Family Of Vector Fields Research Articles

Liouville results for fully nonlinear equations modeled on Hörmander vector fields: II. Carnot groups and Grushin geometries

This paper treats second order fully nonlinear degenerate elliptic equations having a family of subunit vector fields satisfying a full-rank bracket condition. It studies Liouville properties for viscosity sub- and supersolutions in the whole space, namely, under a suitable bound at infinity from above and, respectively, from below, they must be constants. In a previous paper, we proved an abstract result and discussed operators on the Heisenberg group. Here, we consider various families of vector fields: the generators of a Carnot group, with more precise results for those of step 2, in particular H-type groups and free Carnot groups, the Grushin and the Heisenberg-Greiner vector fields. All these cases are relevant in sub-Riemannian geometry and have in common the existence of a homogeneous norm that we use for building Lyapunov-like functions for each operator. We give explicit sufficient conditions on the size and sign of the first and 0-th order terms in the equations and discuss their optimality. We also outline some applications of such results to the problem of ergodicity of multidimensional degenerate diffusion processes in the whole space.

A global bifurcation organizing rhythmic activity in a coupled network

We study a system of coupled phase oscillators near a saddle-node on invariant circle bifurcation and driven by random intrinsic frequencies. Under the variation of control parameters, the system undergoes a phase transition changing the qualitative properties of collective dynamics. Using Ott–Antonsen reduction and geometric techniques for ordinary differential equations, we identify heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we show that heteroclinic bifurcation separates two topologically distinct families of limit cycles: contractible limit cycles before bifurcation from noncontractibile ones after bifurcation. Both families are stable for the model at hand.

Asymptotic analysis for Hamilton-Jacobi-Bellman equations on Euclidean space

The long-time average behavior of the value function in the calculus of variations is known to be connected to the existence of the limit of the corresponding Abel means. Still in the Tonelli case, such a limit is in turn related to the existence of solutions of the critical (or ergodic) Hamilton-Jacobi equation. The goal of this paper is to address similar issues when set on the whole Euclidean space and the Hamiltonian fails to be Tonelli. We first study the convergence of the time-averaged value function as the time horizon goes to infinity, proving the existence of the critical constant (Mañé critical value) for a general control system. Then, we show that the ergodic equation admits solutions for systems associated with a family of vector fields which satisfies the Lie Algebra rank condition. Finally, we construct a solution to the critical HJB equation on the whole space which coincides with its Lax-Oleinik evolution.

Families of vector fields with many numerical invariants

<p style='text-indent:20px;'>We study bifurcations in finite-parameter families of vector fields on <inline-formula><tex-math id="M1">\begin{document}$S^2$\end{document}</tex-math></inline-formula>. Recently, Yu. Ilyashenko, Yu. Kudryashov, and I. Schurov provided examples of (locally generic) structurally unstable <inline-formula><tex-math id="M2">\begin{document}$3$\end{document}</tex-math></inline-formula>-parameter families of vector fields: topological classification of these families admits at least one numerical invariant. They also provided examples of <inline-formula><tex-math id="M3">\begin{document}$(2D+1)$\end{document}</tex-math></inline-formula>-parameter families such that the topological classification of these families has at least <inline-formula><tex-math id="M4">\begin{document}$D$\end{document}</tex-math></inline-formula> numerical invariants and used those examples to construct families with functional invariants of topological classification.</p><p style='text-indent:20px;'>In this paper, we construct locally generic <inline-formula><tex-math id="M5">\begin{document}$4$\end{document}</tex-math></inline-formula>-parameter families with any prescribed number of numerical invariants and use them to construct <inline-formula><tex-math id="M6">\begin{document}$5$\end{document}</tex-math></inline-formula>-parameter families with functional invariants. We also describe a locally generic class of <inline-formula><tex-math id="M7">\begin{document}$3$\end{document}</tex-math></inline-formula>-parameter families with a tail of an infinite number sequence as an invariant of topological classification.</p>

Leibniz cohomology and connections on differentiable manifolds

We show how an affine connection on a Riemannian manifold occurs naturally as a cochain in the complex for Leibniz cohomology of vector fields with coefficients in the adjoint representation. The Leibniz coboundary of the Levi-Civita connection can be expressed as a sum of two terms, one the Laplace-Beltrami operator and the other a Ricci curvature term. The vanishing of this coboundary has an interpretation in terms of eigenfunctions of the Laplacian. Separately, we compute the Leibniz cohomology with adjoint coefficients for a certain family of vector fields on Euclidean Rn corresponding to the affine orthogonal Lie algebra, n≥3.

