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Morris W. Hirsch

Publications and source records attributed to Morris W. Hirsch.

At least 19 recordsLinked to original sources
arXiv2018

Primary singularities of vector fields on surfaces

Unless another thing is stated one works in the $C^\infty$ category and manifolds have empty boundary. Let $X$ and $Y$ be vector fields on a manifold $M$. We say that $Y$ tracks $X$ if $[Y,X]=fX$ for some continuous function $f\colon M\rightarrow\mathbb R$. A subset $K$ of the zero set ${\mathsf Z}(X)$ is an essential block for $X$ if it is non-empty, compact, open in ${\mathsf Z}(X)$ and its Poincaré-Hopf index does not vanishes. One says that $X$ is non-flat at $p$ if its $\infty$-jet at $p$ is non-trivial. A point $p$ of ${\mathsf Z}(X)$ is called a primary singularity of $X$ if any vector field defined about $p$ and tracking $X$ vanishes at $p$. This is our main result: Consider an essential block $K$ of a vector field $X$ defined on a surface $M$. Assume that $X$ is non-flat at every point of $K$. Then $K$ contains a primary singularity of $X$. As a consequence, if $M$ is a compact surface with non-zero characteristic and $X$ is nowhere flat, then there exists a primary singularity of $X$.

math.DS↗
arXiv2018

Monotone local flows with dense periodic orbits

Let X be a connected open set in n-dimensional Euclidean space, partially ordered by a closed convex cone K with nonempty interior: y > x if and only if y-x is nonzero and in K. Theorem: If F is a monotone local flow in X whose periodic points are dense in X, then F is globally periodic.

math.DS↗
arXiv2016

Zero sets of Lie algebras of analytic vector fields on real and complex 2-dimensional manifolds, II

On a real ($\mathbb F=\mathbb R$) or complex ($\mathbb F=\mathbb C$) analytic connected 2-manifold $M$ with empty boundary consider two vector fields $X,Y$. We say that $Y$ {\it tracks} $X$ if $[Y,X]=fX$ for some continuous function $f\colon M\rightarrow\mathbb F$. Let $K$ be a compact subset of the zero set ${\mathsf Z}(X)$ such that ${\mathsf Z}(X)-K$ is closed, with nonzero Poincaré-Hopf index (for example $K={\mathsf Z}(X)$ when $M$ is compact and $χ(M)\neq 0$) and let $\mathcal G$ be a finite-dimensional Lie algebra of analytic vector fields on $M$. \smallskip {\bf Theorem.} Let $X$ be analytic and nontrivial. If every element of $\mathcal G$ tracks $X$ and, in the complex case when ${\mathsf i}_K (X)$ is positive and even no quotient of $\mathcal G$ is isomorphic to ${\mathfrak {s}}{\mathfrak {l}} (2,\mathbb C)$, then $\mathcal G$ has some zero in $K$. \smallskip {\bf Corollary.} If $Y$ tracks a nontrivial vector field $X$, both of them analytic, then $Y$ vanishes somewhere in $K$. \smallskip Besides fixed point theorems for certain types of transformation groups are proved. Several illustrative examples are given.

math.DS↗
arXiv2016

Zero sets of Abelian Lie algebras of vector fields

Assume M is a 3-dimensional real manifold without boundary, A is an abelian Lie algebra of analytic vector fields on M, and X is an element of A. The following result is proved: If K is a locally maximal compact set of zeroes of X and the Poincar'e-Hopf index of X at K is nonzero, there is a point in K at which all the elements of A vanish.

math.DS↗
arXiv2015

Zero sets of Lie algebras of analytic vector fields on real and complex 2-manifolds

Let X be an analytic vector field on a real or complex 2-manifold, and K a compact set of zeros of X whose fixed point index is not zero. Let A denote the Lie algebra of analytic vector fields Y on M such that at every point of M the values of X and [X,Y] are linearly dependent. Then the vector fields in A have a common zero in K. Application: Let G be a connected Lie group having a 1-dimensional normal subgroup. Then every action of G on M has a fixed point.

math.DS↗
arXiv2015

Common zeroes of families of smooth vector fields on surfaces

Let Y and X denote C^k vector fields on a possibly noncompact surface with empty boundary, k >0. Say that Y tracks X if the dynamical system it generates locally permutes integral curves of X. Let K be a locally maximal compact set of zeroes of X. THEOREM Assume the Poincar'e-Hopf index of X at K is nonzero, and the k-jet of X at each point of K is nontrivial. If g is a supersolvable Lie algebra of C^k vector fields that track X, then the elements of g have a common zero in K. Applications are made to attractors and transformation groups.

