AS.110.302 Lecture Notes - Lecture 24: Four Color Theorem, Multivibrator, Leea
Document Summary
Autonomoussystemsandstability let day fix y day a xy st f ghavecontinuouspartialderivatives fix. F g t fct theindependentvariable date fix or i wherextek. eeandseek yj st. Stability for i fix thepointxos. t. sc i o iscalledacriticalpoint. Acriticalpointisstableif given c o i 870 se everysolution ict 4ct forwhich. 114701 folk8 existsfor all ezo andsatisfies 11441 toll c. Acriticalpointisasymptoticallystable if it isstableand 7so cost for all solutionskkk 4ct wehave if1197011 toll so thenit. Real distinct r so ar unstable saddle ri r complex at ib. Fx x y x y l x y x 21g x y l x y. Gcxg xc2ty criticalpointsexiston for x y flxyl o u x y acxy 0 xy a03are thelinesx o y 2 xy fo are y i x andy x xy a o u x y f03 coo codc227 so. Letioc1122beacriticalpointfor i fix theset of allpointsbcr2withthepropertythat atrajectory passingthrough convergesto io as e as iscalled a basinofattractionof io asolutionthatbounds abasinofattractioniscalleda separatrix.
