Introduction to the Modeling and Analysis of Complex Systems

9.1. CHAOS IN DISCRETE-TIME MODELS 1552.5r = 1.82.01.51.00.50.00.50 20 40 60 80 100Figure 9.2: Example of the “butterfly effect,” the extreme sensitivity of chaotic systemsto initial conditions. The two curves show time series generated using Eq. (8.37) withr = 1.8; one with x 0 = 0.1 and the other with x 0 = 0.100001.Exercise 9.1 There are many simple mathematical models that exhibit chaoticbehavior. Try simulating each of the following dynamical systems (shown inFig. 9.3). If needed, explore and find the parameter values with which the systemshows chaotic behaviors.• Logistic map: x t = rx t−1 (1 − x t−1 )• Cubic map: x t = x 3 t−1 − rx t−1• Sinusoid map: x t = r sin x t−1• Saw map: x t = fractional part of 2x t−1Note: The saw map may not show chaos if simulated on a computer, but it will showchaos if it is manually simulated on a cobweb plot. This issue will be discussedlater.predictions for our future!