Center manifold theory

Carr, J., Applications of Center Manifold Theory . Springer, Berlin (1981). 

Note This problem can also be solved by a method called center manifold theory, as explained in Wiggins (1990) and Guckenheimer and Holmes (1983).)... 

Center manifold theory has become an indispensable tool for the study of ODEs. An equilibrium is called hyperbolic if the linearization of the vector field at that equilibrium does not possess spectrum on the imaginary axis. The local dynamics of ODEs near a hyperbolic equilibrium is determined by that linearization. In particular there exist stable and un-... 

Center manifold theory extends to many infinite-dimensional systems, like certain partial differential equations (PDFs). Center manifold reductions can be obtained locally or globally. For local center manifolds of parabolic PDFs see Vanderbauwhede and looss [78]. Dimension reductions via global center manifolds for spatially inhomogeneous planar media have been achieved by Jangle [27, 33] more details will be presented below. 

A. Vanderbauwhede and G. looss. Center manifold theory in infinite dimensions. In Dynamics Reported, New Series 1 125-163, 1992. 

Spectral analysis of the linearized semiflow along a periodic solution is called Floquet theory. The eigenvalues p of the linearized period map are called Floquet multipliers. A Floquet exponent is a complex / such that exp(/3r) is a Floquet multiplier of the system, where t denotes the minimal period. A periodic solution is hyperbolic if, and only if, it possesses only the trivial Floquet multiplier p = 1 on the unit circle, and this multiplier has algebraic multiplicity one. Otherwise it is called non-hyperbolic. In ODEs hyperbolic periodic solutions possess stable and unstable manifolds, similarly to the case of hyperbolic equilibria. Non-hyperbolic periodic solutions possess center manifolds. 

As we have discussed in Section 2 in classical mechanics the transition state is represented by a lower dimensional invariant subsystem, the center manifold. In the quantum world, due to Heisenberg s uncertainty principle, we cannot localize quantum states entirely on the center manifold, so fhere cannot be any invariant quantum subsystem representing the transition states. Instead we expect a finite lifetime for fhe fransition state. The lifetime of fhe fransifion sfate is determined by the Gamov-Siegert resonances, whose importance in the theory of reacfion rafes has been emphasized in fhe liferafure [61, 62]. 

In the traditional theory of the cooperative JT effect, its significant part is one-center JT problem in a low-symmetry mean field (see the last paragraph of Sect. 2.2). In particular, it includes the eigenvalue problem for the Hamiltonian, similar to (7), operating in an infinite manifold of vibrational one-center states. Compared to this relatively complex step, in the OOA, the mean-field approximation is much simpler. In the OOA, one has to solve just a finite-size matrix (2 x 2 in this case) or, for other JT cases, a somewhat larger matrix but finite anyway. In the theory of the cooperative JT effect, this important advantage of the OOA allows to proceed farther than... 

Transitions arising from the Dq level of the 41 electronic configuration of are intensities between the magnetic dipolar Dq Fi ( 590 nm) and the electric dipolar Dq —> p2 ( 610 nm) transitions. The higher this ratio, the closer the local symmetry around Eu " " is to an inversion center. In the standard theory, the spontaneous emission of a integrated coefficient of the transition between two manifolds / and / is given by... 

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