midterm_1_review (dragged) 5

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

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6M A 3 6 6 0 0 M I D T E R M # 1 R E V I E W § 2.9: First Order Diference Equations. A recursive relation the form y n +1 = Γ( n, y n ) is called a frst order diFerence equation . Using Euler’s Method, the diFerential equation y ° = G ( t, y ) has the associated diFerence equation y n +1 = Γ( n, y n ) in terms of Γ( n, y )= y + G ( t 0 + nh, y ) · h. If the diFerential 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 + nh ) h and b n = g ( t 0 + nh ) h , so an equation in the form y n +1 = ρ n y n + b n is called a linear diFerence equation . Otherwise, we call such diFerence equations nonlinear . If the diFerential 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 diFerence 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 =l im t →∞ y n = y L = φ ( y L ) . Such a solution y L is called an equilibrium solution to the diFerence equation. A linear autonomous diFerence equation must have constant coefficients:
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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.

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