Question

6. Consider the system * = -x(x2 + y2 + x - 2) - Y y = -y(x2 + y2 +1 -2) +r. Prove that this system has a periodic orbit. Hint: Convert to polar form.

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Answer 1

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It It Soln dx=-=(x²+y²+x-2)-Y dy = -y (x²+y²+x-2)+x It has periodic orbit, ac²ty²+x-2 =0 =) (x++) y = ( 2 ) As we get in directem field below with trajectories Direction anow from outside and inside towards it it's asymptotically stable. Periodic orbit is stable limit cycle. Convert it to polan form rcoso de -sind.r.do IF TE It y=rsino dy wso.r.do 7 It 자 -roso (r&rcoso-2)-rsino It st!) sind dr -rsino (ritrios9 -2) +ruose ---12) (1) XLOSO +(2)xsino dr _r(oqroso-2) > = -1.5 = coso dr -3.5 1 1 5 -4.5 -3.5 -2.5 -2 1.5 3.5 de = sind.dr =) Loso.dr It sino-rdo teso.rdo = 자 IF = -r dt (1) sino - (2) x woso rdo -r do It L - - It it contains the (0,0) unstable equilibriumpint.