Analytic invariant manifolds for nonautonomous equations

2015 ◽  
Vol 46 (1) ◽  
pp. 29
Author(s):  
Claudià Valls ◽  
Luis Barreira
Keyword(s):  
Invariant Manifolds ◽  
Nonautonomous Equations ◽  
Analytic Invariant

Higher regularity of invariant manifolds for nonautonomous equations

Nonlinearity ◽  
2005 ◽  
Vol 18 (5) ◽  
pp. 2373-2390 ◽  
Author(s):  
Luis Barreira ◽  
Claudia Valls
Keyword(s):  
Invariant Manifolds ◽  
Nonautonomous Equations ◽  
Higher Regularity

scholarly journals Analytic invariant manifolds for sequences of diffeomorphisms

2008 ◽  
Vol 245 (1) ◽  
pp. 80-101
Author(s):  
Luis Barreira ◽  
Claudia Valls
Keyword(s):  
Invariant Manifolds ◽  
Analytic Invariant

PARAMETER DEPENDENCE OF STABLE MANIFOLDS FOR IMPULSIVE EQUATIONS

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.

scholarly journals Invariant manifolds in stratified turbulence

2019 ◽  
Vol 4 (5) ◽  
Author(s):  
N. E. Sujovolsky ◽  
G. B. Mindlin ◽  
P. D. Mininni
Keyword(s):  
Invariant Manifolds ◽  
Stratified Turbulence

scholarly journals ANALYTIC INVARIANT CHARGE IN QCD

2003 ◽  
Vol 18 (30) ◽  
pp. 5475-5519 ◽  
Author(s):  
A. V. NESTERENKO
Keyword(s):  
Perturbative Expansion ◽  
Quantum Field ◽  
Perturbative Approach ◽  
Interaction Processes ◽  
Running Coupling ◽  
Wide Range ◽  
Local Quantum ◽  
Concrete Realization ◽  
Β Function ◽  
Analytic Invariant

This paper gives an overview of recently developed model for the QCD analytic invariant charge. Its underlying idea is to bring the analyticity condition, which follows from the general principles of local Quantum Field Theory, in perturbative approach to renormalization group (RG) method. The concrete realization of the latter consists in explicit imposition of analyticity requirement on the perturbative expansion of β function for the strong running coupling, with subsequent solution of the corresponding RG equation. In turn, this allows one to avoid the known difficulties originated in perturbative approximation of the RG functions. Ultimately, the proposed approach results in qualitatively new properties of the QCD invariant charge. The latter enables one to describe a wide range of the strong interaction processes both of perturbative and intrinsically nonperturbative nature.

scholarly journals Fast and slow invariant manifolds in chemical kinetics

2013 ◽  
Vol 65 (10) ◽  
pp. 1502-1515 ◽  
Author(s):  
V. Bykov ◽  
V. Gol’dshtein
Keyword(s):  
Chemical Kinetics ◽  
Invariant Manifolds

Local Mass Conservation and Velocity Splitting in PV-Based Balanced Models. Part II: Numerical Results

2007 ◽  
Vol 64 (6) ◽  
pp. 1794-1810 ◽  
Author(s):  
Ali R. Mohebalhojeh ◽  
Michael E. McIntyre
Keyword(s):  
Potential Vorticity ◽  
Wave Speed ◽  
Invariant Manifolds ◽  
Mass Conservation ◽  
Flow Speed ◽  
Local Mass ◽  
Potential Vorticity Inversion ◽  
Divergence Equation ◽  
Local Mass Conservation ◽  
Derivatives Of

The effects of enforcing local mass conservation on the accuracy of non-Hamiltonian potential-vorticity- based balanced models (PBMs) are examined numerically for a set of chaotic shallow-water f-plane vortical flows in a doubly periodic square domain. The flows are spawned by an unstable jet and all have domain-maximum Froude and Rossby numbers Fr ∼0.5 and Ro ∼1, far from the usual asymptotic limits Ro → 0, Fr → 0, with Fr defined in the standard way as flow speed over gravity wave speed. The PBMs considered are the plain and hyperbalance PBMs defined in Part I. More precisely, they are the plain-δδ, plain-γγ, and plain-δγ PBMs and the corresponding hyperbalance PBMs, of various orders, where “order” is related to the number of time derivatives of the divergence equation used in defining balance and potential-vorticity inversion. For brevity the corresponding hyperbalance PBMs are called the hyper-δδ, hyper-γγ, and hyper-δγ PBMs, respectively. As proved in Part I, except for the leading-order plain-γγ each plain PBM violates local mass conservation. Each hyperbalance PBM results from enforcing local mass conservation on the corresponding plain PBM. The process of thus deriving a hyperbalance PBM from a plain PBM is referred to for brevity as plain-to-hyper conversion. The question is whether such conversion degrades the accuracy, as conjectured by McIntyre and Norton. Cumulative accuracy is tested by running each PBM alongside a suitably initialized primitive equation (PE) model for up to 30 days, corresponding to many vortex rotations. The accuracy is sensitively measured by the smallness of the ratio ϵ = ||QPBM − QPE||2/||QPE||2, where QPBM and QPE denote the potential vorticity fields of the PBM and the PEs, respectively, and || ||2 is the L2 norm. At 30 days the most accurate PBMs have ϵ ≈ 10−2 with PV fields hardly distinguishable visually from those of the PEs, even down to tiny details. Most accurate is defined by minimizing ϵ over all orders and truncation types δδ, γγ, and δγ. Contrary to McIntyre and Norton’s conjecture, the minimal ϵ values did not differ systematically or significantly between plain and hyperbalance PBMs. The smallness of ϵ suggests that the slow manifolds defined by the balance relations of the most accurate PBMs, both plain and hyperbalance, are astonishingly close to being invariant manifolds of the PEs, at least throughout those parts of phase space for which Ro ≲ 1 and Fr ≲ 0.5. As another way of quantifying the departures from such invariance, that is, of quantifying the fuzziness of the PEs’ slow quasimanifold, initialization experiments starting at days 1, 2, . . . 10 were carried out in which attention was focused on the amplitudes of inertia–gravity waves representing the imbalance arising in 1-day PE runs. With balance defined by the most accurate PBMs, and imbalance by departures therefrom, the results of the initialization experiments suggest a negative correlation between early imbalance and late cumulative error ϵ. In such near-optimal conditions the imbalance seems to be acting like weak background noise producing an effect analogous to so-called stochastic resonance, in that a slight increase in noise level brings PE behavior closer to the balanced behavior defined by the most accurate PBMs when measured cumulatively over 30 days.

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