Normal Forms

Summary

Normal form theory is at the heart of the theory of nonlinear deterministic and random DS. Its aim is to simplify (ultimately linearize) a DS by means of a smooth coordinate transformation. This is fundamental e. g. in bifurcation theory. However, obstructions against transforming away certain terms called “resonances” appear. so that the “simplest possible” form is in general nonlinear.

Here we develop normal form theory for RDS which was initiated by engineers and physicists almost 20 years ago. It turns out that the cohomological equations that need to be solved to eliminate terms are now random affine difference or differential equations, so that we can use our results from Sect. 5.6. Resonances now take the form of integer relations between Lyapunov exponents.

As a preparation we give a brief introduction into deterministic normal form theory in Sect. 8.1. Then normal form theory for discrete time RDS is presented in Sect. 8.2. the main statement being Theorem 8.2.11. The RDE case is dealt with in Sects. 8.3 (Theorem 8.3.7 for the nonresonant case and Theorem 8.3.10 for the resonant case). In Sect. 8.4 we present the random analogue of a very successful procedure for simultaneously obtaining the normal form, eliminating the stable variables from the center equations and determining the center manifold (Theorems 8.4.1 and 8.4.3). We apply this procedure to the noisy Duffing-van der Pol oscillator in Subsect. 8.4.3. The SDE case is treated in Sect. 8.5, where we have to cope with anticipative data (Theorem 8.5.1).