Center Manifold theorem

A more sophisticated form of reduction is obtained when the so-called center manifold theorem is invoked. This says essentially that a subspace of lower dimension than the whole state-space gives a true representation of the essential features of the system and one that can be built on to give a yet more accurate picture. We shall not attempt to go into this here to see the method in action, the reader cannot do better than to read C. Chang and... 

We now state the center manifold theorem of the perturbed system ... 

Armed as we are now with the KAM theorem, the Center Manifold theorem, and the Stable Manifold theorem, we can begin to visualize the phase space of reaction dynamics. Returning to our original system (see Uncoupled Reaction Dynamics in Two Degrees of Freedom ), we now realize that the periodic orbit that sews together the half-tori to make up the separatrix is a hyperbolic periodic orbit, and it is not a fixed point of reflection. From our previous visualization of uncoupled phase-space dynamics, we know that the separatrix is completely nontwisted. In the terminology of Poliak and Pechukas, the hyperbolic periodic orbit is a repulsive PODS. ... 

Application of the center manifold theorem [1] to system (1) leads to a statement of very great generality. With its aid it is possible to characterize the topology of the ensemble of all solutions of system (1). This is done in a phase space ft of n+1 dimensions consisting of the union a U z. The key to the understanding of choking is in the identification of the singular points of system (1) whose coordinates in phase space ft are solutions o, z of the simultaneous equations... 

Center Manifolds. The Center Manifold Theorem (see Carr (1981)) states that all branches of stationary and periodic states in a neighborhood of a bifurcation point are embedded in a sub-manifold of the extended phase space X M that is invariant with respect to the flow generated by the ODE (2.1). All trajectories starting on this so-called center manifold remain on it for all times. All trajectories starting from outside of it exponentially converge towards the center manifold. Specifically, static bifurcations are embedded in a two dimensional center manifold, whereas center manifolds for Hopf bifurcations are three dimensional. Figures 2.1 and 2.2 summarize the geometric properties of the flows inside a center manifold in the case of saddle-node and Hopf bifurcations, respectively. 

The a priori unknown nonlinear transformation (2-3), the existence of which is claimed by the center manifold theorem, has to be generated by some multivariate approximation scheme. Artificial Neural Networks are known to be well suited for those kind of problems. Neural networks can be regarded as multivariate approximation functions consisting of simple process units (neurons). There is a weight vector w specifying the transformation behavior of the network. By means of an optimization algorithm this weights have to be specified in order to achieve the desired approximation feature... 

For the problem under consideration, i.e. Eq. (11), the nonlinear function is odd and v = H(u,n) is at least quadratic in u and n. Thus, Eq. (12) restricted to the center manifold will have contribution from the stable equations of the order ( u ), k > 3, and can be neglected in the first approximation. The above equation can be further simplified either by method of averaging or method of normal forms. It may be noted that the averaged and normal form equations can also be obtained directly from Eq. (11) without employing the center manifold theorem as indicated in Sri Namachchivaya and Chow and Mallet-Paret. ... 

Let X be a set of parameter values for which a solution of eqs. (2), referred to as reference state, loses its stability and gives rise to new branches o7 sol uti ons by a bifurcation mechanism. We want to see how the solution of the master equation, eq. (1), behaves under these conditions, and how this behavior depends on small changes of the parameters X around The answer to this question depends on the kind of bifurcation considered, on the nature of the reference state, and on the number of variables involved in the dynamics. The simplest case is, by far, the pitchfork bifurcation occurring as a first transition from a previously stable spatially uniform stationary state. This transition is characterized by a remarkable universality. First, whatever the number of variables present initially, it is always possible to cast the stochastic dynamics in terms of a single, "critical" variable. This is the probabilistic analog of adiabatic elimination or, in more modern terms, of the center manifold theorem [4,8-10]. Second, the stationary probability distribution of the critical variable can be cast in the form (we set 6X = (p-Xg, rstands for the spatial coordinates) ... 

Using center-manifold theorem and normal form techniques [65,66], we have explicitly reduced our reaction-diffusion system (3) to the Hopf normal form (64) of the single-front solution. Technically we have used the normal form reduction method proposed by Coullet and Spiegel [110] (see also [111]). We refer the reader to [62] and [104], where this lengthy calculation has been carried out step by step. For the sake of simplicity, we will skip this technical part here, and we will focus on the theoretical prediction so obtained (on the critical surface n = 0) for the coefficient k in Equation (64). Our purpose is actually to compare the prediction for the value of Re K, with the measurement of the same quantity from direct simulations of the reaction-diffusion system (3). The numerical estimate is easily obtained from the amplitude of oscillation of the single-front solution in the ( , ") direction (Figure 20). If we write 2 = p e, the real part of Equation (64) yields ... 

