Symmetric Vector Fields Research Articles

SIMPLE VECTOR FIELDS WITH COMPLEX BEHAVIOR

We construct examples of vector fields on a three-sphere, amenable to analytic proof of properties that guarantee the existence of complex behavior. The examples are restrictions of symmetric polynomial vector fields in R4 and possess heteroclinic networks producing switching and nearby suspended horseshoes. The heteroclinic networks in our examples are persistent under symmetry preserving perturbations. We prove that some of the connections in the networks are the transverse intersection of invariant manifolds. The remaining connections are symmetry-induced. The networks lie in an invariant three-sphere and may involve connections exclusively between equilibria or between equilibria and periodic trajectories. The same construction technique may be applied to obtain other examples with similar features.

Embedding and splitting ordinary differential equations in normal form

We introduce an embedding of real or complex n-dimensional space K n as an algebraic variety V which is determined by the action of a linear one-parameter group. Every analytic vector field on K n corresponds to some embedded vector field on V. For a symmetric vector field this embedded vector field splits into a reduced system and a direct sum of non-autonomous linear systems. Examples and applications are mostly concerned with Poincaré–Dulac normal forms. Embeddings provide a natural setting for perturbations of symmetric systems, in particular of systems in normal form up to some degree.

Regular and irregular cycling near a heteroclinic network

Heteroclinic networks are invariant sets containing more than one heteroclinic cycle. Such networks can appear robustly in equivariant vector fields. Previous authors have demonstrated that trajectories near heteroclinic networks can be attracted to one of a number of simultaneously ‘stable’ invariant subsets of the network. None of these invariant sets are asymptotically stable, but satisfy weaker definitions of stability. In this paper we discuss the behaviour of trajectories for one specific symmetric vector field. This vector field contains a robust heteroclinic network, and nearby trajectories display a variety of interesting dynamics. In particular, trajectories are observed to settle into a pattern of excursions around different parts of the network that we call ‘cycling sub-cycles’. Cycling patterns displaying different numbers of loops around the individual component cycles can be stable for the same parameter values, as can combinations of regular and irregular cycling. Analytic results for the regular cycling behaviour agree well with numerical simulations. We show that there exist parameter values where some trajectories display irregular cycling behaviour, in the sense that the numbers of loops around individual sub-cycles form a bounded aperiodic infinite sequence.

Homoclinic Bifurcations in Symmetric Unfoldings of a Singularity with Three–fold Zero Eigenvalue

In this paper we study the singularity at the origin with three–fold zero eigenvalue for symmetric vector fields with nilpotent linear part and 3–jet C∞–equivalent to \( y\frac{\partial } {{\partial x}} + z\frac{\partial } {{\partial y}} + ax^{2} y\frac{\partial } {{\partial z}} \) with a ≠ 0. We first obtain several subfamilies of the symmetric versal unfoldings of this singularity by using the normal form and blow–up methods under some conditions, and derive the local and global bifurcation behavior, then prove analytically the existence of the Sil'nikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of this singularity, by using the generalized Mel'nikov methods of a homoclinic orbit to a hyperbolic or non–hyperbolic equilibrium in a highdimensional space.

A planar model system for the saddle–node Hopf bifurcation with global reinjection

A simultaneous saddle–node and Hopf bifurcation is a basic codimension-two local bifurcation of vector fields with a phase space of dimension at least three. Its local unfoldings are now well known and it has been found as an organizing centre of the dynamics in many vector fields arising in applications. Here we study this very bifurcation but in the presence of a reinjection mechanism that causes trajectories to return to the relevant local neighbourhood in phase space. This happens in applications, for example, in a semiconductor laser subject to optical injection.We propose and study a -symmetric planar vector field with an additional 2π periodicity as a global model for the planar vector field reduction near a saddle–node Hopf (SNH) bifurcation with global reinjection. The phase space of this model vector field is a half-cylinder and the SNH bifurcation reduces to a saddle–node pitchfork (SNP) bifurcation with global reinjection.Two-parameter unfoldings are presented for the different cases of the SNP bifurcation in the presence of global reinjection. New phenomena that we find are periodic, homoclinic and families of heteroclinic orbits that wind around the cylinder. The topological unfoldings are developed hand-in-hand with careful numerical investigations of these global bifurcations. We demonstrate how our results can be applied to a planar model of a semiconductor laser with optical injection. This is followed by a discussion of how to interpret the presented unfoldings in terms of the full three-dimensional dynamics near a SNH bifurcation with global reinjection.

