Analytic Vector Fields Research Articles

Real analytic vector fields with first integral and separatrices

We prove that a germ of analytic vector field at $$({\mathbb {R}}^3,0)$$ that possesses a non-constant analytic first integral has a real formal separatrix. We provide an example which shows that such a vector field does not necessarily have a real analytic separatrix.

A topological characterization of the $\omega$-limit sets of analytic vector fields on open subsets of the sphere

In [ 15 ], V. Jimenez Lopez and J. Llibre characterized, up to homeomorphism, the \begin{document}$ \omega $\end{document} -limit sets of analytic vector fields on the sphere and the projective plane. The authors also studied the same problem for open subsets of these surfaces. Unfortunately, an essential lemma in their programme for general surfaces has a gap. Although the proof of this lemma can be amended in the case of the sphere, the plane, the projective plane and the projective plane minus one point (and therefore the characterizations for these surfaces in [ 15 ] are correct), the lemma is not generally true, see [ 6 ]. Consequently, the topological characterization for analytic vector fields on open subsets of the sphere and the projective plane is still pending. In this paper, we close this problem in the case of open subsets of the sphere.

Solitary trajectories of analytic vector fields

A trajectory $$\gamma $$ of an analytic vector fields $$\xi $$ is said to be solitary, if $$\gamma $$ is asymptotic to a formal curve and can be isolated from any other trajectory of $$\xi $$ by a finite sequence of point centered blow ups. We prove that the collection of all solitary trajectories generates a polynomially bounded o-minimal structure.

A Real-Time Hydrodynamic-Based Obstacle Avoidance System for Non-holonomic Mobile Robots with Curvature Constraints

The harmonic potential field of an incompressible nonviscous fluid governed by the Laplace’s Equation has shown its potential for being beneficial to autonomous unmanned vehicles to generate smooth, natural-looking, and predictable paths for obstacle avoidance. The streamlines generated by the boundary value problem of the Laplace’s Equation have explicit, easily computable, or analytic vector fields as the path tangent or robot heading specification without the waypoints and higher order path characteristics. We implemented an obstacle avoidance approach with a focus on curvature constraint for a non-holonomic mobile robot regarded as a particle using curvature-constrained streamlines and streamline changing via pure pursuit. First, we use the potential flow field around a circle to derive three primitive curvature-constrained paths to avoid single obstacles. Furthermore, the pure pursuit controller is implemented to achieve a smooth transition between the streamline paths in the environment with multiple obstacles. In addition to comparative simulations, a proof of concept experiment implemented on a two-wheel driving mobile robot with range sensors validates the practical usefulness of the integrated system that is able to navigate smoothly and safely among multiple cylinder obstacles. The computational requirement of the obstacle avoidance system takes advantage of an a priori selection of fast computing primitive streamline paths, thus, making the system able to generate online a feasible path with a lower maximum curvature that does not violate the curvature constraint.

On the Dulac's problem for piecewise analytic vector fields

This paper presents a version of Dulac's Problem for piecewise analytic vector fields that states conditions for the number of limit cycles around certain minimal sets. A suitable model-theoretic structure is introduced under which a qualitative investigation of the problem is settled.

An inverse approach to the center problem

We consider analytic or polynomial vector fields of the form $${\mathcal {X}}=\left( -y+X\right) \dfrac{\partial }{\partial x}+\left( x+Y\right) \dfrac{\partial }{\partial y},$$ where $$X=X(x,y))$$ and $$Y=Y(x,y))$$ start at least with terms of second order. It is well-known that $${\mathcal {X}}$$ has a center at the origin if and only if $${\mathcal {X}}$$ has a Liapunov–Poincare local analytic first integral of the form $$H=\dfrac{1}{2}(x^2+y^2)+\sum _{j=3}^ {\infty } H_j$$ , where $$H_j=H_j(x,y)$$ is a homogenous polynomial of degree j. The classical center-focus problem already studied by Poincare consists in distinguishing when the origin of $${\mathcal {X}}$$ is either a center or a focus. In this paper we study the inverse center problem, i.e. for a given analytic function H of the previous form defined in a neighborhood of the origin, we determine the analytic or polynomial vector field $${\mathcal {X}}$$ for which H is a first integral. Moreover, given an analytic function $$V=1+\sum _{j=1}^ {\infty } V_j$$ in a neighborhood of the origin, where $$V_j$$ is a homogenous polynomial of degree j, we determine the analytic or polynomial vector field $${\mathcal {X}}$$ for which V is a Reeb inverse integrating factor. We study the particular case of centers which have a local analytic first integral of the form $$ H=\dfrac{1}{2}(x^2+y^2)\,\left( 1+ \sum _{j=1}^{\infty } \Upsilon _j\right) , $$ in a neighborhood of the origin, where $$\Upsilon _j$$ is a homogenous polynomial of degree j for $$j\ge 1.$$ These centers are called weak centers, they contain the uniform isochronous centers and the isochronous holomorphic centers, but they do not coincide with the class of isochronous centers. We have characterized the expression of an analytic or polynomial differential system having a weak center at the origin We extended to analytic or polynomial differential systems the weak conditions of a center given by Alwash and Lloyd for linear centers with homogeneous polynomial nonlinearities. Furthermore the centers satisfying these weak conditions are weak centers.

