Classical Dynamics of Particles and Systems · Problem 4-16

Gravitation — Marion & Thornton

Physics — Mechanics · logistic-map-stability

Problem

The curve xn+1=f(xn)x_{n+1} = f(x_n) intersects the curve xn+1=xnx_{n+1} = x_n at x0x_0. The expansion of xn+1x_{n+1} about x0x_0 is xn+1−x0=β(xn−x0)x_{n+1} - x_0 = \beta(x_n - x_0) where β=(df/dx)\beta = (df/dx) at x=x0x = x_0. (a) Describe the geometrical sequence that the successive values of xn+1−x0x_{n+1} - x_0 form. (b) Show that the intersection is stable when ∣β∣<1|\beta| < 1 and unstable when ∣β∣>1|\beta| > 1.

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