Stochastic Center Manifolds
Alex Hutt suggests adding a reference to Boxler's "stochastic center manifolds" paper. This seems a good idea to me, however, I was wondering if there were a more modern reference that discusses these ideas and puts them in context. Perhaps the paper
Arnold, L. (1995). Random dynamical systems. Dynamical Systems. L. Arnold, C. Jones, K. Mischaikow and G. Raugel. Berlin, Springer-Verlag. 1609: 1-43.
It is a good idea to cite this volume, but it also talks about other effects in random dynamical systems, not only about stochastic center manifolds. Hence I suggest to cite Boxler and Arnold. In addition, there is the paper by Xu and Roberts (1996)(On the low-dimensional modelling of Stratonovich stochastic differential equations, Physica A 225, 62-80), which gives a very nice introduction to the theorem and shows how to calculate the center manifold.
-- Axel Hutt
I noticed a possible discrepancy between two related contributions, namely Center manifold and Synergetics:
In Center manifold, part Centre manifolds: quote: In more physical terms, the dynamics <http://www.scholarpedia.org/article/Dynamical_Systems> of y follows the dynamics of x and one may say that x enslaves the variable y. This interpretation has been called slaving principle. end quote
If I am not misled, what I learned from synergetics is the reverse: the fast variables are eliminated (in a local approximation) close to the bifurcation, because slow time scales (in the example y) enslave fast ones (here x).
Which may be found in the contribution about Synergetics, part Method of solution: quote The exponentially increasing or neutral solutions characterize the "unstable modes". Their amplitudes or phases become, in the fully nonlinear treatment, which takes also fluctuations into account, the order parameters. The equations of motions are then transformed to these new variables, amplitudes and phases defining order parameters, and the still stable modes. Then, taking into account the fluctuations, the damped (stable) modes are eliminated (slaving principle). end quote
However, as far as the relation between this theorem, hence the existence of the 3 invariant manifolds, stable, unstable, neutral, and reduction to the neutral one, and the slaving principle in physics by Haken is concerned, some areas of shadow remain: the centre theorem states the conditions when one can project the dynamics onto the _neutral _manifold, the slaving states one can eliminate fast variables and keep the _unstable ones _(slowing close to the bifurcation) by appropriate scaling. Is this fully equivalent? Haken (1996), in the paper Slaving principle revisited, Physica D 97, 95-103, aims at demonstrating it. In the present contribution in Scholarpedia, Synergetics, both unstable and neutral modes are included to form the order parameters: "The exponentially increasing or neutral solutions characterize the "unstable modes", not only the neutral modes.
It looks like the slaving principle, a physical principle, contains the centre manifold technique, but also other techniques. Convergence between so called mathematical techniques and physical phenomena are always a subject of deep surprise.
My best regards
Slaving of synergetics versus center manifolds
I suggest that the slaving principle intersects with center manifolds. Look at three types of possibilities:
- for a system with slow dynamics and fast decaying dynamics then slaving and center manifolds are equivalent (although center manifolds have additional theory about emergence/relevance/asymptotic completeness)
- for a system with slow dynamics and fast oscillating dynamics, slaving and the SLOW manifold are equivalent, but the center manifold is all the slow and oscillating dynamics (and in this case normal forms show that the slaving principle generates models that are not necessarily relevant to the original dynamics)
- for a system with oscillating dynamics and decaying dynamics, slaving does not directly apply (although one could develop a variation) whereas center manifolds does model the long lasting oscillations.
Tony Roberts, University of Adelaide
Minor Equation Corrections
- I noticed that in discussion after Equation (3), there is a remark about \(y(0) = h(y_0)\). I believe it should be corrected to \(y(0) = y_0 = h(x_0)\).
- In equation (7) \(u' = Au + f(x,h(u))\) should be corrected to \(u' = Au + f(u,h(u))\).