Set Of Vector Fields Research Articles

Shape and stability of rotation sets for incompressible flows on the torus

AbstractIn this paper we introduce the concept of rotation cones, determined by rotation sets, and use these to describe the stability of rotation sets of incompressible and fixed-point free continuous flows on the torus $$\mathbb{T}^{d}$$ T d , d ⩾ 2. The previous concept is equivalent and extends the stability of rotation numbers (and rotation vectors) in the special case of fixed-point free continuous flows on $$\mathbb{T}^{2}$$ T 2 . We prove that incompressible Lipschitz vector fields with stable rotation sets have convex rotation cones with non-empty interior, and that all extremal edges are collinear with vectors in ℤd. If, in addition, the rotation cones are proper, then these are polyhedral with finitely many edges. Finally we prove that the set of vector fields with stable rotation sets is C0-open and dense among those having proper rotation cones.

Weakly shadowable vector fields on non-oriented surfaces

We say that a vector field has the weakly shadowing property if for any there exists such that for every d-pseudo orbit there exists an exact orbit whose -neighbourhood containing the pseudo orbit. It is proved in Li Ming and Zhongjie Liu [Weak shadowing property for flows on oriented surfaces, Proc. Amer. Math. Soc. 145(6) (2017), pp. 2591–2605.] that vector fields in the C 1-interior of the set of vector fields on an oriented smooth closed surface having the weakly shadowing property are structurally stable. In this paper, we show that the above conclusion does not hold for non-oriented surfaces. Precisely, we construct a non--stable vector field on the Klein bottle which has the weakly shadowing property robustly.

Towards a classification of networks with asymmetric inputs

Coupled cell systems associated with a coupled cell network are determined by (smooth) vector fields that are consistent with the network structure. Here, we follow the formalisms of Stewart et al (2003 SIAM J. Appl. Dyn. Syst. 2 609–646), Golubitsky et al (2005 SIAM J. Appl. Dyn. Syst. 4 78–100) and Field (2004 Dyn. Syst. 19 217–243). It is known that two non-isomorphic n-cell coupled networks can determine the same sets of vector fields—these networks are said to be ordinary differential equation (ODE)-equivalent. The set of all n-cell coupled networks is so partitioned into classes of ODE-equivalent networks. With no further restrictions, the number of ODE-classes is not finite and each class has an infinite number of networks. Inside each ODE-class we can find a finite subclass of networks that minimize the number of edges in the class, called minimal networks. In this paper, we consider coupled cell networks with asymmetric inputs. That is, if k is the number of distinct edges types, these networks have the property that every cell receives k inputs, one of each type. Fixing the number n of cells, we prove that: the number of ODE-classes is finite; restricting to a maximum of n(n − 1) inputs, we can cover all the ODE-classes; all minimal n-cell networks with n(n − 1) asymmetric inputs are ODE-equivalent. We also give a simple criterion to test if a network is minimal and we conjecture lower estimates for the number of distinct ODE-classes of n-cell networks with any number k of asymmetric inputs. Moreover, we present a full list of representatives of the ODE-classes of networks with three cells and two asymmetric inputs.

A family of sharp inequalities on real spheres

We prove a family of sharp multilinear integral inequalities on real spheres involving functions that possess some symmetries that can be described by annihilation by certain sets of vector fields. The Lebesgue exponents involved are seen to be related to the combinatorics of such sets of vector fields. Moreover, we derive some Euclidean Brascamp–Lieb inequalities localized to a ball of radius R, with a blow-up factor of type , where the exponent is related to the aforementioned Lebesgue exponents, and prove that in some cases δ is optimal.

Supersymmetry constraints on U-duality invariant deformations of mathcal{N} \u2265 5 supergravity

Candidate counterterms break E7 type U-duality symmetry of mathcal{N} ≥ 5 supergravity theories in four dimensions [1, 2]. A proposal was made in [3] to restore it, starting with a double set of vector fields and argued that a supersymmetric extension of their proposal should exist. We show that the extra vectors, needed for the deformation, can not be auxiliary fields in an eventual off-shell formulation of mathcal{N} ≥ 5 supergravity, assuming that such a formulation exists. Furthermore we show that these extra vector fields can not be dynamical either since that changes the unitary supermultiplets underlying these theories and requires one to go beyond the standard framework of extended simple supergravities. To show this we list all relevant unitary conformal supermultiplets of SU(2, 2| mathcal{N} + n). We find that doubling of vectors consistent with linearized supersymmetry requires to change the number of scalars, violating the coset structure of the theory, and also to add a finite number of higher spin fields, which do not admit consistent couplings to theories with spins ≤ 2. Thus, the proposed duality restoring deformation along the lines of [3] can not be implemented within the standard framework of extended supergravity theories. We argue therefore that, in the absence of anomalies, E7 type duality together with supersymmetry, might protect mathcal{N} ≥ 5 supergravity from UV divergences, in particular, mathcal{N} = 5 supergravity at 4 loops in d=4.

