Computing Geodesic Level Sets on Global (Un)stable Manifolds of Vector Fields

AbstractWe propose a general procedure to construct noncommutative deformations of an algebraic submanifold M of $\mathbb {R}^{n}$ ℝ n , specializing the procedure [G. Fiore, T. Weber, Twisted submanifolds of$\mathbb {R}^{n}$ ℝ n , arXiv:2003.03854] valid for smooth submanifolds. We use the framework of twisted differential geometry of Aschieri et al. (Class. Quantum Grav. 23, 1883–1911, 2006), whereby the commutative pointwise product is replaced by the ⋆-product determined by a Drinfel’d twist. We actually simultaneously construct noncommutative deformations of all the algebraic submanifolds Mc that are level sets of the fa(x), where fa(x) = 0 are the polynomial equations solved by the points of M, employing twists based on the Lie algebra Ξt of vector fields that are tangent to all the Mc. The twisted Cartan calculus is automatically equivariant under twisted Ξt. If we endow $\mathbb {R}^{n}$ ℝ n with a metric, then twisting and projecting to normal or tangent components commute, projecting the Levi-Civita connection to the twisted M is consistent, and in particular a twisted Gauss theorem holds, provided the twist is based on Killing vector fields. Twisted algebraic quadrics can be characterized in terms of generators and ⋆-polynomial relations. We explicitly work out deformations based on abelian or Jordanian twists of all quadrics in $\mathbb {R}^{3}$ ℝ 3 except ellipsoids, in particular twisted cylinders embedded in twisted Euclidean $\mathbb {R}^{3}$ ℝ 3 and twisted hyperboloids embedded in twisted Minkowski $\mathbb {R}^{3}$ ℝ 3 [the latter are twisted (anti-)de Sitter spaces dS2, AdS2].

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A SURVEY OF METHODS FOR COMPUTING (UN)STABLE MANIFOLDS OF VECTOR FIELDS

World Scientific Series on Nonlinear Science Series B - Modeling and Computations in Dynamical Systems ◽

10.1142/9789812774569_0004 ◽

2006 ◽

pp. 67-95 ◽

Cited By ~ 1

Author(s):

B. KRAUSKOPF ◽

H. M. OSINGA ◽

E. J. DOEDEL ◽

M. E. HENDERSON ◽

J. GUCKENHEIMER ◽

...

Keyword(s):

Vector Fields ◽

Stable Manifolds

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A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

Communications in Mathematical Physics ◽

10.1007/s00220-021-04077-z ◽

2021 ◽

Author(s):

Maximilian Engel ◽

Christian Kuehn

Keyword(s):

Dynamical Systems ◽

Limit Cycle ◽

Level Sets ◽

Return Time ◽

Random Dynamical Systems ◽

Expected Return ◽

Stable Manifolds ◽

Kolmogorov Operator ◽

Definition Of ◽

Stochastic Oscillations

AbstractFor an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point of the orbit as the cross-section with fixed return time under the flow. Equivalently, isochrons can be characterized as stable manifolds foliating neighborhoods of the limit cycle or as level sets of an isochron map. In recent years, there has been a lively discussion in the mathematical physics community on how to define isochrons for stochastic oscillations, i.e. limit cycles or heteroclinic cycles exposed to stochastic noise. The main discussion has concerned an approach finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator. We discuss the problem in the framework of random dynamical systems and introduce a new rigorous definition of stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period. This allows us to establish a random version of isochron maps whose level sets coincide with the random stable manifolds. Finally, we discuss links between the random dynamical systems interpretation and the equal expected return time approach via averaged quantities.

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Twisted submanifolds of $${\mathbb {R}}^n$$

Letters in Mathematical Physics ◽

10.1007/s11005-021-01418-w ◽

2021 ◽

Vol 111 (3) ◽

Author(s):

Gaetano Fiore ◽

Thomas Weber

Keyword(s):

Level Sets ◽

Vector Fields ◽

General Procedure ◽

Killing Vector ◽

Tangent Vector ◽

De Sitter ◽

Killing Vector Fields ◽

Gauss Theorem ◽

Levi Civita Connection ◽

Algebraic Manifolds

AbstractWe propose a general procedure to construct noncommutative deformations of an embedded submanifold M of $${\mathbb {R}}^n$$ R n determined by a set of smooth equations $$f^a(x)=0$$ f a ( x ) = 0 . We use the framework of Drinfel’d twist deformation of differential geometry of Aschieri et al. (Class Quantum Gravity 23:1883, 2006); the commutative pointwise product is replaced by a (generally noncommutative) $$\star $$ ⋆ -product determined by a Drinfel’d twist. The twists we employ are based on the Lie algebra $$\Xi _t$$ Ξ t of vector fields that are tangent to all the submanifolds that are level sets of the $$f^a$$ f a (tangent infinitesimal diffeomorphisms); the twisted Cartan calculus is automatically equivariant under twisted $$\Xi _t$$ Ξ t . We can consistently project a connection from the twisted $${\mathbb {R}}^n$$ R n to the twisted M if the twist is based on a suitable Lie subalgebra $${\mathfrak {e}}\subset \Xi _t$$ e ⊂ Ξ t . If we endow $${\mathbb {R}}^n$$ R n with a metric, then twisting and projecting to the normal and tangent vector fields commute, and we can project the Levi–Civita connection consistently to the twisted M, provided the twist is based on the Lie subalgebra $${\mathfrak {k}}\subset {\mathfrak {e}}$$ k ⊂ e of the Killing vector fields of the metric; a twisted Gauss theorem follows, in particular. Twisted algebraic manifolds can be characterized in terms of generators and $$\star $$ ⋆ -polynomial relations. We present in some detail twisted cylinders embedded in twisted Euclidean $${\mathbb {R}}^3$$ R 3 and twisted hyperboloids embedded in twisted Minkowski $${\mathbb {R}}^3$$ R 3 [these are twisted (anti-)de Sitter spaces $$dS_2,AdS_2$$ d S 2 , A d S 2 ].

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PARAMETER DEPENDENCE OF STABLE MANIFOLDS FOR IMPULSIVE EQUATIONS

Analysis and Applications ◽

10.1142/s0219530514500092 ◽

2014 ◽

Vol 12 (02) ◽

pp. 131-160

Author(s):

LUIS BARREIRA ◽

CLAUDIA VALLS

Keyword(s):

Vector Fields ◽

Invariant Manifolds ◽

Linear Part ◽

Small Perturbations ◽

Stable Manifolds ◽

Class C ◽

Nonautonomous Equations ◽

Parameter Dependent ◽

Nonuniform Exponential Dichotomy ◽

Fiber Contraction

We establish the existence of stable manifolds under sufficiently small perturbations of a linear impulsive equation. Our results are optimal, in the sense that for vector fields of class C1 outside the jumping times, the invariant manifolds are also of class C1 outside these times. We also consider the case of C1 parameter-dependent perturbations and we establish the C1 dependence of the stable manifolds on the parameter. The proof uses the fiber contraction principle. We emphasize that we consider the general case of nonautonomous equations for which the linear part has a nonuniform exponential dichotomy.

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