Partial and complete linearisations at stationary points of infinite-dimensional dynamical systems with foliations and applications

1996 ◽  
Vol 126 (5) ◽  
pp. 1067-1085
Author(s):  
W. M. Rivera
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Linear Operator ◽  
Coupled Systems ◽  
Stationary Points ◽  
Infinite Dimensional ◽  
Infinite Dimensional Dynamical Systems ◽  
Weakly Coupled ◽  
Beam Equations ◽  
Weakly Coupled Systems

In this paper we discuss C1-linearisations of diffeomorphisms and flows on Banach spaces. Strong foliations of the neighbourhood of the fixed point composed of leaves based on successively larger subspaces (similar to those in [14]) are constructed. Generalised gap conditions which involve the width and separation of vertical bands containing the spectrum of a linear operator are imposed to achieve maximal smoothness. The method of proof generalises that of Hartman and of Mora and Solá-Morales. Our theorems apply to weakly coupled systems of damped wave and beam equations.

scholarly journals Periodic perturbations of Hamiltonian systems

2016 ◽  
Vol 5 (4) ◽  
Author(s):  
Alessandro Fonda ◽  
Maurizio Garrione ◽  
Paolo Gidoni
Keyword(s):  
Fixed Point ◽  
Hamiltonian Systems ◽  
Coupled Systems ◽  
Multiplicity Results ◽  
Periodic Perturbations ◽  
Existence And Multiplicity ◽  
Weakly Coupled ◽  
Higher Dimensional ◽  
Dimensional Version ◽  
Weakly Coupled Systems

AbstractWe prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincaré–Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems.

Article

1999 ◽  
Vol 77 (11) ◽  
pp. 1810-1812 ◽  
Author(s):  
Alex D Bain
Keyword(s):  
Spin System ◽  
Coupled System ◽  
Coupled Systems ◽  
Physical Reason ◽  
Strongly Coupled ◽  
System Combination ◽  
First Order ◽  
Weakly Coupled ◽  
Quantum Transitions ◽  
Weakly Coupled Systems

Strongly coupled spin systems provide many curious and interesting effects in NMR spectra, one of which is the presence of unexpected (from a first-order viewpoint) lines. A physical reason is given for the presence of these combination lines. The X part of the spectrum of an ABX spin system is analysed as an example. For an ABX system, it is well known that the AB nuclei give a spectrum consisting of two AB-type spectra, corresponding to the two orientations of the X nucleus. It can also be shown that the X part of the spectrum corresponds to the X nucleus undergoing a transition in the presence of an AB-like spin system. For weakly coupled systems, the four observed lines correspond to the four different orientations of the A and B nuclei. For a strongly coupled system, two additional lines may appear, the combination lines. The resulting six lines correspond to the four spin orientations, plus the two zero-quantum transitions. It is shown that these six lines are such that there is no net excitation of the AB-like spin system associated with the X transitions. There is no AB coherence created directly by a pulse applied to X. AB coherence is created as the system evolves, and this is responsible for many of the curious effects. This is shown to be true for all spin sub-systems, which are weakly coupled to a strongly coupled sub-system.Key words: NMR, strong coupling, second-order spectra, ABX spin system, combination lines, spectral analysis.

Sign in / Sign up

Export Citation Format

Share Document