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Unformatted text preview: y n +1 = ρ y n + b. The general solution is y n = ρ n y + 1 − ρ n 1 − ρ b if ρ ° = 1; y + n b if ρ = 1. • The diFerence equation u n +1 = ρ u n (1 − u n ) is nonlinear autonomous equation. This is known as the logistic diFerence equation . Even though the equilibrium solutions are u L = 0 and u L = ( ρ − 1) /ρ , we have the following limiting values: lim n →∞ u n = ρ − 1 ρ whenever 1 < ρ < 3; ρ + 1 ± ° ( ρ + 1) ( ρ − 3) 2 ρ whenever 3 < ρ < 1 + √ 6. The value ρ = 3 is a point of bi±urcation i.e., when ρ < 3 there is one equilibrium solution, but when ρ > 3 there are two. As ρ increases eve more, the number of periods increases – until there are in±nitely many. This is known as chaotic behavior ....
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue.
 Spring '09
 EdrayGoins
 Equations

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