hw5 - r = 1. What is its period T ? (b) (10 pts) Find that...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Applied Dynamical Systems - ME215A Fall 2010 Homework #5 - Due Wednesday November 3rd in class 1. (20 pts total) Consider the system ˙ x = x + y - x 3 - 6 xy 2 (1) ˙ y = - x 2 + 2 y - 8 y 3 - x 2 y (2) (a) (5 pts) Show that ( x, y ) = (0 , 0) is an unstable fixed point. (b) (10 pts) Let V = x 2 + 2 y 2 . Show that ˙ V 0 on the curve V = 1, and ˙ V 0 on the curve V = 1 / 2. (c) (5 pts) Given that there are no other fixed points besides the one at ( x, y ) = (0 , 0), what does the result in (b) imply about the existence of periodic orbits? 2. (25 pts total) Consider the system ˙ r = μr (1 - r ) , (3) ˙ θ = ω, (4) where ( r, θ ) are standard polar coordinates. (a) (5 pts) Show that there is a periodic orbit with
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: r = 1. What is its period T ? (b) (10 pts) Find that exact Poincar´ e map r n +1 = P ( r n ) using the Poincar´ e section Σ at θ = 0. (c) (5 pts) By evaluating the derivative of P at the periodic orbit, determine, in terms of μ , when the periodic orbit is stable and unstable. (d) (5 pts) Linearize (3) about r = 1 by setting r = 1 + y . Solve the resulting equation for y ( t ). How is y ( T ) related to y (0)? How is this related to the results of (c)? 1...
View Full Document

This note was uploaded on 12/29/2011 for the course ME 215a taught by Professor Moehlis,j during the Fall '08 term at UCSB.

Ask a homework question - tutors are online