Question(s):
1. Verify that the logistic family f (x) = (lamda)*x*(1-x) has
fixed points at x = 0, and x = (lamda - 1)/(lamda).
2. Also compute the slope of the graph of the Logistic map at
the fixed point 0 in terms of lamda.
3. What is the slope for the other fixed point?
1. Symbolically, we will use the symbol "&" in place of lambda to make this easier (no lamda symbol!); the extension to any general lamda is clear.
First, f(0) = (lamda)*0*(1-0) = 0, so that x = 0 is indeed a fixed point.
Second, f((lamda - 1)/lamda) = f((& - 1)/&) = &*(&
- 1)/&*(1-(& - 1)/&) = (& - 1)*(1/&) = (& - 1)/&,
so that x = (lamda - 1)/lamda is in fact a fixed point.
2. Using the same symbolism as in part 1 above, our function is f(x) = &*x*(1-x) = -&x^2 + &x. To find the slope of the map at the fixed points described in part 1 above, we must take the DERIVATIVE of our function and evaluate it at our fixed point value. The derivative is f'(x) = -2&x + &. Thus, the slope of the graph at the fixed point 0 is f'(0) = -2&(0) + & = 0 + & = &. We therefore see that the slope at the fixed point 0 in terms of lamda is simply lamda!
3. Here we calculate f'((lamda-1)/lamda) = f'((& - 1)/&)
= -2&((& - 1)/&) + & = -2(& - 1) + & = -2&
+ 2 + & = 2 - &.
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