### Investigate the presence of homoclinic

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x'=ux-y+x^3, y'=bx-y.

(a). Locate any local bifurcations that occur as u and b are varied. Analyze the bifurcations fully. (To do this you will need to transform to the linearized normal form at the critical point. If the bifurcation occurs in one dimension you then need to calculate an approximation to the center manifold in which it occurs.)

(b).Investigate the presence of limit cycles both analytically and numberically.

(c).Investigate the presence of homoclinic/heteroclinic orbits and global bifurcations numerically.

(d).Present a conclusion of all your findings which includes a bifurcation diagram

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