Introduction to the Modeling and Analysis of Complex Systems

144 CHAPTER 8. BIFURCATIONSWith a = 0.7, b = 0.8, and c = 3, do the following:• Numerically obtain the equilibrium point of this model for several values of z,ranging between -2 and 0. There is only one real equilibrium point in thissystem.• Apply the result obtained above to the Jacobian matrix of the model, andnumerically evaluate the stability of that equilibrium point for each value of z.• Estimate the critical thresholds of z at which a Hopf bifurcation occurs. Thereare two such critical thresholds.• Draw a series of its phase spaces with values of z varied from 0 to -2 toconfirm your analytical prediction.8.4 Bifurcations in Discrete-Time ModelsThe bifurcations discussed above (saddle-node, transcritical, pitchfork, Hopf) are alsopossible in discrete-time dynamical systems with one variable:x t = F (x t−1 ) (8.35)The Jacobian matrix of this system is, again, a 1×1 matrix whose eigenvalue is its contentitself, which is given by dF/dx. Since this is a discrete-time model, the critical condition atwhich a bifurcation occurs is given bydF∣ dx ∣ = 1. (8.36)x=xeqLet’s work on the following example:x t = x t−1 + r − x 2 t−1 (8.37)This is a discrete-time analog of Eq. (8.3). Therefore, it has the same set of equilibriumpoints:x eq = ± √ r (8.38)Next, we calculate dF/dx as follows:dFdx = (r + x − x2 ) ′ = 1 − 2x (8.39)dF∣ dx ∣ = |1 ± 2 √ r| (8.40)√ x=± r