{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

4-1 Conservation

# 4-1 Conservation - Linearization and Conservation Laws...

This preview shows pages 1–5. Sign up to view the full content.

Linearization and Conservation Laws Integral Curves and Trajectories April 1, 2008 Linearization and Conservation Laws

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Linearization: the one variable case For a single first order autonomous equation like y = f ( y ) = ay ( M - y ) , critical points (fixed points, equilibrium points) are found by solving 0 = f ( y ) = ay ( M - y ) giving y = 0 or y = M . Near y = 0, we can approximate f ( y ) f (0) + f (0)( y - 0) = 0 + aM ( y - 0) = aMy , and see that for y near 0, the solution of y = ay ( M - y ) is close to the solution of y = amy ,— y ( t 0 ) e aM ( t - t 0 ) . -1 -0.5 0 0.5 1 0 2 4 6 8 10 12 t y y’=y(10-y) y’=10 y y’=-10 y Linearization and Conservation Laws
For y near M , we approximate f ( y ) f ( M ) + f ( M )( y - M ) = 0 - aM ( y - M ) Let ∆ y = y - M and we get the linearized equation d (∆ y ) dt = d ( y - y 0 ) dt = - aM y with solution ∆ y = ∆ y ( t 0 ) e - aM ( t - t 0 ) . Note: this is equivalent to y = M + ( y ( t 0 ) - M ) e - aM ( t - t 0 ) . -1 -0.5 0 0.5 1 0 2 4 6 8 10 12 t y y’=y(10-y) y’=10 y y’=-10 y Linearization and Conservation Laws

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Return to Lotka-Volterra x = ax - bxy y = - cy + dxy has critical points (0,0) and ( c / d , a / b ). Near (0,0), we saw that
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern