Perturbation of the default system in pplane6 exhibits two bifurcations involving the sadd...

Perturbation of the default system in pplane6 exhibits two bifurcations involving the saddle point at the origin and the equilibrium point near (3/2, −3/2). To be precise we are looking at the system

as the parameter A varies.

a) Show that between A = 2.5 and A = 3, the equilibrium point changes from a sink to a source and spawns a limit cycle, i.e., a Hopf bifurcation occurs.

b) Show that between A = 3 and A = 3.2 the limit cycle disappears. At some point in this transition, the limit cycle becomes a homoclinic orbit for the saddle point at the origin. A homoclinic orbit is one that originates as an unstable orbit for saddle point at t = −∞, and then becomes a stable orbit for the same saddle point as t → +∞. It is extremely difficult to find the value of A for which this occurs, so do not try too hard. The process illustrated here is called a homoclinic bifurcation.

c) The default system has four quadratic terms. Show that if any of the coefficients of these terms is altered up or down by as little as 0.5 from the default, a bifurcation takes place. Show that there is a Hopf bifurcation in one direction, and a homoclinic bifurcation in the other.