Introduction to the Modeling and Analysis of Complex Systems

5.6. ASYMPTOTIC BEHAVIOR OF DISCRETE-TIME LINEAR DYNAMICAL... 89Now we see the complex unit j (yes, Python uses j instead of i to represent the imaginaryunit i) in the result, which means this system is showing oscillation. Moreover, you cancalculate the absolute value of those eigenvalues:Code 5.10:map(abs, eig([[0.5, 1], [-0.5, 1]])[0])Then the result is as follows:Code 5.11:[0.99999999999999989, 0.99999999999999989]This means that |λ| is essentially 1, indicating that the system shows sustained oscillation,as seen in Fig. 4.3.For higher-dimensional systems, various kinds of eigenvalues can appear in a mixedway; some of them may show exponential growth, some may show exponential decay,and some others may show rotation. This means that all of those behaviors are goingon simultaneously and independently in the system. A list of all the eigenvalues is calledthe eigenvalue spectrum of the system (or just spectrum for short). The eigenvaluespectrum carries a lot of valuable information about the system’s behavior, but often, themost important information is whether the system is stable or not, which can be obtainedfrom the dominant eigenvalue.Exercise 5.12 Study the asymptotic behavior of the following three-dimensionaldifference equation model by calculating its eigenvalues and eigenvectors:x t = x t−1 − y t−1 (5.53)y t = −x t−1 − 3y t−1 + z t−1 (5.54)z t = y t−1 + z t−1 (5.55)Exercise 5.13 Consider the dynamics of opinion diffusion among five people sittingin a ring-shaped structure. Each individual is connected to her two nearestneighbors (i.e., left and right). Initially they have random opinions (represented asrandom real numbers), but at every time step, each individual changes her opinionto the local average in her social neighborhood (i.e, her own opinion plus those ofher two neighbors, divided by 3). Write down these dynamics as a linear difference