On the geometry of singular foliations generated by the family of vector fields

On the geometry of singular foliations generated by the family of vector fields

Nilpotent singularities and chaos: Tritrophic food chains

Abstract Local bifurcation theory is used to prove the existence of chaotic dynamics in two well-known models of tritrophic food chains. To the best of our knowledge, the simplest technique to guarantee the emergence of strange attractors in a given family of vector fields consists of finding a 3-dimensional nilpotent singularity of codimension 3 and verifying some generic algebraic conditions. We provide the essential background regarding this method and describe the main steps to illustrate numerically the chaotic dynamics emerging near these nilpotent singularities. This is a general-purpose method and we hope it can be applied to a huge range of models.

Various Equivalence Relations in Global Bifurcation Theory

We discuss various definitions of equivalence for bifurcations of vector fields on the sphere and give a large number of examples (both known and new) that illustrate the advantages and disadvantages of different definitions. In addition to the classical definitions of strong and weak equivalence, we consider new notions of Sing-equivalence and moderate equivalence. These definitions seem to be more relevant to and consistent with the intuitive notion of equivalent bifurcations. They were introduced and used to describe the structural instability of some finite-parameter families of vector fields on the sphere and to study invariants of their classification.

On Geometry of Vector Fields

It is well known that the study of the geometry and topology of the attainability set of a family of vector fields is one of the main tasks of the qualitative control theory, which is closely related to the geometry of orbits of vector fields. In this paper, we present the authors’ results on the geometry of the attainability set of a family of vector fields: results on the geometry of T -attainability sets and the geometry of orbits of Killing vector fields.

The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function

We define and study C1-solutions of the Aronsson equation (AE), a second order quasi linear equation. We show that such super/subsolutions make the Hamiltonian monotone on the trajectories of the closed loop Hamiltonian dynamics. We give a short, general proof that C1-solutions are absolutely minimizing functions. We discuss how C1-supersolutions of (AE) become special Lyapunov functions of symmetric control systems, and allow to find continuous feedbacks driving the system to a target in finite time, except on a singular manifold. A consequence is a simple proof that the corresponding minimum time function is locally Lipschitz continuous away from the singular manifold, despite classical results showing that it should only be Hölder continuous unless appropriate conditions hold. We provide two examples for Hörmander and Grushin families of vector fields where we construct C1-solutions (even classical) explicitly.

Random switching between vector fields having a common zero

Let $E$ be a finite set, $\{F^{i}\}_{i\in E}$ a family of vector fields on $\mathbb{R}^{d}$ leaving positively invariant a compact set $M$ and having a common zero $p\in M$. We consider a piecewise deterministic Markov process $(X,I)$ on $M\times E$ defined by $\dot{X}_{t}=F^{I_{t}}(X_{t})$ where $I$ is a jump process controlled by $X$: ${\mathsf{P}}(I_{t+s}=j|(X_{u},I_{u})_{u\leq t})=a_{ij}(X_{t})s+o(s)$ for $i\neq j$ on $\{I_{t}=i\}$. We show that the behaviour of $(X,I)$ is mainly determined by the behaviour of the linearized process $(Y,J)$ where $\dot{Y}_{t}=A^{J_{t}}Y_{t}$, $A^{i}$ is the Jacobian matrix of $F^{i}$ at $p$ and $J$ is the jump process with rates $(a_{ij}(p))$. We introduce two quantities $\Lambda^{-}$ and $\Lambda^{+}$, respectively, defined as the minimal (resp., maximal) growth rate of $\|Y_{t}\|$, where the minimum (resp., maximum) is taken over all the ergodic measures of the angular process $(\Theta,J)$ with $\Theta_{t}=\frac{Y_{t}}{\|Y_{t}\|}$. It is shown that $\Lambda^{+}$ coincides with the top Lyapunov exponent (in the sense of ergodic theory) of $(Y,J)$ and that under general assumptions $\Lambda^{-}=\Lambda^{+}$. We then prove that, under certain irreducibility conditions, $X_{t}\rightarrow p$ exponentially fast when $\Lambda^{+}<0$ and $(X,I)$ converges in distribution at an exponential rate toward a (unique) invariant measure supported by $M\setminus \{p\}\times E$ when $\Lambda^{-}>0$. Some applications to certain epidemic models in a fluctuating environment are discussed and illustrate our results.

Global Bifurcations in Generic One-parameter Families on $$\mathbb{S}^2$$ S 2

In this paper we prove that generic one-parameter families of vector fields on $$\mathbb{S}^2$$ in the neighborhood of the fields of classes AH, SN, HC, SC (Andronov–Hopf, saddle-node, homoclinic curve, saddle connection) are structurally stable. We provide a classification of bifurcations in these families.

