DOC PREVIEW
MIT 18 034 - Lecture Notes

This preview shows page 1 out of 2 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 2 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 2 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

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


View Full Document

MIT 18 034 - Lecture Notes

Documents in this Course
Exam 3

Exam 3

9 pages

Exam 1

Exam 1

6 pages

Load more
Download Lecture Notes
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Lecture Notes and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Lecture Notes 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?