Question

5. Determine the possible equilibrium states of the system and investigate their stability by the first method of Lyapunov, if it is described by a system of AN = P(x, y); 7 = Q(x, y) equations p dt P(x, y) = 2x²y +1 Q(x, y) = x + y

Answer 1

Som. dx dy dt dt The equilibrum Slates ch the bystem is determined by equeating de dy So we have 2x 2 x2 Al so we have 2xt ()'3 and 1. 2(2)5 Real equilibrium þoint f te bystem is 2. The E (xa,ise) = -:(0 a, -o a) 2V3 2Y3 more equilibsium point of the supkem are Complexx "neember So we will disuss the There are tuso but Stability of the syetem enly around the point E. First Lyapunov method :- Jm this method the sabity analysis f an equilibrum point E(X0, yo] is done the Coreoponding linearized sypem around E(xayi useng

Now afpeying get the correoponding Jacobian mahrix as the limear Stability analysis fos system D we Je= JE = AS 2 xo 1 -2.51985 JE = 1.25992 The eigemalues of the mabix is obtained as and 82= 132748 2.84733 real both the eigenvalees and of opposite sign therefore it is a Saddle juolieh Now is always unstable