constructing integrals of motion-even in nonconservative systems, where no integrals produced by quadrature exist. These phase plane portraits are produced with no propagated error. Phase portraits do not usually display the parametrization along orbits. The use of high resolution...

The goal of this paper is to illustrate the rapid construction of phase plane portraits by a method that has no propagated error. This is done by constructing integrals (constants of motion) for differential equations on the plane-even in non-conservative systems, where energy integrals produced by quadrature do not exist. The correct time paramete...

We generate phase plane portraits of autonomous differential equations by constructing integrals of motion—even in nonconservative systems, where no integrals produced by quadrature exist. These phase plane portraits are produced with no propagated error. Phase portraits do not usually display the parametrization along orbits. The use of high resol...

We report new results on the shadowing of computable nonperiodic pseudo-orbits by computable chaotic orbits. While ordinary machine truncation/roundoff decisions can produce only periodic pseudo-orbits, we show how nonperiodic pseudo-orbits of chaotic maps can be generated by using for the truncation decision an algorithm for a computable irrationa...

We consider one-parameter families of Julia sets arising from Newton's method in the complex domain. We show the existence of bifurcation points where zeros coalesce or change from attractors to repellors, and points where chaotic behavior occurs.

We contrast analytic properties of chaotic maps with the results of fixed-precision computation and then use Turing's ideas of computable irrational numbers to illustrate the computation of chaotic orbits to arbitrary N-bit precision. This leads to the study of chaos theory via integer maps that are automata with long-range site interactions. We al...

We study the dynamics of finitely many vortices in a circular disk and compare the integrability of this problem with that of Kirchhoff's problem of vortices in the plane. The effect of the topology of the phase space on the two Hamiltonian systems is compared. Our goal is to apply topological methods uniformly to investigate the flow of these dyna...

We examine the dynamics problem of finitely many vortices in a circular disk and study stationary configurations of the vortices. For any fixed number of two or more vortices we prove that there are families of stationary vortex configurations in which bifurcation occurs. Sharp lower bounds on the numbers of stationary configurations are obtained b...

The following theorem is proved. THEOREM.For any n≥2, the set of collinear relative equilibria classes of the n-body problem generates by analytical continuation a total of n!(n+3)/2 relative equilibria classes of the n+1 body problem.
Together with Arenstorf's results we state a general theorem for the 4 body problem with 3 arbitrary masses and 1...

For a conservative dynamical system with n deg of freedom we show that the equations of variation along an orbit may be written with respect to an orthonormal moving frame (a generalized Frenet frame) in which the tangential variation is given by a quadrature and the normal and n - 2 binormal variations are solutions of n - 1 coupled second order e...

An old problem of the evolution of finitely many interacting point vortices in the plane is shown to be amenable to investigation by critical point theory in a way that is identical to the study of the planar n-body problem of celestial mechanics. For any choice of positive circulations of the vortices it is shown by critical point theory applied t...

We elaborate a variational method used recently in the proof of Saari's conjecture.

We prove the existence of many homographic solutions of the n-body problem in E4 by topological methods. Homographic solutions are associated with relative equilibria. Homothetic solutions always give rise to central configurations. In Euclidean space E4 central configurations are a proper subset of the relative equilibria for any n ⩾ 3 and for any...

A solution of the Newtoniann-body problem for which the moment of inertia (with respect to the centre of mass) is constant is a solution of relative equilibrium.

Estimates on a minimal classification of relative equilibria in the planarn-body problem of celestial mechanics have been announced in [1], [2]. Our main theorem asserts that these estimates are actually met for anyn≧3 on an open set in IR
inf+supn
. For anyn≧4, this open set is proper.

We announce several theorems on the nature of new relative equilibria classes in the planar n-body problem of celestial mechanics. Their existence is suggested by the creation and annihilation of relative equilibria at degenerate relative equilibria classes. We announced several basic new results in [2], [4]. We refer to these papers for definition...

We announce several theorems on the evolution of relative equilibria classes in the planar n-body problem. In an earlier paper [1] we announced a partial classification of relative equilibria of four equal masses. In [2] we described these new relative equilibria classes and showed the way in which a degeneracy arose in the four body problem. These...

