Science:Math Exam Resources/Courses/MATH103/April 2015/Question 06 (b)

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MATH103 April 2015

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Question 06 (b)
Consider the function $F(x)=\lambda xe^{-x}$ , where $\lambda >0$ is a parameter. Now consider the iterated map $x_{t+1}=F(x_{t})$ for $t=0,1,2,3,\cdots$ . (b) Determine the range of $\lambda$ for which each fixed point (steady state, equilibrium) is stable.

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Hint
A fixed point $x_{0}$ of $F$ is stable if and only if $\|F'(x_{0})\|<1$ .

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Solution
Recall that in part (a), we find the fixed points of $F$ , $x=0,x=\ln \lambda$ . By the definition of a stable fixed point in Hint, we first find the derivative of $F$ : $F'(x)=(\lambda xe^{-x})'=\lambda (e^{-x}-xe^{-x})=\lambda (1-x)e^{-x}$ . Then, we can get $\|F'(0)\|=\|\lambda (1-0)e^{-0}\|=\lambda$ and $\|F'(\ln \lambda )\|=\|\lambda (1-\ln \lambda )e^{-\ln \lambda }\|=\lambda \|1-\ln \lambda \|{\frac {1}{\lambda }}=\|1-\ln \lambda \|.$ . Therefore, $x=0$ is stable if $\lambda <1$ , while $x=\ln \lambda$ is stable if $\|1-\ln \lambda \|<1$ ( $\iff -1<1-\ln \lambda <1\iff 0<\ln \lambda <2\iff 1<\lambda <e^{2}$ .) Answer: for the fixed points $x=0,\ln \lambda$ , the ranges are $\color {blue}0<\lambda <1,1<\lambda <e^{2}$ , respectively