Search results for "vector fields"
showing 7 items of 17 documents
Dynamical analysis of anisotropic inflation
2016
Inflaton coupling to a vector field via the $f^2(\phi)F_{\mu\nu}F^{\mu\nu}$ term is used in several contexts in the literature, such as to generate primordial magnetic fields, to produce statistically anisotropic curvature perturbation, to support anisotropic inflation and to circumvent the $\eta$-problem. Here, I perform dynamical analysis of such a system allowing for most general Bianchi I initial conditions. I also confirm the stability of attractor equilibrium points in phase-space directions that had not been investigated before.
On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields
1986
Consider a fami ly of vector fCelds x~ on the plane. This fami ly depends on a parameter ~ ~ /R A, for some A ~ /~, and is supposed to be 0 ~ in (m,~) 6 /i~ 2X /~A. Suppose that for ~ = O, the vector f i e l d X o has a separatrix loop. This means that X o has an hyperbol ic saddle point s o and that one of the stable separatr ix of 8 o coincides with one of the unstable one. The union of th is curve and s o is the loop ?. A return map is defined on one side of r .
The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations
2010
Abstract In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle V k to a decreasing family of k foliations F i on a manifold M . We have shown that there exists a ( 1 , 1 ) tensor J of V k such that J k ≠ 0 , J k + 1 = 0 and we defined by L J ( V k ) the Lie Algebra of vector fields X on V k such that, for each vector field Y on V k , [ X , J Y ] = J [ X , Y ] . In this note, we study the first Chevalley–Eilenberg Cohomology Group, i.e. the quotient space of derivations of L J ( V k ) by the subspace of inner derivations, denoted by H 1 ( L J ( V k ) ) .
Minimal unit vector fields
2002
We compute the first variation of the functional that assigns each unit vector field the volume of its image in the unit tangent bundle. It is shown that critical points are exactly those vector fields that determine a minimal immersion. We also find a necessary and sufficient condition that a vector field, defined in an open manifold, must fulfill to be minimal, and obtain a simpler equivalent condition when the vector field is Killing. The condition is fulfilled, in particular, by the characteristic vector field of a Sasakian manifold and by Hopf vector fields on spheres.
Harnack estimates for degenerate parabolic equations modeled on the subelliptic $p-$Laplacian
2014
Abstract We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype ∂ t u = − ∑ i = 1 m X i ⁎ ( | X u | p − 2 X i u ) where p ⩾ 2 , X = ( X 1 , … , X m ) is a system of Lipschitz vector fields defined on a smooth manifold M endowed with a Borel measure μ, and X i ⁎ denotes the adjoint of X i with respect to μ. Our estimates are derived assuming that (i) the control distance d generated by X induces the same topology on M ; (ii) a doubling condition for the μ-measure of d-metric balls; and (iii) the validity of a Poincare inequality involving X and μ. Our results extend the recent work in [16] , [36] , to a more general setting including the model cases of (1)…
Poisson-Nijenhuis structures and the Vinogradov bracket
1994
We express the compatibility conditions that a Poisson bivector and a Nijenhuis tensor must fulfil in order to be a Poisson-Nijenhuis structure by means of a graded Lie bracket. This bracket is a generalization of Schouten and Frolicher-Nijenhuis graded Lie brackets defined on multivector fields and on vector valued differential forms respectively.
Multiple Canard Cycles in Generalized Liénard Equations
2001
AbstractThe paper treats multiple limit cycle bifurcations in singular perturbation problems of planar vector fields. The results deal with any number of parameters. Proofs are based on the techniques introduced in “Canard Cycles and Center Manifolds” (F. Dumortier and R. Roussarie, 1996, Mem. Amer. Math. Soc., 121). The presentation is limited to generalized Liénard equations εx+α(x, c)x+β(x, c)=0.