A linear system of differential equations is given by {x_1 = ax_1 + bx_2, x_2 =

Question

A linear system of differential equations is given by {x_1 = ax_1 + bx_2, x_2 = cx_1 + dx_2. Writing this as a matrix equation, we have x = Ax, where A = (a b c d). Recall that the trace of a matrix is the sum of its diagonal entries, so Tr(A) = a + d and that the determinant of a 2 times 2 matrix is det(A) = ad - bc. Find conditions on Tr(A) and det(A) such that the equilibrium of (1) is a stable node, an unstable node, a stable spiral, an unstable spiral or a saddle. For instance, you should find that the equilibrium is a saddle if and only if det(A)

Explanation / Answer

unstable node :if both the eigen values are positive and real then their sum i.e. tr(A)>0 and det(A)>0

stable node : if both the eigen values are negative then tr(A)<0 and detA>0

saddle point : if both eigenvalues are of different sign then detA<0

unstable spiral: if eigen values are complex but the real part is positive then trA and detA>0

stable spiral: if eigen values are complex but the real part is negative then trA<0 and detA>0

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.