1. Linearization

2. Fixed Point  A map f is defined on a metric space X.  Points are mapped to other points in the space.  A fixed point for the map is mapped into itself. X p x1x1 f x2x2 f

3. Attracting Fixed Point  Let f : R 1  R 1 Fixed point f(p) = pFixed point f(p) = p Consider x = p +  xConsider x = p +  x Expand f n (x)Expand f n (x)  The point is stable if the series converges. |f ’(p)| < 1|f ’(p)| < 1 p is an attracting fixed pointp is an attracting fixed point Repelling fixed points have |f ’(p)| > 1Repelling fixed points have |f ’(p)| > 1

4. Hyperbolic Fixed Point  The Jacobian is a partial differential matrix n  m matrixn  m matrix  A hyperbolic matrix has no unit eigenvalues. Unit is in the complex planeUnit is in the complex plane  A hyperbolic fixed point p : Smooth map f on R nSmooth map f on R n Df(p) is hyperbolicDf(p) is hyperbolic  A saddle point p of f ( R 2 ): Real eigenvalues  0 < |  | < 1 <| |

5. Topologically Conjugate  Let F, G be maps F is a map on space X G is a map on space Y  If there exists a homeomorphism h h : Y  X G = h -1 Fh  Then F, G are topologically conjugate. h X Y G h -1 F

6. How To Linearize  Consider a 2-D system. Let p be saddle pointLet p be saddle point  Translate the origin to the point p.  Calculate the matrix L  Define a topologically conjugate map g

7. Eigenvalues  The origin is a fixed point.  For small , there are two approximate solutions.  The generalized variables had mass and length folded into them.

8. Eigenvalue Results  The origin is a fixed point.  For small x, there is an approximate expansion of the conjugate map. next