Search results for "Vector field"
showing 10 items of 164 documents
Cyclicity of common slow–fast cycles
2011
Abstract We study the limit cycles of planar slow–fast vector fields, appearing near a given slow–fast cycle, formed by an arbitrary sequence of slow parts and fast parts, and where the slow parts can meet the fast parts in a nilpotent contact point of arbitrary order. Using the notion slow divergence integral, we delimit a large subclass of these slow–fast cycles out of which at most one limit cycle can perturb, and a smaller subclass out of which exactly one limit cycle will perturb. Though the focus lies on common slow–fast cycles, i.e. cycles with only attracting or only repelling slow parts, we present results that are valid for more general slow–fast cycles. We also provide examples o…
Isotropic stochastic flow of homeomorphisms on Rd associated with the critical Sobolev exponent
2008
Abstract We consider the critical Sobolev isotropic Brownian flow in R d ( d ≥ 2 ) . On the basis of the work of LeJan and Raimond [Y. LeJan, O. Raimond, Integration of Brownian vector fields, Ann. Probab. 30 (2002) 826–873], we prove that the corresponding flow is a flow of homeomorphisms. As an application, we construct an explicit solution, which is also unique in a certain space, to the stochastic transport equation when the associated Gaussian vector fields are divergence free.
Quantitative ergodicity for some switched dynamical systems
2012
International audience; We provide quantitative bounds for the long time behavior of a class of Piecewise Deterministic Markov Processes with state space Rd × E where E is a finite set. The continuous component evolves according to a smooth vector field that switches at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Under regularity assumptions on the jump rates and stability conditions for the vector fields we provide explicit exponential upper bounds for the convergence to equilibrium in terms of Wasserstein distances. As an example, we obtain convergence results for a stochastic version of the Morris-Lecar model of neurobiology.
Highly sensitive superconducting circuits at ∼700 kHz with tunable quality factors for image-current detection of single trapped antiprotons
2016
We developed highly-sensitive image-current detection systems based on superconducting toroidal coils and ultra-low noise amplifiers for non-destructive measurements of the axial frequencies (550$\sim$800$\,$kHz) of single antiprotons stored in a cryogenic multi-Penning-trap system. The unloaded superconducting tuned circuits show quality factors of up to 500$\,$000, which corresponds to a factor of 10 improvement compared to our previously used solenoidal designs. Connected to ultra-low noise amplifiers and the trap system, signal-to-noise-ratios of 30$\,$dB at quality factors of > 20$\,$000 are achieved. In addition, we have developed a superconducting switch which allows continuous tu…
Principal configurations and umbilicity of submanifolds in $\mathbb R^N$
2004
We consider the principal configurations associated to smooth vector fields $\nu$ normal to a manifold $M$ immersed into a euclidean space and give conditions on the number of principal directions shared by a set of $k$ normal vector fields in order to guaranty the umbilicity of $M$ with respect to some normal field $\nu$. Provided that the umbilic curvature is constant, this will imply that $M$ is hyperspherical. We deduce some results concerning binormal fields and asymptotic directions for manifolds of codimension 2. Moreover, in the case of a surface $M$ in $\mathbb R^N$, we conclude that if $N>4$, it is always possible to find some normal field with respect to which $M$ is umbilic and …
Hydrodynamics with spin in bacterial suspensions
2016
We describe a new kind of self-propelling motion of bacteria based on the cooperative action of rotating flagella on the surface of bacteria. Describing the ensemble of rotating flagella in the framework of the hydrodynamics with spin the reciprocal theorem of Stokesian hydrodynamics is generalized accordingly. The velocity of the self-propulsion is expressed in terms of the characteristics of the vector field of flagella orientation and it is shown that unusually high velocities of \textit{Thiovulum majus} bacteria may be explained by the cooperative action of the rotating flagella. The expressions obtained enable us to estimate the torque created by the rotary motors of the bacterium and …
Orientation of a Surface
2012
We know from Chap. 4 that in order to evaluate the flux of a vector field across a regular surface S, we need to choose a unit normal vector at each point of S in such a way that the resulting vector field is continuous. For instance, if we submerge a permeable sphere into a fluid and we select the field of unit normal outward vectors on the sphere, then the flux of the velocity field of the fluid across the sphere gives the amount of fluid leaving the sphere per unit time. However, if we select the field of unit normal inward vectors on the sphere, then the flux of the velocity field of the fluid across the sphere gives the amount of fluid entering the sphere per unit time (which is the ne…
Limit Periodic Sets
1998
As explained at the end of the previous chapter, the most difficult problem in the study of bifurcations in a family of vector fields on a surface of genus 0 is the control of the periodic orbits. In fact, in generic smooth families the periodic orbits will be isolated for each value of the parameter. For analytic families we have two possibilities for each orbit: it may be isolated or belong to a whole annulus of periodic orbits. In this last case and for the parameter values for which the system has infinitely many periodic orbits, the vector field has a local analytic first integral and the nearby vector fields in the family may be studied by the perturbation theory introduced in Chapter…
Spacelike energy of timelike unit vector fields on a Lorentzian manifold
2004
On a Lorentzian manifold, we define a new functional on the space of unit timelike vector fields given by the L2 norm of the restriction of the covariant derivative of the vector field to its orthogonal complement. This spacelike energy is related with the energy of the vector field as a map on the tangent bundle endowed with the Kaluza–Klein metric, but it is more adapted to the situation. We compute the first and second variation of the functional and we exhibit several examples of critical points on cosmological models as generalized Robertson–Walker spaces and Godel universe, on Einstein and contact manifolds and on Lorentzian Berger’s spheres. For these critical points we have also stu…
Relativistic kinematic approach to the classical ideal gas
2019
he necessary and sufficient conditions for a unit time-like vector field to be the unit velocity of a classical ideal gas are obtained. In a recent paper [Coll, Ferrando and S\'aez, Phys. Rev D {\bf 99} (2019)] we have offered a purely hydrodynamic description of a classical ideal gas. Here we take one more step in reducing the number of variables necessary to characterize these media by showing that a plainly kinematic description can be obtained. We apply the results to obtain test solutions to the hydrodynamic equation that model the evolution in local thermal equilibrium of a classical ideal gas. \end{abstract}