MATH-M 344 Chapter Notes - Chapter 9: Saddle Point, Phase Portrait, Exponential Decay
Document Summary
M344 section 9. 1 notes- the phase plane: linear systems. 2-20-17: case 1: real and distinct eigenvalues of same sign, phase portrait- representative set of trajectories (different solutions) Goal- getting qualitative information about solution of linear system : is constant (cid:884) (cid:884) matrix (important to remember) Equilibrium/critical point- for differential equation =(cid:1858)(cid:4666)(cid:4667), (cid:2868) is critical point if (cid:1858)(cid:4666)(cid:2868)(cid:4667)=(cid:882: then (cid:4666)(cid:4667)=(cid:2868) for all is a solution of =(cid:1858)(cid:4666)(cid:4667, for =, (cid:4666)(cid:882),(cid:882)(cid:4667) is critical point, assume det (cid:882) ( invertible); then (cid:4666)(cid:882),(cid:882)(cid:4667) is the only equilibrium. A solution (cid:4666)(cid:4667)=(cid:4666)(cid:2869),(cid:2870)(cid:4667) of = can be viewed as parametric representation (trajectory) for a curve in (cid:2869)(cid:2870) plane (phase plane) (cid:4666)(cid:4667)=(cid:1857)(cid:3118)((cid:2869)(cid:1857)(cid:4666)(cid:3117) (cid:3118)(cid:4667)(cid:2778)+(cid:2870)(cid:2779)), where (cid:2869) (cid:2870)