Linearization of control systems: A Lie series approach

We address the problem of static state linearization of multi-input nonlinear control systems via coordinate transformation. Necessary and sufficient geometric conditions, in terms of certain set of vector fields associated with the system, were obtained in the early eighties stating the fact that such set of vector fields should be commutative and of constant rank. The state linearization problem, i.e., the finding of linearizing coordinates, was thus reduced to solving a set of partial differential equations. The objective of this paper is to provide an algorithm allowing to compute explicitly the linearizing state coordinates. The algorithm is performed using a maximum of n−1 steps (n being the dimension of the system) and is made possible by extending the explicit solvability of the Flow-Box Theorem to a commutative set of vector fields. Examples are provided to illustrate the results. An extension of the method to dynamic feedback linearization is also outlined.

Regularity in the obstacle problem for parabolic non-divergence operators of Hörmander type

In this paper we continue the study initiated in [15] concerning the obstacle problem for a class of parabolic non-divergence operators structured on a set of vector fields X={X1,…,Xq} in Rn with C∞-coefficients satisfying Hörmanderʼs finite rank condition, i.e., the rank of Lie[X1,…,Xq] equals n at every point in Rn. In [15] we proved, under appropriate assumptions on the operator and the obstacle, the existence and uniqueness of strong solutions to a general obstacle problem. The main result of this paper is that we establish further regularity, in the interior as well as at the initial state, of strong solutions. Compared to [15] we in this paper assume, in addition, that there exists a homogeneous Lie group G=(Rn,∘,δλ) such that X1,…,Xq are left translation invariant on G and such that X1,…,Xq are δλ-homogeneous of degree one.

Sculpting dynamical systems for models of neural computation and memory

Dynamical systems are now a mainstay of computational neuroscience. Persistent activity in biological neural networks has been posited to result from dynamical attractors in neural state-space [1], and computation with attractors underlies a variety of models for information processing and memory function in the brain [2-4]. Nevertheless, there exists no general, substrate-free method for the construction of these dynamical systems [2], substrate-free meaning not tied to a particular neural instantiation. We introduce such a method in this work, with a focus on applications to neural computation and memory modeling. We call this method state-space sculpting. The systems it constructs, or sculpts, are realized as ordinary differential equations. Sculpting is therefore substrate free; it is general in the sense that sculpted systems may contain any number of attractors embedded at any locations in the state-space and the basins of those attractors may be shaped with any smooth geometry. We conjecture that the sculpting process is analogous to learning by synaptic modification in some biological and artificial neural networks. The method works as follows. First, a set of attractor locations and types (fixed point, periodic, strange) is defined. Then a set of corresponding vector fields is identified: Each of these has one of the defined attractor types at one of the defined locations in its state-space. Next, the basins of these attractors are delineated by defining smooth boundaries for the corresponding vector fields. Finally, the set of vector fields and boundaries is unified into a single smooth dynamical equation using regularization [5]. We have sculpted various 2-dimensional systems by this method, and their dynamics were confirmed in simulation to exhibit the defined attractors and basins. These systems contained arbitrary combinations of fixed point and periodic attractors, as well as repelling regions, with arbitrary basin geometries. Systems of saddle equilibria were also sculpted to demonstrate the versatility of the method. These can be made to exhibit heteroclinic channels and cycles, which are key features of the Winnerless Competition Principle of neural computation [4]. As a first test of our conjecture on the analogy between sculpting and synaptic modification, the well known Hopfield network [6] was analyzed and characterized in sculpting terms. This analysis demonstrated most notably that the network's basins of attraction are delineated by combinations of codimension-1 hyperplanes containing saddle equilibria. The analogue of state-space sculpting in the Hopfield network is therefore limited, since attractor basins cannot be shaped with any smooth geometry. Moving forward, it is necessary to study more complicated networks to show that they can be cast in the sculpting framework, as well as to extend the sculpting method to higher-dimensional state-spaces and manifolds.

Dynamical systems and σ-symmetries

A deformation of the standard prolongation operation, defined on sets of vector fields in involution rather than on single ones, was recently introduced and christened ‘σ-prolongation’; correspondingly, one has ‘σ-symmetries’ of differential equations. These can be used to reduce the equations under study, but the general reduction procedure under σ-symmetries fails for equations of order 1. In this paper, we discuss how σ-symmetries can be used to reduce dynamical systems, i.e. sets of first-order ODEs in the form .

Congested Traffic Equilibria and Degenerate Anisotropic PDEs

Congested traffic problems on very dense networks lead, at the limit, to minimization problems posed on measures on curves as shown in Baillon and Carlier (Netw. Heterogenous Media 7: 219--241, 2012). Here, we go one step further by showing that these problems can be reformulated in terms of the minimization of an integral functional over a set of vector fields with prescribed divergence. We prove a Sobolev regularity result for their minimizers despite the fact that the Euler-Lagrange equation of the dual is highly degenerate and anisotropic. This somehow extends the analysis of Brasco et al. (J. Math. Pures Appl. 93: 652--671, 2010) to the anisotropic case.

