Question

3. For the following example there is a bifurcation at the origin for 0. Find the evolution equation on the center manifold correct to whatever order is necessary to determine the type of bifurcation. Give a quick sketch of the 2Dflow for μ 0. 2r y - r2 [Hint: Let μ-€2, treating € as a parameter. Write down the system in (u, v, e) variables, where u, v are as in Problem 2. For this problem, it is enough to sketch the flow in u, v variables.] 2. For the following example, find an approximation to the center manifold at the origin and hence determine the local behavior of solutions. 2r y 2 Hint: First solve the problem in transformed variables u, v which make the linearised system into normal form, and then translate back to original vari- ables z, y.]

Answer 1

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