Understanding Affine Difference Equations and Long-Term Behavior
Exploring different slopes in affine difference equations and their impact on the behavior of solutions. Discover how iterating with points relative to fixed points reveals attracting or repelling characteristics. Gain insights into the convergence or divergence of sequences in relation to fixed points in the functions.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
E N D
Presentation Transcript
= ( ) f x y Affine Difference Equations---Slope bigger than 1 = y x n a
= ( ) f x y Affine Difference Equations---Slope bigger than 1 = y x n a
Affine Difference Equations---Slope less than -1 = y x n a = ( ) f x y
Affine Difference Equations---Slope smaller than 1 = y x = ( ) f x y n a fixed pt What if we start iterating with a point that lies to the left of the fixed point?
Affine Difference Equations---Slope in (-1,0). = y x n a fixed pt = ( ) f x y
Affine Difference Equations---Slope equal to1 = ( ) f x x + = ( 1) ( ) A n A n = + ( ) f x x k + = + ( 1) ( ) A n A n k
Conclusions: Long term behavior of solutions to affine difference equations: A( 1) ( ) iterating n kA n b + = + = kx b + ( ) f x | | 1 k If , the sequence (A(n)) , n = 1, 2, 3,. . . blows up . That is, The fixed point is a repelling fixed point. If , the sequence (A(n)) , n = 1, 2, 3,. . . Converges to the fixed point of the function. That is, ( ) A n as n 0 | | 1 k b ( ) A n as n 1 k The fixed point is an attracting fixed point.
