MIT18_03S10_c39 - 18.03 Class 39, May 12, 2010 Dangers of...

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18.03 Class 3 9, May 12, 2010 Dangers of linearization; limit cycles; chaos; next steps in mathematics 1. Stability 2. Limits of linearization 3. Limit cycles 4. Strange attractors 5. Essential skills 6. Next steps [1] Let's say that an equilibrium is -- "stable" if all nearby trajectories stay near to it and converge to it as t --> infinity -- "unstable" otherwise 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. The facts: (1) If the linearization is not on the borderline of the stable quadrant, it correctly predicts the stability of the equilibrium. (2) If the eigenvalues of the Jacobian have nonzero real parts - so (tr,det) is neither on the horizontal axis nor on the tr = 0, det > 0 line -- then the phase portrait is a small deformation of the linear phase portrait. Example: x' = -y - y^2, y' = x + y^2 . The linearization at
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This note was uploaded on 10/21/2011 for the course MATH 18.03 taught by Professor Vogan during the Spring '09 term at MIT.

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MIT18_03S10_c39 - 18.03 Class 39, May 12, 2010 Dangers of...

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