scholarly journals Extraction of Distinguished Hyperbolic Trajectories for 2D Time‐Dependent Vector Field Topology

2020 ◽  
Vol 39 (3) ◽  
pp. 303-315
Author(s):  
Lutz Hofmann ◽  
Filip Sadlo
Keyword(s):  
Vector Field ◽  
Time Dependent ◽  
Vector Field Topology ◽  
Dependent Vector ◽  
Hyperbolic Trajectories

scholarly journals Local Extraction of 3D Time‐Dependent Vector Field Topology

2021 ◽  
Vol 40 (3) ◽  
pp. 111-122
Author(s):  
Lutz Hofmann ◽  
Filip Sadlo
Keyword(s):  
Vector Field ◽  
Time Dependent ◽  
Vector Field Topology ◽  
Dependent Vector

Moments of certain stochastic integrals occurring in mathematical physics

1992 ◽  
Vol 120 (3-4) ◽  
pp. 267-282 ◽  
Author(s):  
Lieven Smits
Keyword(s):  
Brownian Motion ◽  
Vector Field ◽  
Mathematical Physics ◽  
Time Dependent ◽  
Regularity Conditions ◽  
Stochastic Integrals ◽  
Cauchy Problems ◽  
Dependent Vector ◽  
Set Up

SynopsisWe give an expression for the n-th moment of certain Itô integrals. The integrands considered are nonanticipating functionals of the form s↦a(s, Xs), where a is a measurable time-dependent vector field in space satisfying mild regularity conditions, and Xs is standard translated Brownian motion. The expressions are similar to the Dyson-Phillips terms for magnetic Schrödinger semigroups.We use these expressions to establish properties of the solutions of certain Cauchy problems and we relate our results to the framework of generalised Dyson expansions as set up by Johnson and Lapidus.

A completeness result for time-dependent vector fields and applications

2015 ◽  
Vol 17 (04) ◽  
pp. 1450040 ◽  
Author(s):  
Stefano Biagi ◽  
Andrea Bonfiglioli
Keyword(s):  
Vector Field ◽  
Lie Groups ◽  
Vector Fields ◽  
Time Dependent ◽  
Sufficient Condition ◽  
Invariance Condition ◽  
The Third ◽  
Completeness Result ◽  
Dependent Vector ◽  
Hörmander Vector Fields

We provide a sufficient condition for the completeness of a time-dependent vector field in ℝN, generalizing the well-known left-invariance condition on Lie groups. This result can be applied to the construction of Lie groups associated to suitable families X of Hörmander vector fields, without the need to use the Third Fundamental Theorem of Lie. Further applications are given to the control-theoretic distance related to X, and to the existence of the relevant geodesics.

Existence and Computation of Hyperbolic Trajectories of Aperiodically Time Dependent Vector Fields and Their Approximations

2003 ◽  
Vol 13 (06) ◽  
pp. 1449-1457 ◽  
Author(s):  
Ning Ju ◽  
Des Small ◽  
Stephen Wiggins
Keyword(s):  
Vector Fields ◽  
A Priori ◽  
Sufficient Conditions ◽  
Time Dependent ◽  
Time Interval ◽  
Scale Separation ◽  
Time Scale Separation ◽  
Infinite Time Interval ◽  
Dependent Vector ◽  
Hyperbolic Trajectories

In this paper we give sufficient conditions for the existence of hyperbolic trajectories in aperiodically time dependent vector fields. These conditions do not require the a priori introduction of hyperbolicity into the dynamics of the vector field or assumptions of "time scale separation". The hyperbolic trajectory is obtained as a solution of an integral equation over an infinite time interval. We give an expression for the error obtained when the solution is approximated over a finite time interval. Finally, we show how the method can be numerically implemented in a specific example.

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