## Citations

... Rather than search for the most general class of motions that 'preserve' the vortex strengths, we shall consider only rigidly rotating configurations, which yield constant values of the individual components zz, P0 ..... p~-i of the righthand sides of equations (11). Recent studies of stationary vortex configurations using geometric methods have been carried out by Palmore [ 1982], Koiller et al. [1985], and Kirwan [1988], among others. ...

... Central configurations are well-studied objects (cf. [2,3,8,10,11,12,15]), dating back to Euler and Lagrange, a simple question whether the number of central configurations is finite, for any given set of n positive masses, remain a major open problem in mathematics (cf. Smale [12], Albouy and Kaloshin [1], Hampton & Moeckel [4] ). ...

... Equivalences between singular vector fields, singular differential forms, singular orientable quadratic differentials and singular flat structures Vector fields related to complex analytic functions are very interesting and useful mathematical objects, both from the point of view of pure mathematics as from that of applications. They arise in multiple contexts: many physical phenomena can be modelled by vector fields (electric fields, magnetic fields, velocity fields, to name a few); and there are many interesting applications concerning the geometry and dynamics associated to them ( [4], [5], [9], [21], [35], [37], [56], [57], [60], [62], [69]). Moreover, the visualization of vector fields, besides being beautiful in itself, can be of great help towards the understanding of certain theoretical concepts. ...

... This technique, using Φ X , was originally presented by H. E. Benzinger, S. A. Burns and J. I. Palmore [9], [18], [62] in order to visualize rational vector fields on C. In this work we show that the technique can be extended to work on singular complex analytic vector fields, even those that have essential singularities or accumulation points of poles and zeros. We do this by i) extending the visualization method, originally presented by H. E. Benzinger, S. A. Burns and J. I. Palmore for rational vector fields in C, to work on all Newton vector fields on an arbitrary Riemann surface, and ii) since all singular complex analytic vector fields are in fact Newton vector fields, this provides a framework in which we can actually obtain a solution of (1) for all singular complex analytic vector fields and hence can be visualized. ...

... For the Euclidean case κ = 0 (planar relative equilibria), very few degenerate relative equilibria are known. For n = 3 all relative equilibria are non-degenerate [16], so we can extend them to spaces of constant curvature for all mass values. In 1973 J. Palmore found the first example of degenerate relative equilibrium in a planar 4-body problem. ...

... This coordinate-free way is adopted in the study of relative equilibria in the Newtonian n-body problem in higher dimensions, see [5,19]. ...

... Note that this potential is not the solution of the Laplace equation in R 4 , and so is not the standard model in mathematical physics. However the induced dynamics is interesting and was and is considered for theoretical significance (see, for instance, [PJ80,PJ81a,PJ81b,AC98,OV06,Ch13]). We observe that the dynamics under the R 4 gravitational potential, that is V (r) = −k/r 2 , k > 0, also known as the Jacobi potential (see [Al15] and references within), presents intriguing degeneracies and we defer its detailed study for future projects. ...

Reference: On the n-body problem in R^4

... Equivalences between singular vector fields, singular differential forms, singular orientable quadratic differentials and singular flat structures Vector fields related to complex analytic functions are very interesting and useful mathematical objects, both from the point of view of pure mathematics as from that of applications. They arise in multiple contexts: many physical phenomena can be modelled by vector fields (electric fields, magnetic fields, velocity fields, to name a few); and there are many interesting applications concerning the geometry and dynamics associated to them ( [4], [5], [9], [21], [35], [37], [56], [57], [60], [62], [69]). Moreover, the visualization of vector fields, besides being beautiful in itself, can be of great help towards the understanding of certain theoretical concepts. ...

... [4] studied the number of equivalence classes in the restricted four body problem. [5] extended the model considered by Arenstorff and studied the collinear relative equilibria of the planar N-body problem. [6] investigated the restricted 3-body problem by supposing both the primaries as solar radiation effects. ...

... Uma metodologia promissora para criptografiaé a aplicação de sistemas caóticos na criação de chaves criptográficas, trabalhos sobre este tópico são amplamente encontrados na literatura, sejam eles atuais [7,13], ou da ultima década [18]. No fim da década de 1980, Henring e Palmore [5] afirmaram que geradores de números pseudo-aleatórios são sistemas dinâmicos caóticos [11]. Tal afirmação, reforçam que sistemas como o de Lorenz [9], possuem propriedades favoráveis para gerar uma chave criptográfica. ...