Introduction to the Modeling and Analysis of Complex Systems

92 CHAPTER 5. DISCRETE-TIME MODELS II: ANALYSISNow, what if F is a multidimensional nonlinear function? Such an F can be spelledout as a set of multiple scalar functions, as follows:x 1,t = F 1 (x 1,t−1 , x 2,t−1 , . . . , x n,t−1 ) (5.63)x 2,t = F 2 (x 1,t−1 , x 2,t−1 , . . . , x n,t−1 ) (5.64).x n,t = F n (x 1,t−1 , x 2,t−1 , . . . , x n,t−1 ) (5.65)Using variable replacement similar to Eq. (5.58), these equations are rewritten as follows:x 1,eq + ∆x 1,t = F 1 (x 1,eq + ∆x 1,t−1 , x 2,eq + ∆x 2,t−1 . . . , x n,eq + ∆x n,t−1 ) (5.66)x 2,eq + ∆x 2,t = F 2 (x 1,eq + ∆x 1,t−1 , x 2,eq + ∆x 2,t−1 . . . , x n,eq + ∆x n,t−1 ) (5.67).x n,eq + ∆x n,t = F n (x 1,eq + ∆x 1,t−1 , x 2,eq + ∆x 2,t−1 . . . , x n,eq + ∆x n,t−1 ) (5.68)Since there are many ∆x i ’s in this formula, the Taylor expansion might not apply simply.However, the assumption that they are extremely small helps simplify the analysis here.By zooming in to an infinitesimally small area near the equilibrium point, each F i looks likea completely flat “plane” in a multidimensional space (Fig. 5.14) where all nonlinear interactionsamong ∆x i ’s are negligible. This means that the value of F i can be approximatedby a simple linear sum of independent contributions coming from the n dimensions, eachof which can be calculated in a manner similar to Eq. (5.61), asF i (x 1,eq + ∆x 1,t−1 , x 2,eq + ∆x 2,t−1 . . . , x n,eq + ∆x n,t−1 )≈ F i (x eq ) + ∂F i∂x 1∣ ∣∣∣xeq∆x 1,t−1 + ∂F i∂x 2∣ ∣∣∣xeq∆x 2,t−1 + . . . + ∂F i∂x n∣ ∣∣∣xeq∆x n,t−1 . (5.69)This linear approximation allows us to rewrite Eqs. (5.66)–(5.68) into the following,very concise linear equation:⎛x eq + ∆x t ≈ F (x eq ) + ⎜⎝∂F 1 ∂F 1∂x 1∂F 2 ∂F 2∂x 1.∂F n∂x 1∂F 1∂x n∂F 2∂x n∂x 2. . .∂x 2. . ... .. .∂F n ∂F∂x 2. . . n∂x n⎞⎟∆x t−1 (5.70)⎠∣x=x eqThe coefficient matrix filled with partial derivatives is called a Jacobian matrix of theoriginal multidimensional function F . It is a linear approximation of the nonlinear function