Bifurcation of critical periods for plane vector fields
Carmen
Chicone;
Marc
Jacobs
433-486
Abstract: A bifurcation problem in families of plane analytic vector fields which have a nondegenerate center at the origin for all values of a parameter $\lambda \in {{\mathbf{R}}^N}$ is studied. In particular, for such a family, the period function $(\xi ,\lambda) \mapsto P(\xi ,\lambda)$ is defined; it assigns the minimum period to each member of the continuous band of periodic orbits (parametrized by $\xi \in {\mathbf{R}}$) surrounding the origin. The bifurcation problem is to determine the critical points of this function near the center with $\lambda$ as bifurcation parameter. Generally, if the function $\rho$, given by $ \xi \mapsto P(\xi ,{\lambda_\ast}) - P(0,{\lambda_\ast})$, vanishes to order $2k$ at the origin, then it is shown that the period function, after a perturbation of ${\lambda_\ast}$, has at most $k$ critical points near the origin. If $ \rho$ vanishes to infinite order, i.e., the center is isochronous, it is shown that the number of critical points of $P$ for perturbations of ${\lambda_\ast}$ depends on the number of generators of the ideal of all Taylor coefficients of $\rho (\xi ,\lambda)$, where the coefficients are considered elements of the ring of convergent power series in $\lambda$. Specifically, if the ideal is generated by the first $2k$ Taylor coefficients, then a perturbation of ${\lambda_\ast}$ produces at most $k$ critical points of $P$ near the origin. These theorems are applied to the quadratic systems with Bautin centers and to one degree of freedom "kinetic+potential" Hamiltonian systems with polynomial potentials. For the quadratic systems a complete solution of the bifurcation problem is obtained. For the Hamiltonian systems a number of results are proved independent of the degree of the potential and a complete solution is obtained for potentials of degree less than seven. Aside from their intrinsic interest, monotonicity properties of the period function are important in the question of existence and uniqueness of autonomous boundary value problems, in the study of subharmonic bifurcation of periodic oscillations, and in the analysis of the problem of linearization. In this regard it is shown that a Hamiltonian system with a polynomial potential of degree larger than two cannot be linearized. However, while these topics are the subject of a large literature, the spirit of this paper is more akin to that of N. Bautin's treatment of the multiple Hopf bifurcation for quadratic systems and the work on various forms of the weakened Hilbert's 16th problem first posed by V. Arnold.
On the reconstruction of topological spaces from their groups of homeomorphisms
Matatyahu
Rubin
487-538
Abstract: For various classes $ K$ of topological spaces we prove that if $ {X_1},{X_2} \in K$ and ${X_1},{X_2}$ have isomorphic homeomorphism groups, then ${X_1}$ and ${X_2}$ are homeomorphic. Let $G$ denote a subgroup of the group of homeomorphisms $H(X)$ of a topological space $X$. A class $K$ of $ \langle X,G\rangle$'s is faithful if for every $\langle {X_1},{G_1}\rangle ,\langle {X_2},{G_2}\rangle \in K$, if $\varphi :{G_1} \to {G_2}$ is a group isomorphism, then there is a homeomorphism $h$ between ${X_1}$ and ${X_2}$ such that for every $g \in {G_1}\;\varphi (g) = hg{h^{ - 1}}$. Theorem 1: The following class is faithful: $ \{ \langle X,H(X)\rangle \vert(X$ is a locally finite-dimensional polyhedron in the metric or coherent topology or $X$ is a Euclidean manifold with boundary) and for every $ x \in X\;x$ is an accumulation point of $\{ g(x)\vert g \in H(X)\} \} \cup \{ \langle X,G\rangle \vert X$ is a differentiable or a $ PL$-manifold and $ G$ contains the group of differentiable or piecewise linear homeomorphisms$ \}$ $\cup \{ \langle X,H(X)\rangle \vert X$ is a manifold over a normed vector space over an ordered field$\}$. This answers a question of Whittaker $[{\text{W}}]$, who asked about the faithfulness of the class of Banach manifolds. Theorem 2: The following class is faithful: $\{ \langle X,G\rangle \vert X$ is a locally compact Hausdorff space and for every open $T \subseteq X$ and $x \in T\;\{ g(x)\vert g \in H(X)$ and $ g \upharpoonright (X - T) = \operatorname{Id}\}$ is somewhere dense$ \}$. Note that this class includes Euclidean manifolds as well as products of compact connected Euclidean manifolds. Theorem 3: The following class is faithful: $\{ \langle X,H(X)\rangle \vert$ (1) $X$ is a 0-dimensional Hausdorff space; (2) for every $x \in X$ there is a regular open set whose boundary is $\{ x\}$; (3) for every $x \in X$ there are ${g_{1,}}{g_2} \in G$ such that $x \ne {g_1}(x) \ne {g_2}(x) \ne x$, and (4) for every nonempty open $V \subseteq X$ there is $g \in H(X) - \{ \operatorname{Id}\}$ such that $g \upharpoonright (X - V) = \operatorname{Id}\}$. Note that (2) is satisfied by 0-dimensional first countable spaces, by order topologies of linear orderings, and by normed vector spaces over fields different from $ {\mathbf{R}}$. Theorem 4: We prove (Theorem 2.23.1) that for an appropriate class ${K^T}$ of trees $\{ \langle \operatorname{Aut}(T),T; \leq , \circ ,\operatorname{Op}\rangle \vert T \in {K^T}\}$ is first-order interpretable in $\{ \operatorname{Aut}(T)\vert T \in {K^T}\}$.
