Session overview Attraction and repulsion April 11, 2008 CSSE/MA - PowerPoint PPT Presentation

Session overview � Attraction and repulsion April 11, 2008 CSSE/MA 325 Lecture #18 1

Characterization of Linear Systems � Last question on quiz. Answers? April 11, 2008 CSSE/MA 325 Lecture #18 2

Attracting fixed points � x 0 is an attracting fixed point if there exists an interval I (containing x 0 ) ∋ F n (x) ∈ I ∀ n > 0 and x ∈ I, and F n (x) → x 0 as n → ∞ April 11, 2008 CSSE/MA 325 Lecture #18 3

Repelling fixed points � x 0 is a repelling fixed point if ∀ small intervals I (containing x 0 ) there exists x ∈ I and N > 0 ∋ F n (x) ∉ I ∀ n > N April 11, 2008 CSSE/MA 325 Lecture #18 4

Neutral fixed points � x 0 is a neutral fixed point if there exists an interval I (containing x 0 ) ∋ F n (x) ∈ I ∀ n > 0 and (x ∈ I but F n (x) does not approach x 0 ) as n → ∞ April 11, 2008 CSSE/MA 325 Lecture #18 5

Example � For the linear map, F(x) = ax + b: � x 0 is attracting when |a| < 1 � x 0 is repelling when |a| > 1 � x 0 is neutral when |a| = 1 April 11, 2008 CSSE/MA 325 Lecture #18 6

Linearization � Let ε >0 be a small number � Let F(x) be differentiable on the interval I = (x 0 - ε , x 0 + ε ) � Then F(x) ≈ F(x 0 ) + F’(x 0 )(x-x 0 ) for x ∈ I � In other words, all smooth curves look linear if looked at up close April 11, 2008 CSSE/MA 325 Lecture #18 7

Local representation � Suppose x 0 is a fixed point of F � We can represent the smooth map locally on I as F(x) ≈ F’(x 0 )x + F(x 0 ) - x 0 F’(x 0 ) � associate m with F’(x 0 ) � associate b with F(x 0 ) - x 0 F’(x 0 ) � Recall that � |m| < 1 ⇒ ___________, � |m| > 1 ⇒ ___________, � If |F’(x 0 )| < 1 then x 0 is an attracting fixed point, and if |F’(x 0 )| > 1 then x 0 is a repelling fixed point April 11, 2008 CSSE/MA 325 Lecture #18 8

Periodic points � Periodic points are also classified as attracting or repelling � Suppose x 0 is a periodic point of period p � If x 0 is an attracting fixed point of F p , then… � x 0 is an attracting periodic point of period p � Similarly for repelling and neutral April 11, 2008 CSSE/MA 325 Lecture #18 9

Slopes of periodic points � |(F p )’(x 0 )| < 1 ⇒ � x 0 is an attracting fixed point of F p ⇒ � x 0 is an attracting periodic point of F � Similarly, |(F p )’(x 0 )| > 1 ⇒ x 0 is a repelling periodic point of F April 11, 2008 CSSE/MA 325 Lecture #18 10

How do you compute (F p )’(x 0 )? � (Chain Rule; on board) April 11, 2008 CSSE/MA 325 Lecture #18 11

How do you compute (F p )’(x 0 )? � Use the chain rule: (F ° G)’(x 0 ) = F’(G(x 0 ))G’(x 0 ) � So (F 2 )’(x 0 ) = F’(F(x 0 ))F’(x 0 ) = F’(x 1 )F’(x 0 ) � … � (F n )’(x 0 ) = F’(x n-1 )F’(x n-2 )…F’(x 1 )F’(x 0 ) April 11, 2008 CSSE/MA 325 Lecture #18 12

Example � F(x) = -x 3 � x 0 = 1 is a period-2 point � its orbit is { 1, -1, 1, -1, … } � Is this point attracting or repelling? April 11, 2008 CSSE/MA 325 Lecture #18 13

Quiz � Analyze the logistic map, f(x) = ax(1-x) � More interesting than the linear map April 11, 2008 CSSE/MA 325 Lecture #18 14