Riemannian manifolds carrying a pair of skew symmetric conformal vector fields

We deal with a Riemannian manifoldM carrying a pair of skew symmetric conformal vector fields (X, Y). The existence of such a pairing is determined by an exterior differential system in involution (in the sense of Cartan). In this case,M is foliated by 3-dimensional totally geodesic submanifolds. Additional geometric properties are proved.

Hamiltonian Systems Near Relative Periodic Orbits

We give explicit differential equations for a symmetric Hamiltonian vector field near a relative periodic orbit. These decompose the dynamics into periodically forced motion in a Poincare section transversal to the relative periodic orbit, which in turn forces motion along the group orbit. The structure of the differential equations inherited from the symplectic structure and symmetry properties of the Hamiltonian system is described, and the effects of time reversing symmetries are included. Our analysis yields new results on the stability and persistence of Hamiltonian relative periodic orbits and provides the foundations for a bifurcation theory. The results are applied to a finite dimensional model for the dynamics of a deformable body in an ideal irrotational fluid.

On space–time carrying a total hyperbolic skew symmetric Killing vector field

The complex vectorial formalism on a general space–time ( M, g) was constructed by Cahen, Debever and Defrise. This formalism is based on the local isomorphism I: L(4)→SO 3( C) , where L(4) is the four-dimensional Lorentz group acting on the tangent spaces T p M and SO 3( C) is the three-dimensional complex rotation group. In this framework, the congruence of Debever plays a distinguished role. Its properties determine the general space–time M, in terms of Petrov’s classification. In the present paper, we assume that any hyperbolic vector field X on M is a skew symmetric Killing vector field having a spatial vector field Y as generative. The existence of such a vector field X is determined by an exterior differential system in involution. It is shown that M is the local Riemannian product M= M h× M s, where M h (resp. M s) is a totally geodesic and totally pseudo-isotropic hyperbolic (resp. spatial) surface (the Gauss map is ametric). Any such M is a space–time of type D in Petrov’s classification. It is proved that the congruence of Debever is of electric type; in particular, it is geodesic and shear 1-free. Other geometric properties on such a general space–time are obtained.

Symmetric and reversible families of vector fields near a partially hyperbolic singularity

We study smooth local models of families of symmetric and reversible vector fields near a partially hyperbolic singularity. Special attention is given to the question of whether the involved changes of variables commute with the symmetry.

On space-time manifolds carrying two exterior concurrent skew symmetric killing vector fields

On space-time manifolds carrying two exterior concurrent skew symmetric killing vector fields

On space-time manifolds carrying a skew symmetric killing vector field

We analyse the structural properties, from a geometrical point of view, of space-time manifolds carrying a skew symmetric Killing vector field.

Codimension 4 singularities of reflectionally symmetric planar vector fields

The paper deals with the topological classification of singularities of vector fields on the plane which are invariant under reflection with respect to a line. As it has been proved in previous papers, such a classification is necessary to determine the different topological types of singularities of vector fields on ${\mathbb R}^3$ whose linear part is invariant under rotations. To get the classification we use normal form theory and the blowing-up method.

Dual number method, rank of a screw system and generation of Lie sub-algebras

The classical method used to estimate the rank of a set χ of m screws, that is to calculate the rank of matrix of dimension m×6, is not often practical especially in singular cases. The aim of this paper is to explain a general method to evaluate the rank of such a set by the determination of the rank of a three-dimensional system with the help of dual numbers and of the representation of screws by elements of the Lie algebra of skew symmetric vector fields. With this method, we propose a new method of classification of screw sets based on the concept of free maximal list of χ. Moreover, we also study the subspaces generated by the set χ using the Lie brackets of ith order. We prove that the subspaces generated by the 4th order bracket is always a Lie subalgebra. Finally, we give a complete demonstration of the mobility conditions of the Bennett mechanisms without using assumptions used in the previous proof by Ogino in [9].

Codimension-three unfoldings of reflectionally symmetric planar vector fields

We consider unfoldings of a codimension-three singularity of reflectionally symmetric planar vector fields that provide a link between the different codimension-two unfoldings known in the literature. This work has been motivated by and can be applied to the study of bifurcations at infinity of planar vector fields with an even rotational symmetry. The codimension-three singularity is determined by its 4-jet, which gives eight classes. We analyse the 4-jet of the unfoldings, of which there are 22 classes. The unfoldings are (conjecturally) versal for 16 classes. The other six classes of unfoldings contain centres, and it is necessary to consider a jet of higher order, which is only indicated here. In our analysis we use that the 4-jet can be reduced to a quadratic vector field.

