Fig 4. Fibonacci numbers.

Left panel: In red the map x′ = x − ux (f*(x) with z = 1 and u = ln ϕ with $\varphi =(1+\sqrt{5})/2$). In blue the trajectory x_t = x₀ exp(−ut) initiated at $x_0=-\sqrt{5}$. Its positions approximate asymptotically (minus) the reciprocals of the Fibonacci numbers. Right panel: In red the map x′ = x + ux (f*(x) with Z = 1 and u = ln ϕ). In blue the trajectory x_t = x₀ exp(ut) initiated at $x_0=1/\sqrt{5}$. Its positions approximate asymptotically the Fibonacci numbers. Identity lines in green.