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midterm_1_review (dragged) 5

midterm_1_review (dragged) 5 - y n 1 = ρ y n b The general...

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6 MA 36600 MIDTERM #1 REVIEW § 2.9: First Order Di ff erence Equations. A recursive relation the form y n +1 = Γ ( n, y n ) is called a first order di ff erence equation . Using Euler’s Method, the di ff erential equation y = G ( t, y ) has the associated di ff erence equation y n +1 = Γ ( n, y n ) in terms of Γ ( n, y ) = y + G ( t 0 + n h, y ) · h. If the di ff erential equation is a linear equation then G ( t, y ) = g ( t ) p ( t ) y . Similarly, Γ ( n, y ) = ρ n y + b n in terms of ρ n = 1 p ( t 0 + n h ) h and b n = g ( t 0 + n h ) h , so an equation in the form y n +1 = ρ n y n + b n is called a linear di ff erence equation . Otherwise, we call such di ff erence equations nonlinear . If the di ff erential equation is an autonomous equation then G ( t, y ) = f ( y ). Similarly, Γ ( n, y ) = y + f ( y ) h does not involve n , so an equation in the form y n +1 = φ ( y n ) is called an an autonomous di ff erence equation . Say that we have a sequence { y 0 , y 1 , y 2 , . . . , y n , . . . } is a solution, and denote the “limiting value” y L = lim t →∞ y n = y L = φ ( y L ) . Such a solution y L is called an equilibrium solution to the di ff erence equation. A linear autonomous di ff erence equation must have constant coe
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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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