Bifurcation of a unique stable periodic orbit from a homoclinic orbit in infinite-dimensional systems
Shui-Nee
Chow;
Bo
Deng
539-587
Abstract: Under some generic conditions, we show how a unique stable periodic orbit can bifurcate from a homoclinic orbit for semilinear parabolic equations and retarded functional differential equations. This is a generalization of a result of Šil'nikov for ordinary differential equations.
Two-dimensional Riemann problem for a single conservation law
Tong
Zhang;
Yu Xi
Zheng
589-619
Abstract: The entropy solutions to the partial differential equation $\displaystyle (\partial /\partial t)u(t,x,y) + (\partial /\partial x)f(u(t,x,y)) + (\partial /\partial y)g(u(t,x,y)) = 0,$ with initial data constant in each quadrant of the $(x,y)$ plane, have been constructed and are piecewise smooth under the condition $f''(u) \ne 0, g''(u) \ne 0, (f''(u)/g''(u))\prime \ne 0$. This problem generalizes to several space dimensions the important Riemann problem for equations in one-space dimension. Although existence and uniqueness of solutions are well known, little is known about the qualitative behavior of solutions. It is this with which we are concerned here.
Topological types of finitely-$C\sp 0$-$K$-determined map-germs
Takashi
Nishimura
621-639
Abstract: In this article, we investigate the following two problems Problem 1. Is finite- $ {C^0}{\text{-}}K$-determinacy a topological invariant among analytic map-germs? Problem 2. Do the topological types of all finitely- $ {C^0}{\text{-}}K$-determined map-germs have topological moduli, i.e. do they have infinitely many topological types with the cardinal number of continuum? Problem $ 1$ is solved affirmatively in the complex case. Problem $2$ is solved negatively in the complex case; and affirmatively in the real case.
Sharp estimates for the nontangential maximal function and the Lusin area function in Lipschitz domains
Rodrigo
Bañuelos;
Charles N.
Moore
641-662
Abstract: Let $u$ be a harmonic function on a domain of the form $D = \{ (x,y):x \in {{\mathbf{R}}^n},y \in {\mathbf{R}},y > \phi (x)\}$ where $\phi :{{\mathbf{R}}^n} \to {\mathbf{R}}$ is a Lipschitz function. The authors show a good-$\lambda$ inequality between $ Au$, the Lusin area function of $u$, and $Nu$, the nontangential maximal function of $ u$. This leads to an $ {L^p}$ inequality of the form $\left\Vert Au\right\Vert _p \leq C_p\left\Vert Nu\right\Vert _p$ which is sharp in the sense that $ {C_p}$ is of the smallest possible order in $p$ as $p \to \infty $. For $P \in \partial D$ and $t > 0$ we also consider the functions $Au(P + (0,t))$ and $ Nu(P + (0,t))$ and show that a corollary of the good-$\lambda$ inequality is a law of the iterated logarithm involving these two functions as $ t \to 0$. If $ n = 1$ and $\phi$ has a small Lipschitz constant the above results are shown valid with the roles of $ Nu$ and $Au$ interchanged.
Piecewise linearization of real-valued subanalytic functions
Masahiro
Shiota
663-679
Abstract: We show that for a subanalytic function $f$ on a locally compact subanalytic set $ X$ there exists a unique subanalytic triangulation (a simplicial complex $ K$, a subanalytic homeomorphism $\pi :\vert K\vert \to X$) such that $f \circ \pi {\vert _\sigma }, \sigma \in K$, are linear.
