Introduction to the Modeling and Analysis of Complex Systems

8.3. HOPF BIFURCATIONS IN 2-D CONTINUOUS-TIME MODELS 141y = dx/dt to make it into a 2-D first-order system, as follows:dxdt = y (8.21)dydt = −r(x2 − 1)y − x (8.22)From these, we can easily show that the origin, (x, y) = (0, 0), is the only equilibrium pointof this system. The Jacobian matrix of this system at the origin is given as follows:(0 1−2rxy − 1 −r(x 2 − 1)) ∣∣∣(x,y)=(0,0) ( ) 0 1=−1 r(8.23)The eigenvalues of this matrix can be calculated as follows:∣ 0 − λ 1−1 r − λ ∣ = 0 (8.24)− λ(r − λ) + 1 = λ 2 − rλ + 1 = 0 (8.25)λ = r ± √ r 2 − 42The critical condition for a bifurcation to occur is(8.26)Re(λ) = 0, (8.27)whose left hand side can be further detailed as⎧⎪⎨ r ± √ r 2 − 4if r 2 ≥ 4, orRe(λ) =r2⎪⎩if r 2 < 4.2(8.28)The first case can’t be zero, so the only critical condition for a bifurcation to occur is thesecond case, i.e.r = 0, when Re(λ) = 0 and Im(λ) = ±i. (8.29)This is a Hopf bifurcation because the eigenvalues have non-zero imaginary parts whenthe stability change occurs. We can confirm this analytical prediction by numerical simulationsof the model with systematically varied r, as follows:Code 8.2: van-del-pol-Hopf-bifurcation.pyfrom pylab import *