Question: Consider the system x' = y + x(x 2 + y 2 ) y'= x + y(x 2 + y 2 ) Show that the

Consider the system x' = y + x(x² + y² ) y'= x + y(x² + y² )

Show that the origin is a centre in the linearized system. Then let r² = x² + y² (the polar coordinate r), and use rr' = xx' + yy' to find values of for which r = 0 (i.e., the origin) is unstable, stable, or asymptotically stable (if r starts near 0, does it grow or shrink?).

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