Introduction to the Modeling and Analysis of Complex Systems

5.6. ASYMPTOTIC BEHAVIOR OF DISCRETE-TIME LINEAR DYNAMICAL... 83where n is the dimension of the state space (i.e., A is an n × n matrix). Most real-worldn × n matrices are diagonalizable and thus have n linearly independent eigenvectors, sohere we assume that we can use them as the basis vectors to represent any initial statex 0 2 . If you replace x 0 with this new notation in Eq. (5.36), we get the following:x t = A t (b 1 v 1 + b 2 v 2 + . . . + b n v n ) (5.40)= b 1 A t v 1 + b 2 A t v 2 + . . . + b n A t v n (5.41)= b 1 λ t 1v 1 + b 2 λ t 2v 2 + . . . + b n λ t nv n (5.42)This result clearly shows that the asymptotic behavior of x t is given by a summation ofmultiple exponential terms of λ i . There are competitions among those exponential terms,and which term will eventually dominate the others is determined by the absolute value ofλ i . For example, if λ 1 has the largest absolute value (|λ 1 | > |λ 2 |, |λ 3 |, . . . |λ n |), thenx t = λ t 1(b 1 v 1 + b 2(λ2λ 1) tv 2 + . . . + b n(λnλ 1) tv n), (5.43)lim x t ≈ λ t 1b 1 v 1 . (5.44)t→∞This eigenvalue with the largest absolute value is called a dominant eigenvalue, and itscorresponding eigenvector is called a dominant eigenvector, which will dictate which direction(a.k.a. mode in physics) the system’s state will be going asymptotically. Here isan important fact about linear dynamical systems:If the coefficient matrix of a linear dynamical system has just one dominant eigenvalue,then the state of the system will asymptotically point to the direction given byits corresponding eigenvector regardless of the initial state.This can be considered a very simple, trivial, linear version of self-organization.Let’s look at an example to better understand this concept. Consider the asymptoticbehavior of the Fibonacci sequence:x t = x t−1 + x t−2 (5.45)2 This assumption doesn’t apply to defective (non-diagonalizable) matrices that don’t have n linearlyindependent eigenvectors. However, such cases are rather rare in real-world applications, because anyarbitrarily small perturbations added to a defective matrix would make it diagonalizable. Problems with suchsensitive, ill-behaving properties are sometimes called pathological in mathematics and physics. For moredetails about matrix diagonalizability and other related issues, look at linear algebra textbooks, e.g. [28].