The presence of symplectic strata improves the Gevrey regularity for sums of squares

We consider a class of operators of the type sum of squares of real analytic vector fields satisfying the Hormander bracket condition. The Poisson-Treves stratification is associated to the vector fields. We show that if the deepest stratum in the stratification, i.e., the stratum associated to the longest commutators, is symplectic, then the Gevrey regularity of the solution is better than the minimal Gevrey regularity given by the Derridj-Zuily theorem.

Quasianalytic Ilyashenko Algebras

AbstractWe construct a quasianalytic field of germs at +∞ of real functions with logarithmic generalized power series as asymptotic expansions, such that is closed under differentiation and log-composition; in particular, is a Hardy field. Moreover, the field o (−log) of germs at 0+ contains all transition maps of hyperbolic saddles of planar real analytic vector fields.

Doubly-resonant saddle-nodes in (\protect \mathbb{C}^3,0) and the fixed singularity at infinity in Painlevé equations: analytic classification

In this work, we consider germs of analytic singular vector fields in ℂ 3 with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector fields come from irregular two-dimensional differential systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at infinity in Painlevé equations (P j ) j=I,⋯,V for generic values of the parameters. Under suitable assumptions, we prove a theorem of analytic normalization over sectorial domains, analogous to the classical one due to Hukuhara–Kimura–Matuda for saddle-nodes in ℂ 2 . We also prove that these sectorial normalizing maps are in fact the Gevrey-1 sums of the formal normalizing map, the existence of which has been proved in a previous paper. Finally we provide an analytic classification under the action of fibered diffeomorphisms, based on the study of the so-called Stokes diffeomorphisms obtained by comparing consecutive sectorial normalizing maps à la Martinet–Ramis / Stolovitch for 1-resonant vector fields.

Trajectories in interlaced integral pencils of 3-dimensional analytic vector fields are o-minimal

Let ξ \xi be an analytic vector field at ( R 3 , 0 ) (\mathbb {R}^3,0) and I \mathcal {I} be an analytically non-oscillatory integral pencil of ξ \xi ; i.e., I \mathcal {I} is a maximal family of analytically non-oscillatory trajectories of ξ \xi at 0 all sharing the same iterated tangents. We prove that if I \mathcal {I} is interlaced, then for any trajectory Γ ∈ I \Gamma \in \mathcal {I} , the expansion R a n , Γ \mathbb {R}_{\mathrm {an},\Gamma } of the structure R a n \mathbb {R}_{\mathrm {an}} by Γ \Gamma is model-complete, o-minimal and polynomially bounded.

On geodesibility of algebrizable planar vector fields

Geodesibility of vector fields was studied by Gluck and Sullivan in the 1970s. For the case of complex analytical vector fields, Jenkins shed light on the subject from the end of the 1950s. After the 1970s, multiple authors have studied the subject, such as K. Strebel, and Mucino-Raymundo and Valero-Valdez. In this paper, we consider planar vector fields which are $$\mathbb {A}$$ -algebrizable (differentiable in the sense of Lorch for some associative and commutative algebra $$\mathbb {A}$$ with unit e). We give rectifications of these vector fields and metrics under which they are geodesible.

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

Let$M$be an analytic connected 2-manifold with empty boundary, over the ground field$\mathbb{F}=\mathbb{R}$or$\mathbb{C}$. Let$Y$and$X$denote differentiable vector fields on$M$. We say that$Y$tracks$X$if$[Y,X]=fX$for some continuous function$f:\,M\rightarrow \mathbb{F}$. A subset$K$of the zero set$\mathsf{Z}(X)$is an essential block for$X$if it is non-empty, compact and open in$\mathsf{Z}(X)$, and the Poincaré–Hopf index$\mathsf{i}_{K}(X)$is non-zero. Let${\mathcal{G}}$be a finite-dimensional Lie algebra of analytic vector fields that tracks a non-trivial analytic vector field$X$. Let$K\subset \mathsf{Z}(X)$be an essential block. Assume that if$M$is complex and$\mathsf{i}_{K}(X)$is a positive even integer, no quotient of${\mathcal{G}}$is isomorphic to$\mathfrak{s}\mathfrak{l}(2,\mathbb{C})$. Then${\mathcal{G}}$has a zero in$K$(main result). As a consequence, if$X$and$Y$are analytic,$X$is non-trivial, and$Y$tracks$X$, then every essential component of$\mathsf{Z}(X)$meets$\mathsf{Z}(Y)$. Fixed-point theorems for certain types of transformation groups are proved. Several illustrative examples are given.

