Introduction to the Modeling and Analysis of Complex Systems

84 CHAPTER 5. DISCRETE-TIME MODELS II: ANALYSISAs we studied already, this equation can be turned into the following two-dimensionalfirst-order model:x t = x t−1 + y t−1 (5.46)y t = x t−1 (5.47)This can be rewritten by letting (x t , y t ) ⇒ x t and using a vector-matrix notation, as( ) 1 1x t = x t−1 . (5.48)1 0So, we just need to calculate the eigenvalues and eigenvectors of the above coefficientmatrix to understand the asymptotic behavior of this system. Eigenvalues of a matrix Acan be obtained by solving the following equation for λ:det(A − λI) = 0 (5.49)Here, det(X) is the determinant of matrix X. For this Fibonacci sequence example, thisequation is( )1 − λ 1det= −(1 − λ)λ − 1 = λ 2 − λ − 1 = 0, (5.50)1 −λwhich givesλ = 1 ± √ 52(5.51)as its solutions. Note that one of them ((1 + √ 5)/2 = 1.618 . . .) is the golden ratio! It isinteresting that the golden ratio appears from such a simple dynamical system.Of course, you can also use Python to calculate the eigenvalues and eigenvectors (or,to be more precise, their approximated values). Do the following:Code 5.6:from pylab import *eig([[1, 1], [1, 0]])The eig function is there to calculate eigenvalues and eigenvectors of a square matrix.You immediately get the following results:Code 5.7:(array([ 1.61803399, -0.61803399]), array([[ 0.85065081, -0.52573111],[ 0.52573111, 0.85065081]]))