Introduction to the Modeling and Analysis of Complex Systems

7.5. LINEAR STABILITY ANALYSIS OF NONLINEAR DYNAMICAL SYSTEMS 125Exercise 7.12Show that the unstable points with det(A) < 0 are saddle points.Exercise 7.13• dx ( −1 2dt = 2 −2• dx ( 0.5 −1.5dt = 1 −1Determine the stability of the following linear systems:)x)xExercise 7.14 Confirm the analytical result shown in Fig. 7.5 by conducting numericalsimulations in Python and by drawing phase spaces of the system for severalsamples of A.7.5 Linear Stability Analysis of Nonlinear Dynamical SystemsFinally, we can apply linear stability analysis to continuous-time nonlinear dynamical systems.Consider the dynamics of a nonlinear differential equationdxdt= F (x) (7.64)around its equilibrium point x eq . By definition, x eq satisfies0 = F (x eq ). (7.65)To analyze the stability of the system around this equilibrium point, we do the same coordinateswitch as we did for discrete-time models. Specifically, we apply the followingreplacementx(t) ⇒ x eq + ∆x(t) (7.66)to Eq. (7.64), to obtaind(x eq + ∆x)dt= d∆xdt= F (x eq + ∆x). (7.67)