Diffeomorphisms as time one maps

Arango, Jaime; Gómez, Adriana

doi:10.1007/pl00013195

Cited by 5 publications

(4 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Not all diffeomorphisms can be presented as a time one map of an ODE. This is a wellstudied question in mathematics -see [Katok and Hasselblatt, 1995;Arango and Gómez, 2002]. Without going too deep into the theory of dynamical systems, one can easily find a necessary condition.…”

Section: Ode-based Methodsmentioning

confidence: 99%

Normalizing Flows: An Introduction and Review of Current Methods

Kobyzev,

Prince,

Brubaker

2019

Preprint

View full text Add to dashboard Cite

show abstract

Section: Ode-based Methodsmentioning

confidence: 99%

Normalizing Flows: An Introduction and Review of Current Methods

Kobyzev,

Prince,

Brubaker

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…A different approach to the problem of embeddability was presented by Arango and Gómez [4]. They showed that for a mapping F ∈ C 3 0 (U ) satisfying (i) and (3) with r = 2, δ = 0 and dF (0) = S possessing a real logarithm there exists the limit…”

Section: Definitionmentioning

confidence: 99%

“…Iteration groups of fixed point free homeomorphisms on the plane. 4. Embedding of interval homeomorphisms with two fixed points in a regular iteration group.…”

Section: Introductionmentioning

confidence: 99%

Recent results on iteration theory: iteration groups and semigroups in the real case

Zdun

Solarz

2013

Aequat. Math.

View full text Add to dashboard Cite

Abstract. In this survey paper we present some recent results in iteration theory. Mainly, we focus on the problems concerning real iteration groups (flows) and semigroups (semiflows) such as existence, regularity and embeddability. We also discuss some issues associated to the problem of embeddability, i.e. iterative roots and approximate iterability. The topics of this paper are: (1) measurable iteration semigroups; (2) embedding of diffeomorphisms in regular iteration semigroups in the space R n ; (3) iteration groups of fixed point free homeomorphisms on the plane; (4) embedding of interval homeomorphisms with two fixed points in a regular iteration group; (5) commuting functions and embeddability; (6) iterative roots; (7) the structure of iteration groups on an interval; (8) iteration groups of homeomorphisms of the circle; (9) approximately iterable functions; (10) set-valued iteration semigroups; (11) iterations of mean-type mappings; (12) Hayers-Ulam stability of the translation equation. Most of the results presented here were obtained by means of functional equations. We indicate the relations between iteration theory and functional equations.

show abstract

“…This can be seen as a converse of the classical question of whether a given diffeomorphism can be represented as the time-one flow of an autonomous vector field (see e.g. [5,6]).…”

mentioning

confidence: 99%

On expansions for nonlinear systems, error estimates and convergence issues

Beauchard,

Borgne,

Marbach

2020

Preprint

View full text Add to dashboard Cite

Explicit formulas expressing the solution to non-autonomous differential equations are of great importance in many application domains such as control theory or numerical operator splitting. In particular, intrinsic formulas allowing to decouple time-dependent features from geometry-dependent features of the solution have been extensively studied.First, we give a didactic review of classical expansions for formal linear differential equations, including the celebrated Magnus expansion (associated with coordinates of the first kind) and Sussmann's infinite product expansion (associated with coordinates of the second kind). Inspired by quantum mechanics, we introduce a new mixed expansion, designed to isolate the role of a time-invariant drift from the role of a time-varying perturbation.Second, in the context of nonlinear ordinary differential equations driven by regular vector fields, we give rigorous proofs of error estimates between the exact solution and finite approximations of the formal expansions. In particular, we derive new estimates focusing on the role of time-varying perturbations. For scalar-input systems, we derive new estimates involving only a weak Sobolev norm of the input.Third, we investigate the local convergence of these expansions. We recall known positive results for nilpotent dynamics and for linear dynamics. Nevertheless, we also exhibit arbitrarily small analytic vector fields for which the convergence of the Magnus expansion fails, even in very weak senses. We state an open problem concerning the convergence of Sussmann's infinite product expansion.Eventually, we derive approximate direct intrinsic representations for the state and discuss their link with the choice of an appropriate change of coordinates.

show abstract

Diffeomorphisms as time one maps

Cited by 5 publications

References 6 publications

Normalizing Flows: An Introduction and Review of Current Methods

Normalizing Flows: An Introduction and Review of Current Methods

Recent results on iteration theory: iteration groups and semigroups in the real case

On expansions for nonlinear systems, error estimates and convergence issues

Contact Info

Product

Resources

About