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Register a new userInvariant vector fields and groupoids
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Authors
Eugene Lerman
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We use the notion of isomorphism between two invariant vector fields to shed new light on the issue of linearization of an invariant vector field near a relative equilibrium. We argue that the notion is useful in understanding the passage from the space of invariant vector fields in a tube around a group orbit to the space invariant vector fields on a slice to the orbit. The notion comes from Hepworths study of vector fields on stacks.
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This paper describes a relationship between essentially finite groupoids and 2-vector spaces. In particular, we show to construct 2-vector spaces of Vect-valued presheaves on such groupoids. We define 2-linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation--a weak functor--from Span(Gpd) (the bicategory of groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail.
For a smooth near identity map, we introduce the notion of an interpolating vector field written in terms of iterates of the map. Our construction is based on Lagrangian interpolation and provides an explicit expressions for autonomous vector fields which approximately interpolate the map. We study properties of the interpolating vector fields and explore their applications to the study of dynamics. In particular, we construct adiabatic invariants for symplectic near identity maps. We also introduce the notion of a Poincare section for a near identity map and use it to visualise dynamics of four dimensional maps. We illustrate our theory with several examples, including the Chirikov standard map and a symplectic map in dimension four, an example motivated by the theory of Arnold diffusion.
In this work a theorical framework to apply the Poincare compactification technique to locally Lipschitz continuous vector fields is developed. It is proved that these vectors fields are compactifiable in the n-dimensional sphere, though the compactified vector field can be identically null in the equator. Moreover, for a fixed projection to the hemisphere, all the compactifications of a vector field, which are not identically null on the equator are equivalent. Also, the conditions determining the invariance of the equator for the compactified vector field are obtained. Up to the knowledge of the authors, this is the first time that the Poincare compactification of locally Lipschitz continuous vector fields is studied. These results are illustrated applying them to some families of vector fields, like polynomial vector fields, vector fields defined as a sum of homogeneous functions and vector fields defined by piecewise linear functions.
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 $fcolon Mrightarrowmathbb R$. 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 Poincare-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 $X$.
We study the number and distribution of the limit cycles of a planar vector field whose component functions are random polynomials. We prove a lower bound on the average number of limit cycles when the random polynomials are sampled from the Kostlan-Shub-Smale ensemble. For the related Bargmann-Fock ensemble of real analytic functions we establish an asymptotic result for the average number of empty limit cycles (limit cycles that do not surround other limit cycles) in a large viewing window. Concerning the special setting of limit cycles near a randomly perturbed center focus (where the perturbation has i.i.d. coefficients) we prove that the number of limit cycles situated within a disk of radius less than unity converges almost surely to the number of real zeros of a certain random power series. We also consider infinitesimal perturbations where we obtain precise asymptotics on the global count of limit cycles for a family of models. The proofs of these results use novel combinations of techniques from dynamical systems and random analytic functions.
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