A generalization of λ-symmetry reduction for systems of ODEs: σ-symmetries

We consider a deformation of the prolongation operation, defined on sets of vector fields and involving a mutual interaction in the definition of prolonged ones. This maintains the ‘invariants by differentiation’ property, and can hence be used to reduce ODEs satisfying suitable invariance conditions in a fully algorithmic way, similarly to what happens for standard prolongations and symmetries.

A robustly chain transitive attractor with singularities of different indices

AbstractGiven a 4-manifold, we build a non-empty ${C}^{1} $-open set of vector fields having a (chain transitive) attractor containing singularities of different indices. Then, we begin the study of the hyperbolic properties of such a robust singular attractor.

Combinatorial differential geometry and ideal Bianchi–Ricci identities II – the torsion case

This paper is a continuation of [2], dealing with a general, not-necessarily torsion-free, connection. It characterizes all possible systems of generators for vector-field valued operators that depend naturally on a set of vector fields and a linear connection, describes the size of the space of such operators and proves the existence of an ‘ideal’ basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi–Ricci identities without corrections.

On the transversality of functions at the core of the transverse function approach to control

The transverse function approach to control, introduced by Morin and Samson in the early 2000s, is based on functions that are transverse to a set of vector fields in a sense formally similar to, although strictly speaking different from, the classical notion of transversality in differential topology. In this paper, a precise link is established between transversality and the functions used in the transverse function approach. It is first shown that a smooth function \({f : M \longrightarrow Q}\) is transverse to a set of vector fields which locally span a distribution D on Q if, and only if, its tangent mapping T f is transverse to D, where D is regarded as a submanifold of the tangent bundle T Q. It is further shown that each of these two conditions is equivalent to transversality of T f to D along the zero section of T M. These results are then used to rigorously state and prove that if M is compact and D is a distribution on Q, then the set of mappings of M into Q that are transverse to D is open in the strong (or “Whitney C∞-”) topology on C∞(M, Q).

Combinatorial differential geometry and ideal Bianchi–Ricci identities

Abstract We apply the graph complex approach of [Markl, Differential Geom. Appl. 27: 257–278, 2009] to vector fields depending naturally on a set of vector fields and a linear symmetric connection. We characterize all possible systems of generators for such vector-field valued operators including the classical ones given by normal tensors and covariant derivatives. We also describe the size of the space of such operators and prove the existence of an ‘ideal’ basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi–Ricci identities without the correction terms. The proofs given in this paper combine the classical methods of normal coordinates with the graph complex method.

Vector fields with stable shadowable property on closed surfaces

We prove that the set of vector fields satisfying the C1 stable shadowable property on closed surfaces is characterized as the set of Morse-Smale vector fields. Hence, the vector fields satisfying shadowing property on closed surfaces are C1 dense.

Topological equivalence of vector fields after blow-up

We give a theorem of characterization for the property of being topologically equivalent after blow-up in the set of germs of three-dimensional hyperbolic vector fields. Given ξ, ξ ′ two such a germs and Φ a finite sequence of blow-ups of the ambient space, we find, under non-resonance conditions associated to Φ, a criterion that permits to determine if there is a topological equivalence between ξ and ξ ′ that lifts to Φ. We deduce that there are only finitely many possible classes of Φ-topological equivalence in the considered set of vector fields.

Sets of vector fields with various shadowing properties of pseudotrajectories

Sets of vector fields with various shadowing properties of pseudotrajectories

On Achieving Periodic Joint Motion for Redundant Robots

The consequence of the loss of involutivity of a specific set of vector fields on the periodicity of the joint motion is examined for redundant robots. An output task, defined as a one dimensional periodic closed curve embedded in a two dimensional working surface, is realized through the computation of joint velocities in the configuration space. Depending on the manner in which the joint velocity is computed from the end-effector velocity, the resulting joint motion can become unpredictable and of a chaotical nature, even though the end-effector movement is periodic and predictable. The paper proposes an improvement over classical pseudo-inverse computation of the joint motion by first suitably selecting two involutive vector fields (used as a basis for parameterization) in the tangent bundle of the output manifold. It also presents a sufficient condition for the periodicity of all the joint configuration based on the involutivity of two vector fields in the tangent bundle of the joint space. The results are illustrated on a five-link rotary redundant robot (5R robot).

On the Structure of Nilpotent Normal Form Modules

The set of vector fields that are in normal form with respect to a given linear part has the structure of a module and is best described by giving the Stanley decomposition of that module. An algorithm is presented that produces a Stanley decomposition for the module of equivariants of the flow of a nilpotent linear vector field, given a Stanley decomposition for the ring of invariants. This reduces the study of nilpotent normal forms to classical invariant theory plus this one additional algorithm. Both the inner product normal form (Elphick) and the sl(2) normal form (Cushman–Sanders) are covered within a single theory, and simplified (non-equivariant) versions of each normal form are presented.