Introduction to the Modeling and Analysis of Complex Systems

86 CHAPTER 5. DISCRETE-TIME MODELS II: ANALYSISWe can now look at the dominant eigenvector that corresponds to the dominant eigenvalue,which is (0.85065081, 0.52573111). This eigenvector tells you the asymptotic directionof the system’s state. That is, after a long period of time, the system’s state (x t , y t )will be proportional to (0.85065081, 0.52573111), regardless of its initial state.Let’s confirm this analytical result with computer simulations.Exercise 5.11 Visualize the phase space of Eq. (5.48).The results are shown in Fig. 5.12, for 3, 6, and 9 steps of simulation. As you can seein the figure, the system’s trajectories asymptotically diverge toward the direction given bythe dominant eigenvector (0.85065081, 0.52573111), as predicted in the analysis above.6301504201002105000021050420100610 5 0 5 103040 20 0 20 40150200 150 100 50 0 50 100 150 200t = 0 − 3 t = 0 − 6 t = 0 − 9Figure 5.12: Phase space visualizations of Eq. (5.48) for three different simulationlengths.Figure 5.13 illustrates the relationships among the eigenvalues, eigenvectors, and thephase space of a discrete-time dynamical system. The two eigenvectors show the directionsof two invariant lines in the phase space (shown in red). Any state on each ofthose lines will be mapped onto the same line. There is also an eigenvalue associatedwith each line (λ 1 and λ 2 in the figure). If its absolute value is greater than 1, the correspondingeigenvector component of the system’s state is growing exponentially (λ 1 , v 1 ),whereas if it is less than 1, the component is shrinking exponentially (λ 2 , v 2 ). In addition,for discrete-time models, if the eigenvalue is negative, the corresponding eigenvectorcomponent alternates its sign with regard to the origin every time the system’s state isupdated (which is the case for λ 2 , v 2 in the figure).Here is a summary perspective for you to understand the dynamics of linear systems: