Richard McGehee

Bifurcations from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study

We consider a two-parameter family of maps of the plane to itself. Each map has a fixed point in the first quadrant and is a diffeomorphism in a neighborhood of this point. For certain parameter values there is a Hopf bifurcation to an invariant circle, which is smooth for parameter values in a neighborhood of the bifurcation point. However, computer simulations show that the corresponding invariant set fails to be even topologically a circle for parameter values far from the bifurcation point. This paper is an attempt to elucidate some of the mechanisms involved in this loss of smoothness and alteration of topological type.

Triple collision in the collinear three-body problem

によって記述される.ここで∇qiU は qiに関するU の勾配である. 粒子の位置 (q1, . . . , qn)は、ある i = jに対して qi = qjのとき衝突と呼ばれる.上の方程式 系は衝突を除いていたるところ定義されている.時刻 t = 0に粒子の位置と運動量が与えら れているものとする.衝突から始めない限り、微分方程式の標準的定理により、ある極大区間 [0, t∗)の上で方程式 (1.2)の解の存在と一意性が保証されている.t∗ <∞なら解は t∗で特異性 (singularity)を経験するといわれる. 特異性に近づくときの解のふるまいは完全には理解されておらず、いくつかの可能性が知ら れている.t → t∗のときすべての粒子が極限的位置に近づくなら、この極限位置が衝突であ ることをしめすのは難しくない [12,17].特異性はこのとき衝突特異性と呼ばれ、解は衝突で 終わるといわれる.m個の粒子が一致し、残りは異なる位置にあるとき、衝突はm重衝突と 呼ばれる.衝突以外の特異性があるかどうか知られていない. 二体問題の場合、変数変換によって二体衝突を方程式の正則点に変換できる [6].このよう な変換は二体衝突の正則化 (regularization)と呼ばれる.このとき解は特異性を越えて延長で きる.この延長は物理的には弾性反発に対応する. Sundman[14]は、二体衝突が三体問題でも正則化できることを示した.つまり、解が新しい 時間変数の解析関数として二体衝突を通って接続できるように変数を変換できるのである.こ の場合も延長は弾性反発に対応する. 二体より多い粒子を含む衝突はもちろんもっと複雑であるが、そのふるまいのいくつかの局 面は知られている.粒子の配位 (configuration)を、物理的には慣性モーメントに対応するノ ルムで割った位置と定義する.三体問題の三体衝突において、配位がいわゆる中心図形のひと つに近づくことを Sundman[15]は示している ([12,17]、また以下の 6、7節参照).Wintner[17]

Coexistence of species competing for shared resources

Abstract In this paper we develop a mathematical model in which any number of competing species can coexist on four resources which regenerate according to an algebraic relationship. We show that previous attempts to prove that n species cannot coexist on fewer that n resources (the “competitive exclusion principle”) all make use of the very restrictive assumption that the specific growth rates of all competing species are linear functions of resource densities. When this restriction is relaxed, it becomes possible to find situations in which n species can coexist on fewer than n resources. On the basis of this and other observations we conclude that the competitive exclusion principle should be considered to apply only to coexistence at fixed densities.

Some mathematical problems concerning the ecological principle of competitive exclusion

The ecological principle of competitive exclusion asserts that two species cannot indefinitely occupy the same niche [Sj. Attempts have been made to state this principle as a mathematical theorem. A standard example is due to Volterra [14]. (See Example 2.1 below.) He constructed a model of two species competing for a single resource and showed that one of the species must go extinct. Although the concept of niche is rather vague, it is generally agreed that Volterra’s model is an example of two species competing for the same niche and is therefore an illustration of the principle of competitive exclusion. The generalizations of Volterra’s model have dealt largely with the introduction of more realistic assumptions about the interactions between species. These assumptions have led to some interesting mathematical problems, many of them unsolved. The purpose of this paper is to state these problems precisely, to give proofs of the known results, and to indicate the unanswered questions. The basic object of study is an idealized ecological community consisting of a certain number of species living together in an isolated geographical area. We shall assume that the community interactions are independent of both space and time and that each species is distributed uniformly over the region. The basic variables are the population densities of each of the species. The

A stable manifold theorem for degenerate fixed points with applications to celestial mechanics

Consider the classical three-body problem, i.e. the motion of three point masses under the laws of classical mechanics. We shall say an orbit is parabolic if two of the particles remain bounded for all positive time while the third approaches infinity with zero velocity. One can conjecture that the set of all parabolic orbits forms a smooth submanifold of the phase space. In this paper we prove this conjecture for three special cases of the threebody problem. The first is the well-known “restricted three-body problem” [2], where one of the masses is zero and the other two move in circular orbits. The second is a problem discussed by Sitnikov [5], where two equal masses move in a plane and the third moves on a line perpendicular to the plane through the center of mass of the first two. The third is the collinear three-body problem, where the particles are confined to a line. In each case the problem has only two degrees of freedom. For these examples we prove that the parabolic orbits form an analytically immersed submanifold of the energy surface. Our method is to introduce at infinity a periodic orbit. This orbit has as its asymptotic set the set of parabolic orbits. We study the PoincarC map of this periodic orbit. This map is a diffeomorphism of the plane to itself leaving the origin fixed. The points asymptotic to the origin correspond to parabolic orbits. For a diffeomorphism f: R* + R* leaving the origin fixed, define the local asymptotic set of f as:

Double collisions for a classical particle system with nongravitational interactions

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Conley decomposition for closed relations

This paper presents a theory of dynamics of closed relations on compact Hausdorff spaces. It contains an investigation of set valued maps and establishes generalizations for some topological aspects of dynamical systems theory, including recurrence, attractor–repeller structure and the Conley decomposition theorem.

A new proof of the stable manifold theorem

We give a new proof of the stable manifold theorem for hyperbolic fixed points of smooth maps. This proof shows that the local stable and unstable manifolds are projections of a relation obtained as a limit of the graphs of the iterates of the map. The same proof generalizes to the setting of stable and unstable manifolds for smooth relations.

Twist mappings and their applications

An overview of the study of twist maps - an important area of dynamical systems theory - which covers the new tools used to study the field and their applications to dynamical systems. It presents articles which use such innovations to shed light on the question of stability in mechanical systems.

Triple collision in Newtonian gravitational systems

Abstract : The author describes the motions of Newtonian gravitational systems near triple collision.