MATH-M 344 Chapter Notes - Chapter 9: Phase Portrait, Simply Connected Space, Saddle Point
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M344 section 9. 7 notes- periodic solutions and limit cycles. Consider 2-dimensional autonomous system {(cid:3031)(cid:3051)(cid:3031)=(cid:1832)(cid:4666)(cid:1876),(cid:1877)(cid:4667) (cid:3031)(cid:3052)(cid:3031)=(cid:1833)(cid:4666)(cid:1876),(cid:1877)(cid:4667: assume (cid:1832),(cid:1833) have continuous first derivatives in entire (cid:1876)(cid:1877) plane, seek periodic solutions (cid:4666)(cid:1876),(cid:1877)(cid:4667) where there exists period >(cid:882) such that, for all , (cid:4666)(cid:1876),(cid:1877)(cid:4667)(cid:4666)+(cid:4667)=(cid:4666)(cid:1876),(cid:1877)(cid:4667)(cid:4666)(cid:4667) Recall solutions of linear autonomous system =(cid:1827) periodic iff e-values pure imaginary; If e-values not pure imaginary, no non-constant periodic solutions to linear system critical point at origin is center. Predator-prey problems in 9. 5 all behave similarly, though nonlinear- all solutions in quadrant 1 periodic: corresponding trajectories are closed curves in phase plane. Ex. solutions a point (0-dimensional), not closed curve (1-dimensional), but still periodic. Represent events that occur repeatedly or (cid:862)fi(cid:374)al state(cid:863) approa(cid:272)hed (cid:271)y all (cid:862)(cid:374)eigh(cid:271)ori(cid:374)g(cid:863) Constant solution corresponding to a critical point (ie. (cid:4666)(cid:1876),(cid:1877)(cid:4667)=(cid:4666)(cid:1876)(cid:2868),(cid:1877)(cid:2868)(cid:4667)) is periodic; trajectory is (cid:3031)(cid:3051)(cid:3031)= (cid:1877),(cid:3031)(cid:3052)(cid:3031)=(cid:1876: (cid:4666)(cid:1876),(cid:1877)(cid:4667)(cid:4666)(cid:4667)=(cid:1827)(cid:4666)cos, sin(cid:4667)+(cid:1828)(cid:4666)sin,cos(cid:4667, all trajectories circles; all solutions periodic with =(cid:884) Ex. circle: direction of orientation: starting in first quadrant, if (cid:1876)=(cid:882), (cid:3031)(cid:3052)(cid:3031)=(cid:882), (cid:3031)(cid:3051)(cid:3031)= (cid:1855)
