Question

10.1.11 (Cubic map) Consider the map x,* -3x,-x a) Find all the fixed points and classify. their stability. b) Draw a cobweb staring at x1.9. c) Draw a cobweb starting at x 2.1. d) Try to explain the dramatic difference between the orbits found in parts (b) and (c). For instance, can you prove that the orbit in (b) will remain bounded for all n ? Or that I → oo in (c)?

Answer 1

"x = 0 is one solution Consider, 1 - 3x^2 (or) x = plusorminus squareroot 2 Consider next the stability of these fixed points f(x) = 3x - x^3 f prime (x) = 3 - 3x^2 f Prime (0) = 3 & F prime (plusorminus squareroot 2) = 3 - 6 = -3 All of these points (f(x)) > 1 & then are unstable The column stays with in the range [-2, 2] for all times hence settle to a fixed point [There are unstable] The column diagram set does not stay within a finite range it but rather indicated that x_n rightarrow infinity sketch the function y = x & y = f(x) = 3 times -2 x = squareroot 3 & has minimum at X = -squareroot 3 also, 3x - x^2 rightarrow -infinity as x rightarrow infinity & 3x - x^3 rightarrow -infinity as x rightarrow infinity case 1: To show that