MATH-M 344 Chapter Notes - Chapter 9: Haplogroup R1B, Saddle Point, Potential Energy
Document Summary
M344 section 9. 2 notes- autonomous systems and stability exists. Focusing on systems of 2 equations with form {(cid:3031)(cid:3051)(cid:3031)=(cid:1832)(cid:4666)(cid:1876),(cid:1877)(cid:4667) (cid:3031)(cid:3052)(cid:3031)=(cid:1833)(cid:4666)(cid:1876),(cid:1877)(cid:4667: assume (cid:1832),(cid:1833) continuous and have continuous partial derivatives in some domain (cid:1830) in (cid:1876)(cid:1877) plane. Solution written =(cid:4666)(cid:4667)=[(cid:4666)(cid:4667)(cid:2032)(cid:4666)(cid:4667)] representing curve traced by moving point in (cid:1876)(cid:1877) plane: autonomous system- (cid:1832),(cid:1833) independent of (only depend on (cid:1876) and (cid:1877)) If is constant (cid:884) (cid:884) matrix, = is 2-dimensional autonomous system. If any entries of depend on , system nonautonomous: system has associated direction field independent of time; only 1 trajectory passing through (cid:4666)(cid:1876)(cid:2868),(cid:1877)(cid:2868)(cid:4667) All solutions that satisfy initial conditions (cid:1876)(cid:4666)(cid:2868)(cid:4667)=(cid:1876)(cid:2868),(cid:1877)(cid:4666)(cid:2868)(cid:4667)=(cid:1877)(cid:2868) lie on same trajectory, regardless of time (cid:2868) at which they pass through (cid:4666)(cid:1876)(cid:2868),(cid:1877)(cid:2868)(cid:4667) All solutions that start sufficiently close (within distance of (cid:2012)) to stay close (within distance of. Trajectory does not have to approach critical point as ; only must stay within circle of radius. Correspond to constant (equilibrium) solutions of system.
