18.03 Lecture Notes - Ken Wilber, Phase Portrait, Limit Cycle
Document Summary
Dangers of linearization; limit cycles; chaos; next steps in mathematics: stability, limits of linearization, limit cycles, strange attractors, essential skills, next steps. - "stable" if all nearby trajectories stay near to it and converge to it as t --> infinity. We classified linear phase portraits using the (tr,det) plane. In the (tr,det) plane, the stable region is the upper left quadrant. It is bounded by the straight half-lines where either the eigenvalues are purely imaginary, or one is zero and the other is negative. [2] the method we sketched on friday and monday works well "generically," i. e. almost all the time. It lets you predict what the phase portrait of an autonomous system looks like near equilibrium, most of the time, and whether the equilibrium is stable or not. Example: x" = -y - y^2, y" = x + y^2 . Other examples might show stable spirals instead, or centers. You can"t tell which you get from the linearization alone.