The topology of symmetric, second-order 3D tensor fields

The authors study the topology of symmetric, second-order tensor fields. The results of the study can be readily extended to include general tensor fields through linear combination of symmetric tensor fields and vector fields. The goal is to represent their complex structure by a simple set of carefully chosen points, lines, and surfaces analogous to approaches in vector field topology. They extract topological skeletons of the eigenvector fields and use them for a compact, comprehensive description of the tensor field. Their approach is based on the premise: analyze, then visualize. The basic constituents of tensor topology are the degenerate points, or points where eigenvalues are equal to each other. Degenerate points play a similar role as critical points in vector fields. In tensor fields they identify two kinds of elementary degenerate points, which they call wedge points and trisector points. They can combine to form more familiar singularities-such as saddles, nodes, centers, or foci. However, these are generally unstable structures in tensor fields. Based on the notions developed for 2D tensor fields, they extend the theory to include 3D degenerate points. Examples are given on the use of tensor field topology for the interpretation of physical systems.

SCALING VIOLATION IN O(N) VECTOR MODELS

We investigate O(N)-symmetric vector field theories in the double scaling limit. Our model describes branched polymeric systems in D dimensions, whose multicritical series interpolates between the Cayley tree and the ordinary random walk. We give explicit forms of residual divergences in the free energy, analogous to those observed in the strings in one dimension.

N-DIMENSIONAL CANONICAL CHUA'S CIRCUIT

In this paper we present an n-dimensional canonical piecewise-linear electrical circuit. It contains 2n two-terminal elements: n linear dynamic elements (capacitors and inductors), n - 1 linear resistors and one nonlinear (piecewise-linear) resistor. This circuit can realize any prescribed eigenvalue pattern, except for a set of measure zero, associated with (i) any n-dimensional two-region continuous piecewise-linear vector fields and (ii) any n-dimensional three-region symmetric (with respect to the origin) piecewise-linear continuous vector fields. We also proved a theorem that specifies the conditions for a vector field, realized with our canonical circuit, to have two or three equilibrium points.

Codimension 2 Symmetric Homoclinic Bifurcations and Application to 1:2 Resonance

In this paper we study a codimension 3 form of the 1:2 resonance. It was first noted by Arnold [3] that the study of bifurcations of symmetric vector fields under a rotation of order q yields information about Hopf bifurcation for a fixed point of a planar diffeomorphism F with eigenvalues . The map Fq can be identified to arbitrarily high order with the flow map of a symmetric vector field having a double-zero eigenvalue ([3], [4], [10], [23]). The resonance of order 2 (also called 1:2 resonance) considered here is the case of a pair of eigenvalues —1 with a Jordan block of order 2. The diffeomorphism then has normal form around the origin given by [4]:

Saddle quantities and applications

In this paper we make the connection between the theoretical study of the generalized homoclinic loop bifurcation (GHB ∗) and the practical computational aspects. For this purpose we first compare the Dulac normal form with the Joyal normal form. These forms were both used to prove the GHB ∗ theorem. But the second one is far more practical from the algorithmic point of view. We then show that the information carried by these normal forms can be computed in a much simpler way, using what we shall call dual Lyapunov constants. The coefficients of a normal form or the dual Lyapunov quantities are particular cases of what we shall call saddle quantities. We calculate the saddle quantities for quadratic systems, and we show that no more than three limit cycles can appear in a homoclinic loop bifurcation. We also study the homoclinic loop bifurcation of order 5, appearing in a 6-parameter family close to a Hamiltonian system. To our knowledge, this is the first time that one can find a complete description of a GHB ∗ of such high order. Finally we calculate the saddle quantities for a symmetric cubic vector field, and we deduce a bound for the number of limit cycles that appear in a GHB ∗ .

Axially symmetric vector fields and their complex potentials

The main result of this paper is the following: if a is a fixed direction in a euclidean space and r is the variable vector, then the polynomials are regular in the sense of R. Delanghe, when the coefficients Pl,j are calculated by the following algorithm These regular polynomials are complex potentials of axial regular vector fields. The paper also refers the usual approach involving partial differential equations to axial vector fields.