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nll' LTE -' MAT 332, FALL 2018. ASSIGNMENT 2' DUE ON OCTOBER 18, IN CLASS. 1. Consider the non-linear system :c' = y + am? + if) y' = I + aytrz + 92) depending on the parameter a. a) Lincarizc the system at the origin and verify that {0, 0) is a centre for the linearized system. b) Use 91 Maple plot for a value a # 0 to verify that (0, 0) is not a centre for the non-luicar systcnr c) Rewrite the original system in polar coordinates I:Tcos, y:rsin9 and simplify. (Hint: the system will take a very simple form if you do this correctly). d) Finally, using the system in polar coordinates do a study of the trajectories of the system. Identify all possible scenarios for the phase portrait of the system depending on the value of a, and illustrate with phase plots in (I, y) coordinates. 2) Invent your own nonlinear system describing two competing biologi- cal species similar to the Lotkerolterra equations on page 145, Study its behaviour and illustrate with a Maple plot