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Visualizing Iteration

Discover the behavior of solutions to affine difference equations by visualizing iteration using cobweb diagrams. Analyze slopes bigger than 1 or less than -1, and explore starting points and slopes between -1 and 1. Understand conclusions on long-term behavior based on fixed points being repelling or attracting.

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Visualizing Iteration

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Presentation Transcript

  1. Visualizing Iteration “Cobweb” diagrams

  2. Affine Difference Equations---Slope bigger than 1

  3. Affine Difference Equations---Slope bigger than 1

  4. Affine Difference Equations---Slope less than -1

  5. Affine Difference Equations---Slope smaller than 1 What if we start iterating with a point that lies to the left of the fixed point?

  6. Affine Difference Equations---Slope in (-1,0).

  7. Affine Difference Equations---Slope equal to1

  8. Conclusions: Long term behavior of solutions to affine difference equations: • 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, • The fixed point is an attracting fixed point.

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