Introduction to the Modeling and Analysis of Complex Systems

5.3. COBWEB PLOTS FOR ONE-DIMENSIONAL ITERATIVE MAPS 69following:1. Draw a square on your paper. Label the bottom edge as the axis for x t−1 , and theleft edge as the axis for x t . Label the range of their values on the axes (Fig. 5.4).x maxx tx minx minx t-1x maxFigure 5.4: Drawing a cobweb plot (1).2. Draw a curve x t = f(x t−1 ) and a diagonal line x t = x t−1 within the square (Fig. 5.5).Note that the system’s equilibrium points appear in this plot as the points where thecurve and the line intersect.3. Draw a trajectory from x t−1 to x t . This can be done by using the curve x t = f(x t−1 )(Fig. 5.6). Start from a current state value on the bottom axis (initially, this is theinitial value x 0 , as shown in Fig. 5.6), and move vertically until you reach the curve.Then switch the direction of the movement to horizontal and reach the left axis. Youend up at the next value of the system’s state (x 1 in Fig. 5.6). The two red arrowsconnecting the two axes represent the trajectory between the two consecutive timepoints.4. Reflect the new state value back onto the horizontal axis. This can be done as asimple mirror reflection using the diagonal line (Fig. 5.7). This completes one stepof the “manual simulation” on the cobweb plot.