Consider a logistic map defined by the recursive relation Xn+1 = n (1 - n), (1)...

 

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Consider a logistic map defined by the recursive relation Xn+1 = n (1 - n), (1) defined within the parameter range 0 < < 4. (a) Write (and attach) PYTHON code to simulate the logistic map. For an ap- propriate resolution inu (an starting from below -3), compute and plot the bifurcation diagram and estimate the critical value for u for the transition to chaos (to at least 3 decimal digits accuracy). Also estimate the Feigenbaum number from your numerical analysis. How does it compare with the number for the oscillator? (b) Compute and plot trajectories in the n+1 VS. Xn diagram for the period-1, period-4 and chaotic regimes choosing suitable values of u as determined in part (a); use xo 0.05 as starting value and make one plot (including the identity line) for each u value. Quote the numerical values for the (stable) fixpoints in the regular regimes. - (c) Work out analytically the polynomials gn(x) = f(x) for N=1,4 with f(x) = x (1 x) and write them out sorted by powers of x. Numerically compute and plot the continuous curves g(x) (for a -value in the period-1 regime) and 94(x) (for a -value in the period-4 regime), and determine the intersections with the identity line, id(x): = x. Classify the intersections as stable or unstable fixpoints by evaluating the pertinent values for the derivative of 91,4(x). (d) Estimate the Lyapunov exponent for -3.3 and 3.8. Do this by running very close initial conditions, xo (e.g., deviating by O(10-4) or so) and averaging An over several runs for the same but (slightly) different xo. Then fit a straight line to the semi-log plot of Axn| vs. n (eyeball fit is fine). Consider a logistic map defined by the recursive relation Xn+1 = n (1 - n), (1) defined within the parameter range 0 < < 4. (a) Write (and attach) PYTHON code to simulate the logistic map. For an ap- propriate resolution inu (an starting from below -3), compute and plot the bifurcation diagram and estimate the critical value for u for the transition to chaos (to at least 3 decimal digits accuracy). Also estimate the Feigenbaum number from your numerical analysis. How does it compare with the number for the oscillator? (b) Compute and plot trajectories in the n+1 VS. Xn diagram for the period-1, period-4 and chaotic regimes choosing suitable values of u as determined in part (a); use xo 0.05 as starting value and make one plot (including the identity line) for each u value. Quote the numerical values for the (stable) fixpoints in the regular regimes. - (c) Work out analytically the polynomials gn(x) = f(x) for N=1,4 with f(x) = x (1 x) and write them out sorted by powers of x. Numerically compute and plot the continuous curves g(x) (for a -value in the period-1 regime) and 94(x) (for a -value in the period-4 regime), and determine the intersections with the identity line, id(x): = x. Classify the intersections as stable or unstable fixpoints by evaluating the pertinent values for the derivative of 91,4(x). (d) Estimate the Lyapunov exponent for -3.3 and 3.8. Do this by running very close initial conditions, xo (e.g., deviating by O(10-4) or so) and averaging An over several runs for the same but (slightly) different xo. Then fit a straight line to the semi-log plot of Axn| vs. n (eyeball fit is fine).

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a Here is a Python code to simulate the logistic map and plot the bifurcation diagram The critical v... View the full answer