Transcription of 9.3-4: Phase Plane Portraits - Colorado State University
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9.3-4: Phase Plane Portraits - Colorado State University
: Phase Plane PortraitsClassification of 2d Systems:x =Ax,A=[a bc d]:T=a+d,D=ad bc,p( ) = 2 T +DCase A:T2 4D >0 real distinct eigenvalues 1,2= (T T2 4D)/2 General Solution:(v1,v2: eigenvectors)x(t) =c1e 1tv1+c2e 2tv2L1,2: Full lines generated byv1,2 Half line trajectories:ifc2= 0 x(t) =c1e 1tv1 trajectory is half lineH1+={x= v1| >0}ifc1>0H1 ={x= v1| <0}ifc1<0 Same forH2 ifc1= 0,c2>0 or<0 The 4 half line trajectories separate4 regions ofR2y x v1 v2 H1+ H1 H2 H2+ 2v2 v2 v1 2v1 v2 2v2 v1 2v1 x y Phase portrait:Sketch trajectories. Indicatedirection of motionby arrows point-ing in the direction of increasingtDirection of Motion on Half LineTrajectories: If 1>0 thenx(t) =c1e 1tv1 moves out to fort (outwards arrow onH1+) approaches0fort If 1<0 thenx(t) =c1e 1tv1 approaches0fort (inwards arrow onH1+) moves out to fort 1 Subcases of Case ASaddle 1>0> 2 Half line trajectoriesy x L1 L2 Generic TrajectoriesL1 L2 x y Generic trajectoryineach region appro
9.3-4: Phase Plane Portraits ... At each trajectory the direction of motion should be indicated by an arrow. • In the case of centers, sketch a few closed trajectories with the right direction of rotation. For spirals, one generic trajectory is sufficient.
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