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.

Patchy solution of a Francis–Byrnes–Isidori partial differential equation

“…We will therefore limit our considerations to solving the center manifold equation for (12). It will be assumed that the w-dynamics have w = 0 as a Lyapunov stable and non-attractive equilibrium.…”

“…It will be assumed that the w-dynamics have w = 0 as a Lyapunov stable and non-attractive equilibrium. By Theorem 2.1, there exists a unique analytic mapping φ(w 1 , w 2 ), defined locally about w = 0, solving the center manifold PDE associated to (12).…”

“…In this example we take a linear dynamical system of the form (12), whose center manifold is easily computed, perform a nonlinear change of coordinates and arrive at a nonlinear system on which we apply our method. The true solution for the nonlinear system is then readily available and we can compare the approximations produced by our method with the true soluiton.…”

“…Roughly speaking, if the zero dynamics of (1) has a hyperbolic equilibrium at the origin then a solution to the FBI equations exists by the center manifold theorem [2]. It is known, however, that center manifolds suffer from a number of subtle properties associated with uniqueness and differentiability [12]. Despite these difficulties, a C ∞ dynamical system possess a C k center manifold for each k ≥ 1, and moreover, it is possible to obtain approximate solutions of arbitrarily high-order via Taylor series [2].…”

“…Although this simplification is significant, solutions to center manifolds suffer from subtleties associated with uniqueness and differentiability [12]. For example, it is known that an analytic dynamical system does not generally posses an analytic center manifold, thereby forcing one to seek a center manifold solution that is only C k (k = 2, 3, .…”