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arXiv2015

Smale Strategies for Network Prisoner's Dilemma Games

Smale's approach \cite{Smale80} to the classical two-players repeated Prisoner's Dilemma game is revisited here for $N$-players and Network games in the framework of Blackwell's approachability, stochastic approximations and differential inclusions.

math.OC↗
arXiv2014

Fixed points of local actions of nilpotent Lie groups on surfaces

Let $G$ be connected nilpotent Lie group acting locally on a real surface $M$. Let $φ$ be the local flow on $M$ induced by a $1$-parameter subgroup. Assume $K$ is a compact set of fixed points of $φ$ and $U$ is a neighborhood of $K$ containing no other fixed points. Theorem: If the Dold fixed-point index of $φ_t|U$ is nonzero for sufficiently small $t>0$, then ${\rm Fix} (G) \cap K \ne \emptyset$.

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arXiv2012

Common zeros of inward vector fields on surfaces

A vector field X on a manifold M with possibly nonempty boundary is inward if it generates a unique local semiflow $Φ^X$. A compact relatively open set K in the zero set of X is a block. The Poincaré-Hopf index is generalized to an index for blocks that may meet the boundary. A block with nonzero index is essential. Let X, Y be inward $C^1$ vector fields on surface M such that $[X,Y]\wedge X=0$ and let K be an essential block of zeros for X. Among the main results are that Y has a zero in K if X and $Y$ are analytic, or Y is $C^2$ and $Φ^Y$ preserves area. Applications are made to actions of Lie algebras and groups.

math.DS↗
arXiv2012

Chain transitive sets for smooth strongly monotone dynamical systems

Let K denote a compact invariant set for a strongly monotone semiflow in an ordered Banach space E, satisfying standard smoothness and compactness assumptions. Suppose the semiflow restricted to K is chain transitive. The main result is that either K is unordered, or else K is contained in totally ordered, compact arc of stationary points; and the latter cannot occur if the semiflow is real analytic and dissipative. As an application, entropy is 0 when E = R^3 . Analogous results are proved for maps. The main tools are results of Mierczynski [27 ] and Terescak [37 ]

math.DS↗
arXiv2012

Actions of Lie groups and Lie algebras on manifolds

Questions of the following sort are addressed: Does a given Lie group or Lie algebra act effectively on a given manifold? How smooth can such actions be? What fxed-point sets are possible? What happens under perturbations? Old results are summarized, and new ones presented, including: For every integer n there are solvable (in some cases, nilpotent) Lie algebras g that have effective C-infinity actions on all n-manifolds, but on some (in many cases, all) n-manifolds, g does not have effective analytic actions

math.GR↗
arXiv2007

Attractors in coherent systems of differential equations

Attractors of cooperative dynamical systems are particularly simple; for example, a nontrivial periodic orbit cannot be an attractor. This paper provides characterizations of attractors for the wider class of coherent systems, defined by the property that no directed feedback loops are negative. Several new results for cooperative systems are obtained in the process.

math.DS↗
arXiv2006

Prevalent Behavior of Strongly Order Preserving Semiflows

Classical results in the theory of monotone semiflows give sufficient conditions for the generic solution to converge toward an equilibrium or towards the set of equilibria (quasiconvergence). In this paper, we provide new formulations of these results in terms of the measure-theoretic notion of prevalence. For monotone reaction-diffusion systems with Neumann boundary conditions on convex domains, we show that the set of continuous initial data corresponding to solutions that converge to a spatially homogeneous equilibrium is prevalent. We also extend a previous generic convergence result to allow its use on Sobolev spaces. Careful attention is given to the measurability of the various sets involved.

math.DS↗