(20 points) Consider the system (for a R) x' = y ax (a) , y' = -ay + 14m (a) What is the largest subset of Ra on which the hypotheses of the main existence and uniqueness theorem hold? (b) For each qualitatively different value of a, find all equilibrium points. When the Hartman-Grobman theorem applies, classify each equilibrium point. (c) Describe all bifurcations that occur as a varies and find the bifurcation values of a at which they occur. (Hint: we may haven't given a name to the type of bifurcation occurs at some of the bifurcation values. Just describe what happens, e.g., change of number or type of equilibrium points.) (20 points) Consider the system (for a R) x' = y ax (a) , y' = -ay + 14m (a) What is the largest subset of Ra on which the hypotheses of the main existence and uniqueness theorem hold? (b) For each qualitatively different value of a, find all equilibrium points. When the Hartman-Grobman theorem applies, classify each equilibrium point. (c) Describe all bifurcations that occur as a varies and find the bifurcation values of a at which they occur. (Hint: we may haven't given a name to the type of bifurcation occurs at some of the bifurcation values. Just describe what happens, e.g., change of number or type of equilibrium points.)