Theorem 7.2.2 provides no information about the stabilit
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Theorem 7.2.2 provides no information about the stability of a critical point of an almost linear system if that point i
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Theorem 7.2.2 provides no information about the stability of a critical point of an almost linear system if that point is a center of the corresponding linear system. The systems dx dt = y + x(x2 + y2 ), (i) dy dt = ?x + y(x2 + y2 ) and dx dt = y ? x(x2 + y2 ), (ii) dy dt = ?x ? y(x2 + y2 ) show that this must be so. (a) Show that (0, 0) is a critical point of each system and, furthermore, is a center of the corresponding linear system. (b) Show that each system is almost linear. (c) Let r2 = x2 + y2, and note that x dx?dt + y dy?dt = r dr?dt. For system (ii), show that dr?dt < 0 and that r ? 0 as t ? ?; hence the critical point is asymptotically stable. For system (i), show that the solution of the initial value problem for r with r = r0 at t = 0 becomes unbounded as t ? 1 2 r0 2; hence the critical point is unstable.
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