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Unformatted text preview: Normal Forms Theory CDS140B Lecturer: Wang Sang Koon Winter, 2004 1 Normal Form Theory Introduction. To find a coordinate system where the dynamical system take the “simplest” form. • The method is local in the sense that the coordinate transforms are generated near a know solution, such as a fixed point. • The coordinate transformation will be nonlinear, but these transformation are found by solv- ing a sequence of linear problem. • The structure of the normal form is determined entirely by the nature of the linear part of the problem. Preliminary Preparation. Consider ˙ w = G ( w ) where w ∈ R n , G is C r , and the system has a fixed point at w = w . Then it can be written as (*) ˙ x = Jx + F ( x ) = Jx + F 2 ( x ) + F 3 ( x ) + · · · + F r- 1 ( x ) + O ( | x | r ) where F i ( x ) represent the order i terms in the Taylor expansion of F ( x ). 1 1.1 Simplification of the Second Order Terms Introduce the coordinate transformation x = y + h 2 ( y ) where h 2 ( y ) is second order in...
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- Fall '10
- Geometry, Normal Form, Homological Equation