Zhifu Xie

A simpler proof of regular polygon solutions of the N-body problem☆

Abstract In this Letter we give simpler proofs of the famous results of Perko–Walter–Elmabsout theorem published in Proc. Amer. Math. Soc. 94 (1985) 301 and Celest. Mech. 41 (1988) 131.

Long-time behavior and Turing instability induced by cross-diffusion in a three species food chain model with a Holling type-II functional response.

In this paper, we study a strongly coupled reaction-diffusion system describing three interacting species in a food chain model, where the third species preys on the second one and simultaneously the second species preys on the first one. An intra-species competition b2 among the second predator is introduced to the food chain model. This parameter produces some very interesting result in linear stability and Turing instability. We first show that the unique positive equilibrium solution is locally asymptotically stable for the corresponding ODE system when the intra-species competition exists among the second predator. The positive equilibrium solution remains linearly stable for the reaction diffusion system without cross diffusion, hence it does not belong to the classical Turing instability scheme. But it becomes linearly unstable only when cross-diffusion also plays a role in the reaction-diffusion system, hence the instability is driven solely from the effect of cross diffusion. Our results also exhibit some interesting combining effects of cross-diffusion, intra-species competitions and inter-species interactions. Numerically, we conduct a one parameter analysis which illustrate how the interactions change the existence of stable equilibrium, limit cycle, and chaos. Some interesting dynamical phenomena occur when we perform analysis of interactions in terms of self-production of prey and intra-species competition of the middle predator. By numerical simulations, it illustrates the existence of nonuniform steady solutions and new patterns such as spot patterns, strip patterns and fluctuations due to the diffusion and cross diffusion in two-dimension.

Collinear central configurations in the n-body problem with general homogeneous potential

In this paper we investigate the central configurations of collinear n-body problem given by the general law of attraction of the form f(r)=1/rα. A method involving analysis skills of some elementary algebra and calculus is presented to study the central configurations in the collinear n-body problem. It is well known that for given n positive masses, there are precisely n!/2 collinear central configurations for Newton’s law of gravitation of α=2. However, it is not true that there is always a position that causes a central configuration for any given ordered particles with some positive masses and that there may exist more than one position that make it central for some α 0. Examples that Moulton’s theorem does not work are also provided.

Nested regular polygon solutions of 2N-body problem☆

Abstract In this Letter we study the necessary conditions for the masses of the nested regular polygon solutions of the planar 2 N -body problem.

Isosceles trapezoid central configurations of the Newtonian four-body problem

We use a simple direct and basic method to prove that there is a unique isosceles trapezoid central configuration of the planar Newtonian four-body problem when two pairs of equal masses are located at adjacent vertices of a trapezoid. Such isosceles trapezoid central configurations are an exactly one-dimensional family. Explicit expressions for masses are given in terms of the size of the quadrilateral.

Super central configurations of the n-body problem

In this paper, we consider the inverse problem of central configurations of the n-body problem. For a given q=(q1,q2,…,qn)∊(Rd)n, let S(q) be the admissible set of masses by S(q)={m=(m1,…,mn)∣mi∊R+, q is a central configurationfor m}. For a given m∊S(q), let Sm(q) be the permutational admissible set about m=(m1,m2,…,mn) by Sm(q)={m′∣m′∊S(q), m′≠m and m′ is apermutation of m}. Here, q is called a super central configuration if there exists m such that Sm(q) is nonempty. For any q in the planar four-body problem, q is not a super central configuration as an immediate consequence of a theorem proved by MacMillan and Bartky [“Permanent configurations in the problem of four bodies,” Trans. Am. Math. Soc. 34, 838 (1932)]. The main discovery in this paper is the existence of super central configurations in the collinear three-body problem. We proved that for any q in the collinear three-body problem and any m∊S(q), Sm(q) has at most one element and the detailed classification of Sm(q) is provided.

Super central configurations of the three-body problem under the inverse integer power law

In this paper, we consider the problem of central configurations of the n-body problem with the general homogenous potential 1/rα, where α is a positive integer. A configuration q = (q1, q2, ⋅⋅⋅, qn) is called a super central configuration if there exists a positive mass vector m = (m1, ⋅⋅⋅, mn) such that q is a central configuration for m with mi attached to qi and q is also a central configuration for m′, where m′≠mandm′ is a permutation of m. The main result in this paper is the existence and classifications of super central configurations in the rectilinear three-body problem with general homogenous potential. Our results extend the previous work [Xie, Z., J. Math. Phys. 51, 042902 (2010)]10.1063/1.3345125 from the case in which α = 2 to the case in which α is a positive integer. Descartes’ rule of sign is extensively used in the proof of the main theorem.

Central Configurations, Super Central Configurations, and Beyond in the n-Body Problem

In this survey, we review our recent understandings on central configurations, super central configurations, and their applications. At the end, some challenging problems and further possible extensions are presented.