Introduction to the Modeling and Analysis of Complex Systems

5.7. LINEAR STABILITY ANALYSIS OF DISCRETE-TIME NONLINEAR... 91around its equilibrium point x eq . By definition, x eq satisfiesx eq = F (x eq ). (5.57)To analyze the stability of the system around this equilibrium point, we switch our perspectivefrom a global coordinate system to a local one, by zooming in and capturing asmall perturbation added to the equilibrium point, ∆x t = x t − x eq . Specifically, we applythe following replacementx t ⇒ x eq + ∆x t (5.58)to Eq. (5.56), to obtainx eq + ∆x t = F (x eq + ∆x t−1 ). (5.59)The right hand side of the equation above is still a nonlinear function. If x t is scalarand thus F (x) is a scalar function, the right hand side can be easily approximated usingthe Taylor expansion as follows:F (x eq + ∆x t−1 ) = F (x eq ) + F ′ (x eq )∆x t−1 + F ′′ (x eq )2!∆x 2 t−1 + F ′′′ (x eq )∆x 3 t−1 + . . .3!(5.60)≈ F (x eq ) + F ′ (x eq )∆x t−1 (5.61)This means that, for a scalar function F , F (x eq + ∆x) can be linearly approximated by thevalue of the function at x eq plus a derivative of the function times the displacement fromx eq . Together with this result and Eq. (5.57), Eq. (5.59) becomes the following very simplelinear difference equation:∆x t ≈ F ′ (x eq )∆x t−1 (5.62)This means that, if |F ′ (x eq )| > 1, ∆x grows exponentially, and thus the equilibrium pointx eq is unstable. Or if |F ′ (x eq )| < 1, ∆x shrinks exponentially, and thus x eq is stable.Interestingly, this conclusion has some connection to the cobweb plot we discussed before.|F ′ (x eq )| is the slope of function F at an equilibrium point (where the function curvecrosses the diagonal straight line in the cobweb plot). If the slope is too steep, eitherpositively or negatively, trajectories will diverge away from the equilibrium point. If theslope is less steep than 1, trajectories will converge to the point. You may have noticedsuch characteristics when you drew the cobweb plots. Linear stability analysis offers amathematical explanation of that.