Consider the system x _ = _ 1 1 1 _ x. a. Solve the system for = 1 2 . What are the

Chapter 7, Problem 23

(choose chapter or problem)

Consider the system x _ = _ 1 1 1 _ x. a. Solve the system for = 1 2 . What are the eigenvalues of the coefficient matrix? Classify the equilibrium point at the origin as to type. b. Solve the system for = 2. What are the eigenvalues of the coefficient matrix? Classify the equilibrium point at the origin as to type. c. In parts a and b, solutions of the system exhibit two quite different types of behavior. Find the eigenvalues of the coefficient matrix in terms of , and determine the value of between 1 2 and 2 where the transition from one type of behavior to the other occurs. This value of is called a bifurcation value for this problem. Electric Circuits. 24 and 25 are concerned with the electric circuit described by the system of differential equations in of Section 7.1: d dt _I V _ = R1 L 1L 1C 1 CR2 _I V _, I (0) = I0, V(0) = V0. (37) 2