18.034, Honors Differential Equations Prof. Jason Starr Lectures 33 + 34 4/28 + 4/30/04 1. Began to discuss nonlinear systems and the relation to linear systems, e.g. for a pendulum, =, but for Θ and t small, approximately the same as = . ()θωsin2θω2Ö Ö 2. Defined “structurally stable” (I’m not certain what the conventional term is). Given a nonlinear system (or just = (txFx ,' =)()xFfor an autonomous system), a property of the system is structurally stable if for every continuous (or just G(x) for an autonomous system), there exists (txG ,)()GF,00εε= such that the property holds for GFxε+=', 0εε<. Gave the example of the number of equilibrium points. Prop: Let R C IRn be a bounded closed region. If (1) these are no equilibrium points or R∂ (2) these are only finitely many equilibrium points in Int (R), (3) every equilibrium point is nondegenerate, i.e. ε≠∂∂jiRF at the equilibrium point, then 0ε∃ >0 such that for all 00εεε<<−, (1), (2) +(3) hold for GFε+. Moreover the number of equilibrium points is constant. Pf :- Use the implicit function theorem for GFε+ = R ×IR W → 3. Defined orbits + orbital portrait for a system (obviously closely related to the orbital portraits from Ch.3) Defined nullclines. Worked through the example xyx=' 22' yxy −= Guess these are solutions mxy=and solve to get 212=m Orbits look roughly like the contour curves of a monkey saddle. 18.034, Honors Differential Equations Page 1 of 2 Prof. Jason Starr4. Gave algorithm for sketching an orbital portrait for an autonomous 2D system. Step 1 : Find all equilibrium points. Step 2 : For each equilibrium point, draw the “local picture” (if it is nondegenerate + structurally stable) Step 3 : Draw the nullclines and other “fences” (i.e. curves that help to determine basins of attraction). Step 4 : Interpolate between the local pictures to give a rough sketch. Went through the steps for ()1'−=yxx ()1'−=xyy Separatrix separating basins of attraction. 18.034, Honors Differential Equations Page 2 of 2 Prof. Jason
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