A unified theory for real vs. complex rational Chebyshev approximation on an interval
Arden
Ruttan;
Richard S.
Varga
681-697
Abstract: A unified approach is presented for determining all the constants $ {\gamma _{m,n}}\;(m \geq 0,n \geq 0)$ which occur in the study of real vs. complex rational Chebyshev approximation on an interval. In particular, it is shown that ${\gamma _{m,m + 2}} = 1/3\;(m \geq 0)$, a problem which had remained open.
Hypergraphs with finitely many isomorphism subtypes
Henry A.
Kierstead;
Peter J.
Nyikos
699-718
Abstract: Let $\mathcal{H} = (H,E)$ be an $n$-uniform infinite hypergraph such that the number of isomorphism types of induced subgraphs of $\mathcal{H}$ of cardinality $\lambda$ is finite for some infinite $ \lambda$. We solve a problem due independently to Jamison and Pouzet, by showing that there is a finite subset $K$ of $H$ such that the induced subgraph on $H - K$ is either empty or complete. We also characterize such hypergraphs in terms of finite (not necessarily uniform) hypergraphs.
A Poisson-Plancherel formula for the universal covering group with Lie algebra of type $B\sb n$
Peter
Dourmashkin
719-738
Abstract: A proof is given for the Poisson-Plancherel formula for Lie groups of type ${B_n}$ using the recurrence relations for the Plancherel function on adjacent Cartan subalgebras given in [12] and the recurrence relations for the discrete series constants which determine a $ G$-invariant generalized function on $ {{\mathbf{g}}^\ast }$ appearing in the formula.
Higher-dimensional analogues of the modular and Picard groups
C.
Maclachlan;
P. L.
Waterman;
N. J.
Wielenberg
739-753
Abstract: Clifford algebras are used to describe arithmetic groups which are generalizations of the modular and Picard groups.
Cohomology equations and commutators of germs of contact diffeomorphisms
Augustin
Banyaga;
Rafael
de la Llave;
C. Eugene
Wayne
755-778
Abstract: We study the group of germs of contact diffeomorphisms at a fixed point. We prove that the abelianization of this group is isomorphic to the multiplicative group of real positive numbers. The principal ingredient in this proof is a version of the Sternberg linearization theorem in which the conjugating diffeomorphism preserves the contact structure.
Uniform analyticity of orthogonal projections
R. R.
Coifman;
Margaret A. M.
Murray
779-817
Abstract: Let $X$ denote the circle $T$ or the interval $[ - 1,1]$, and let $d\mu$ denote a nonnegative, absolutely continuous measure on $X$ . Under what conditions does the Gram-Schmidt procedure in the weighted space ${L^2}(X,{\omega ^2}\;d\mu)$ depend analytically on the logarithm of the weight function $ \omega$? In this paper, we show that, in numerous examples of interest, $\log \omega \in BMO$ is a sufficient (often necessary!) condition for analyticity of the Gram-Schmidt procedure. These results are then applied to establish the local analyticity of certain infinite-dimensional Toda flows.
Nonuniqueness for solutions of the Korteweg-de Vries equation
Amy
Cohen;
Thomas
Kappeler
819-840
Abstract: Variants of the inverse scattering method give examples of nonuniqueness for the Cauchy problem for $ {\text{KdV}}$. One example gives a nontrivial $ {C^\infty }$ solution $ u$ in a domain $\{ (x,t):0 < t < H(x)\}$ for a positive nondecreasing function $H$ , such that $u$ vanishes to all orders as $t \downarrow 0$ . This solution decays rapidly as $x \to + \infty $ , but cannot be well behaved as $x$ moves left. A different example of nonuniqueness is given in the quadrant $x \geq 0,t \geq 0$, with nonzero initial data.
Finitely presented lattices: canonical forms and the covering relation
Ralph
Freese
841-860
Abstract: A canonical form for elements of a lattice freely generated by a partial lattice is given. This form agrees with Whitman's canonical form for free lattices when the partial lattice is an antichain. The connection between this canonical form and the arithmetic of the lattice is given. For example, it is shown that every element of a finitely presented lattice has only finitely many minimal join representations and that every join representation can be refined to one of these. An algorithm is given which decides if a given element of a finitely presented lattice has a cover and finds them if it does. An example is given of a nontrivial, finitely presented lattice with no cover at all.