Properties of Center Manifolds

Abstract

Abstract.The center manifold has a number of puzzling properties associated with the basic questions of existence, uniqueness, differentiability and analyticity which may cloud its profitable application in e.g. bifurcation theory. This paper aims to deal with some of these subtle properties.Regarding existence and uniqueness, it is shown that the cut-off function appearing in the usual existence proof is responsible for the selection of a single center manifold, thereby hiding the inherent nonuniqueness. Conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection. This intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium. In the case of nonuniqueness the difference between any two center manifolds will be less than c, exp(c2A_1) with c, and c2 constants, which explains why the formal Taylor expansions of different center manifolds are the same, while the expansions do not converge.The differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost.Center manifolds are usually not analytic; however, an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant. Using this manifold, a new and very simple proof of the Lyapunov subcenter theorem is given.

Global Lipschitz invariant center manifolds for ODEs with generalized trichotomies

“…then equation (7) admits a global Lipschitz invariant center manifold, i.e., there is N ∈ ]0, 1[ and a unique ϕ ∈ A N such that Proof. It is obvious that for this type of bounds (24) is equivalent to (28). Moreover, from (26) and (27) we have…”

“…A good expository paper for the case of autonomous differential equations in finite dimension was written by Vanderbauwhede [29] (see also Vanderbauwhede and Gils [31]) and for the case of autonomous differential equations in infinite dimension we recommend Vanderbauwhede and Iooss [30]. For more details in the finite dimensional case see Chow, Liu and Yi [11,10] and for the infinite dimensional case see Sijbrand [28], Mielke [23], Chow and Lu [12,13] and Chicone and Latushkin [9].…”

Neural network-based systems for handprint OCR applications

“…We effectively reduce the dimension of the problem using the center manifold approach [47]. This approach is similar to the Lyapunov-Schmidt technique [48] which reduces the dimension of the system from to the dimension of the center manifold, which in numerical calculations is equal to the number of calculable nonzero eigenvalues of the Jacobian.…”

“…Therefore, the Jacobian of the inner iteration is singular within computational error [43]. When the Jacobian has large numbers of near-zero eigenvalues, the optimization is dominated by center manifold dynamics [44], [47]. Changing the activation function to a sinusoidal function creates a significant change in the dynamics of the training since the activation function has significant derivatives for all possible input signals.…”

The invariant manifold of beam deformations

“…That this is possible is a reflection of the fact that the precise length of the rod is immaterial to much of the analysis -so long as the rod is "long enough." The surface boundary conditions (19)(20)(21) also are transformed to become, on r --R,…”