A note on $$\varphi $$ φ -analytic conformal vector fields

Taking clue from the analytic vector fields on a complex manifold, $$\varphi \hbox {-analytic}$$ conformal vector fields are defined on a Riemannian manifold (Deshmukh and Al-Solamy in Colloq. Math. 112(1):157–161, 2008). In this paper, we use $$\varphi \hbox {-analytic}$$ conformal vector fields to find new characterizations of the n-sphere $$ S^{n}(c)$$ and the Euclidean space $$(R^{n},\left\langle ,\right\rangle )$$ .

Parameterization of Invariant Manifolds for Periodic Orbits (II): A Posteriori Analysis and Computer Assisted Error Bounds

In this paper we develop mathematically rigorous computer assisted techniques for studying high order Fourier–Taylor parameterizations of local stable/unstable manifolds for hyperbolic periodic orbits of analytic vector fields. We exploit the numerical methods developed in Castelli et al. (SIAM J Appl Dyn Syst 14(1):132–167, 2015) in order to obtain a high order Fourier–Taylor series expansion of the parameterization. The main result of the present work is an a-posteriori theorem which provides mathematically rigorous error bounds. The hypotheses of the theorem are checked with computer assistance. The argument relies on a sequence of preliminary computer assisted proofs where we validate the numerical approximation of the periodic orbit, its stable/unstable normal bundles, and the jets of the manifold to some desired order M. We illustrate our method by implementing validated computations for two dimensional manifolds in the Lorenz equations in $$\mathbb {R}^3$$ and a three dimensional manifold of a suspension bridge equation in $$\mathbb {R}^4$$ .

Dynamics of singular complex analytic vector fields with essential singularities I

We tackle the problem of understanding the geometry and dynamics of singular complex analytic vector fields $X$ with essential singularities on a Riemann surface $M$ (compact or not). Two basic techniques are used. (a) In the complex analytic category on $M$, we exploit the correspondence between singular vector fields $X$, differential forms $\omega _{X}$ (with $\omega _{X}(X)\equiv 1$), orientable quadratic differentials $\omega _{X} \otimes \omega _{X}$, global distinguished parameters $\Psi _{X} (z) = \int ^z \omega _{X}$, and the Riemann surfaces $\mathcal {R}_{X}$ of the above parameters. (b) We use the fact that all singular complex analytic vector fields can be expressed as the global pullback via certain maps of the holomorphic vector fields on the Riemann sphere, in particular, via their respective $\Psi _{X}$. We show that under certain analytical conditions on $\Psi _{X}$, the germ of a singular complex analytic vector field determines a decomposition in angular sectors; center $C$, hyperbolic $H$, elliptic $E$, parabolic $P$ sectors but with the addition of suitable copies of a new type of entire angular sector $\mathscr {E}$, stemming from $X(z)=\mathrm{e}^z \frac {\partial }{\partial z}$. This extends the classical theorems of A. A. Andronov et al. on the decomposition in angular sectors of real analytic vector field germs. The Poincaré–Hopf index theory for $\mathfrak {Re}\left (X\right )$ local and global on compact Riemann surfaces, is extended so as to include the case of suitable isolated essential singularities. The inverse problem: determining which cyclic words $\mathcal {W}_{X}$, comprised of hyperbolic, elliptic, parabolic and entire angular sectors, it is possible to obtain from germs of singular analytic vector fields, is also answered in terms of local analytical invariants. We also study the problem of when and how a germ of a singular complex analytic vector field having an essential singularity (not necessarily isolated) can be extended to a suitable compact Riemann surface. Considering the family of entire vector fields $\mathcal {E}(d) =\{X(z)= \lambda \mathrm{e}^{P(z)}\frac {\partial }{\partial z}\}$ on the Riemann sphere, where $P(z)$ is a polynomial of degree $d$ and $\lambda \in \mathbb {C}^*$, we completely characterize the local and global dynamics of this class of vector fields, compute analytic normal forms for $d=1, 2, 3$, and show that for $d\geq 3$ there are an infinite number of topological (phase portrait) classes of $\mathfrak {Re}(X)$, for $X\in \mathcal {E}(d)$. These results are based on the work of R. Nevanlinna, A. Speisser and M. Taniguchi on entire functions $\Psi _{X}$. Finally, on the topological decomposition of real vector fields into canonical regions, we extend the results of L. Markus and H. E. Benzinger to meromorphic on $\mathbb {C}$ vector fields $X$, with an essential singularity at $\infty \in \widehat {\mathbb {C}}$, whose $\Psi _{X}^{-1}$ have $d$ logarithmic branch points over $d$ finite asymptotic values and $d$ logarithmic branch points over $\infty$.