Global bifurcations in the two-sphere: a new perspective

We construct an open set of structurally unstable three parameter families whose weak and so called moderate topological classification defined below has a numerical invariant that may take an arbitrary positive value. Here and below "families" are "families of vector fields in the two-sphere". This result disproves an Arnold's conjecture of 1985. Then we construct an open set of six parameter families whose moderate topological classification has a functional invariant. This invariant is an arbitrary germ of a smooth map $(\mathbb R_+, a)\to(\mathbb R_+, b)$. More generally, for any positive integers $d$ and $d'$, we construct an open set of families whose topological classification has a germ of a smooth map $\left(\mathbb R_+^d, a\right)\to\left(\mathbb R_+^{d'}, b\right)$ as an invariant. Any smooth germ of this kind may be realized as such an invariant. These results open a new perspective of the global bifurcation theory in the two sphere. This perspective is discussed at the end of the paper.

Evolution systems of measures and semigroup properties on evolving manifolds

An evolving Riemannian manifold $(M,g_t)_{t\in I}$ consists of a smooth $d$-dimensional manifold $M$, equipped with a geometric flow $g_t$ of complete Riemannian metrics, parametrized by $I=(-\infty,T)$. Given an additional $C^{1,1}$ family of vector fields $(Z_t)_{t\in I}$ on $M$. We study the family of operators $L_t=\Delta_t +Z_t $ where $\Delta_t$ denotes the Laplacian with respect to the metric $g_t$. We first give sufficient conditions, in terms of space-time Lyapunov functions, for non-explosion of the diffusion generated by $L_t$, and for existence of evolution systems of probability measures associated to it. Coupling methods are used to establish uniqueness of the evolution systems under suitable curvature conditions. Adopting such a unique system of probability measures as reference measures, we characterize supercontractivity, hypercontractivity and ultraboundedness of the corresponding time-inhomogeneous semigroup. To this end, gradient estimates and a family of (super-)logarithmic Sobolev inequalities are established.

Attractors met X-elliptic operators

We consider degenerate parabolic and damped hyperbolic equations involving an operator L, that is X-elliptic with respect to a family of locally Lipschitz continuous vector fields X={X1,…,Xm}. The local well-posedness is established under subcritical growth restrictions on the nonlinearity f, which are determined by the geometry and functional setting naturally associated with the family of vector fields X. Assuming additionally that f is dissipative, the global existence of solutions follows, and we can characterize their longtime behavior using methods from the theory of infinite dimensional dynamical systems.

Numerical Data in Weakly-Dissipative KAM-Theory

In the article are presented as figures numerical data calculation of the basic thermodynamic variables such as dependence of the thermodynamic potentials from the temperature and pressure, and the geometric characteristics of the dynamics like the center of mass of trial particle. The dynamics is described by a simple Euler discretization of family of vector fields arising in the Bogdanov-Takens bifurcation.

Remarks on multi-marginal symmetric Monge-Kantorovich problems

Symmetric Monge-Kantorovich transport problems involving a cost function given by a family of vector fields were used by Ghoussoub-Moameni to establish polar decompositions of such vector fields into $m$-cyclically monotone maps composed with measure preserving $m$-involutions ($m\geq 2$). In this note, we relate these symmetric transport problems to the Brenier solutions of the Monge and Monge-Kantorovich problem, as well as to the Gangbo-Świech solutions of their multi-marginal counterparts, both of which involving quadratic cost functions.

Some Results About Global Asymptotic Stability

We study the global asymptotic stability of the origin for the continuous and discrete dynamical system associated to polynomial maps in R n (especially when n = 3)oftheform F = λ I + H,with F(0) = 0,where λisarealnumber,Itheidentity map, and H a map with nilpotent Jacobian matrix JH . We distinguish the cases when the rows of JH are linearly dependent over R and when they are linearly independent over R. In the linearly dependent case we find non-linearly triangularizable vector fields F for which the origin is globally asymptotically stable singularity (respectively fixed point) for continuous (respectively discrete) systems generated by F.I n the independent continuous case, we present a family of maps that have orbits escaping to infinity. Finally, in the independent discrete case, we show a large family of vector fields that have a periodic point of period 3.

Thermodynamics in weakly-dissipative Kolmogorov-Arnold-Moser theory

In the article there are presented the results of the numerical data calculation of the basic thermodynamic variables such as dependence of the thermodynamic potentials from the temperature and pressure, and the geometric characteristics of the dynamics like the center of mass of trial particle. The dynamics is described by a simple Euler discretization of family of vector fields arising in the Bogdanov-Takens bifurcation.

Generalized Jacobi Identities and Ball-Box Theorem for Horizontally Regular Vector Fields

We consider a family of vector fields and we assume a horizontal regularity on their derivatives. We discuss the notion of commutator showing that different definitions agree. We apply our results to the proof of a ball-box theorem and Poincar\'e inequality for nonsmooth H\"ormander vector fields.