The key methods in our presentation of local bifurcations are based on the center manifold theorem and on the invariant foliation technique (see Sec. 5.1. of Part I). The assumption that there are no characteristic exponents to the right of the imaginary axis (or no multipliers outside the unit circle) allows us to conduct a smooth reduction of the system to a very convenient standard form. We use this reduction throughout this book both in the study of local bifurcations on the stability boundaries themselves and in the study of global bifurcations on the route over the stability boundaries (Chap. 12).These... 

In Sec. 13.5 we consider the bifurcation of the homoclinic loop of a saddle without any restrictions on the dimensions of its stable and imstable manifolds. We prove a theorem which gives the conditions for the birth of a single periodic orbit from the loop [134], and also formulate (without proof) a theorem on complex dynamics in a neighborhood of a homoclinic loop to a saddle-focus. Here, we show how the non-local center manifold theorem (Chap. 6 of Part I) can be used for simple saddles to reduce our analysis to known results (Theorem 13.6). 

In essence, this is the case for two-dimensional systems, as well as for a number of high-dimensional systems when, for example, they can be reduced to two-dimensional ones by a center manifold theorem (local or global, see Chaps. 5 and 6). 

Hopf had in fact considered the high-dimensional case. However, applying the center manifold theorem, we may restrict our consideration to the two-dimensional case. 

C.3. 24. Prove the following theorem, which is analogous to the center manifold theorem ... 

It must be underlined that the central manifold theorem, extending the linear center manifold into the nonlinear regime, is way less powerful than its stable/ unstable counterpart. There is no limit t —> oo and even no unicity of nonlinear center manifolds. Consequently, it is not well known how this whole beautiful stmcture bifurcates and disappears as E > E. There has been virtually no study of the bifurcation stmcture (see, however, Ref. 55), and the transition from threshold behavior to far-above-threshold behavior is an open question, as far as I am aware. 

We briefly comment on the basic steps in the proof of theorem 1. For homogeneous lighting of intermediate strength a rigidly rotating spiral wave solution was assumed to be given by u . The manifold is close to the unperturbed normally hyperbolic center manifold SE(2)u given by the... 

Under the spectral assumptions of section 3.2.3, theorem 1, it is now possible to reduce the perturbed dynamics to a three-dimensional center manifold which is modeled over the group SE 2) itself. In Palais coordinates... 

As was justified in section 3.2.4, the angle a denotes the phase and. 2 the position of the spiral tip. The Palais section coordinate n U is absent here, because the critical spectrum is now three-dimensional, only, and is accounted for by the three-dimensional group SE 2) itself. Therefore the center manifold M. is a graph over the group coordinates e ° ,z) G SE 2). A rigorous derivation of the reduced equation (3.22) has indeed been achieved in [25, 33], under the assumption that the unperturbed spiral wave n (-) is spectrally stabie with the exception of a triple critical eigenvalue due to symmetry see theorem 1. Note that the nonlinearities -y a,z,s) and h a,z,s) obey the lattice symmetry relic of full Euclidean symmetry, namely... 

Theorem 2. [16, center manifold] Under the above assumptions on the normal velocity U = U k) as a function of curvature k, all solutions U = U s) of (3.45), (3.46) with bounded normal velocity U and nowhere vanishing curvature, for all values of the rotation frequency u) > 0, are contained in the bifurcation diagram of Figure 3.6. 

The results of theorem 2 are based on a center manifold analysis after regularization of the degenerate pendulum system (3.45). Rewriting (3.45) as a system for U and V =Us, and multiplying the right hand side by the Euler multiplier k = r(U) to eliminate the vanishing denominator T(c) = 0, we obtain... 

It turns out that all solutions of theorem 2 must lie in this center manifold, under just the boundedness and sign conditions imposed there. As an easy consequence all such solutions are asymptotically Archimedean in the far field s —> +oo. Indeed this follows directly from the first order ODE... 

The strongly stable manifold is one of the leaves of a -smooth foliation which is transverse to the center manifold. As we have shown in Chap. 5 the following reduction theorem holds ... 

Theorem 9.1. If the equilibrium state is Lyapunov stable in the center manifold then the equilibrium state of the original system (9.1.1) is Lyapunov stable as well Moreover if the equilibrium state is asymptotically stable in the center manifold, then the equilibrium state of the original system is also asymptotically stable. 

Theorem 9.3. If all Lyapunov values are equal to zero, then the associated analytic system has an analytic invariant (center) manifold which is filled with closed trajectories around the origin, as shown in Fig. 9.3.3. On the center manifold the system has a holomorphic integral of the type... 

The tool kit used for studying bifurcational problems consists of three pieces the theorem on center manifold, the reduction theorem, and the method of normal forms. 

The above theorem is related to the map on the center manifold. Reconstructing the behavior of trajectories of the original map (11.6.2) is relatively simple. Here, if L < 0, then the fixed point is stable when /i < 0. When /i > 0 it becomes a saddle-focus with an m-dimensional stable manifold (defined by T = 0) and with a two-dimensional unstable manifold which consists of a part of the plane y = 0 bounded by the stable invariant curve C,... 

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