A Distributed Feedback Motion Planning Protocol for Multiple Unicycle Agents of Different Classes

This paper presents a novel feedback method for the motion planning and coordination of multiple agents that belong to two classes, namely class-A and class-B. All agents are modeled via unicycle kinematics. Agents of class-B do not share information with agents of class-A and do not participate in ensuring safety, modeling thus agents with failed sensing/communication systems, agents of higher priority, or moving obstacles with known upper bounded velocity. The method is built upon a family of 2-D analytic vector fields, which under mild assumptions are proved to be safe feedback motion plans with a unique stable singular point. The conditions which ensure collision free and almost global convergence for a single agent and the analytical form of the vector fields are then utilized in the design the proposed distributed, semi-cooperative multi-agent coordination protocol. Semi-cooperative coordination has been defined in prior work as the ad hoc prioritization and conflict resolution among agents of the same class ; more specifically, participation in conflict resolution and collision avoidance for each agent is determined on-the-fly based on whether the agent's motion results in decreasing its distance with respect to its neighbor agents; based on this condition, the agent decides to either ignore its neighbors, or adjust its velocity and avoid the neighbor agent with respect to which the rate of decrease of the pairwise inter agent distance is maximal. The proposed coordination protocol builds upon this logic and addresses the case of multiple agents of distinct classes (class-A and class-B) in conflict. Guarantees on the safety of the multi-agent system and the almost global convergence of the agents to their destinations are proved. The efficacy of the proposed methodology is demonstrated via simulation results in static and dynamic environments.

On families of differential equations on two-torus with all phase-lock areas

We consider two-parametric families of non-autonomous ordinary differential equations on the two-torus with coordinates (x, t) of the type . We study its rotation number as a function of the parameters (A, B). The phase-lock areas are those level sets of the rotation number function that have non-empty interiors. Buchstaber, Karpov and Tertychnyi studied the case when in their joint paper. They observed the quantization effect: for every smooth periodic function f(t) the family of equations may have phase-lock areas only for integer rotation numbers. Another proof of this quantization statement was later obtained in a joint paper by Ilyashenko, Filimonov and Ryzhov. This implies a similar quantization effect for every and rotation numbers that are multiples of . We show that for every other analytic vector field v(x) (i.e. having at least two Fourier harmonics with non-zero non-opposite degrees and nonzero coefficients) there exists an analytic periodic function f(t) such that the corresponding family of equations has phase-lock areas for all the rational values of the rotation number.

Geometry of conformal vector fields

It is well known that the Euclidean space (Rn,〈,〉), the n-sphere Sn(c) of constant curvature c and Euclidean complex space form (Cn,J,〈,〉) are examples of spaces admitting conformal vector fields and therefore conformal vector fields are used in obtaining characterizations of these spaces. In this article, we study the conformal vector fields on a Riemannian manifold and present the existing results as well as some new results on the characterization of these spaces. Taking clue from the analytic vector fields on a complex manifold, we define φ-analytic conformal vector fields on a Riemannian manifold and study their properties as well as obtain characterizations of the Euclidean space (Rn,〈,〉) and the n-sphere Sn(c) of constant curvature c using these vector fields.

Conformal transformation in an almost Hermitian manifold

In this paper, we have studied a conformal transformation between two almost Hermitian manifolds and shown that the associated Nijenhuis tensor is conformally invariant under this transformation. We have also discussed the properties of contravariant almost analytic vector field, covariant almost analytic vector field and some other properties in almost Hermitian manifold under this transformation.

A Geometric Proof of Bourgain’s $$L^2$$ L 2 Estimate of Maximal Operators Along Analytic Vector Fields

Bourgain (A remark on the maximal function associated to an analytic vector field. Analysis at Urbana, Cambridge University Press, Cambridge, 1989) proved that the maximal operator associated with an analytic vector field is bounded on \(L^2\). In the present paper, we give a geometric proof of Bourgain’s result by using the tools developed by Lacey and Li in (Trans Am Math Soc 358(9):4099–4117, 2006) and (Mem Am Math Soc 205 (965):viii+72, 2010).