Question

1. Linear Equations. Each of the following systems of ODEs involves a real parameter a. Investigate how the nature of the equilibrium point at 0,0) changes as this parameter varies along the real line. dt dt [6 marks] dt dy dt [6 marks]

Answer 1

(a) [dx/dt dy/dt] = [1 1 a 2] [x y] A = [1 1 a 2] characteristics equation of A is lambda^2 - 3 lambda + (2 - a) = 0 lambda = 3 plusminus squareroot 9 - 4(2 - a)/2 lambda = 3 plusminus squareroot 1 + 49/2 if a = -1/4 implies lambda = 3/2 eigenvalue repeated two times implies (0, 0) is source if a < - 1/4 implies lambda is imaginary (complex) whose repl part is positive implies (0, 0) spiral source if - 1/4 lessthanorequalto u < 2 lambda is positive eigenvalue (both) (0, 0) is source if a > 2 implies lambda, is positive & lambda_2 is negative eigenvalue of A implies (0, 0) is saddle if a = 2 if lambda = 0 (0, 0) is point of inflection Hence, a < - 1/4 implies (0, 0) is spiral source - 1/4 lessthanorequalto a < 2 implies (0, 0) is source