140b_hw1

# 140b_hw1 - 1 CDS 140b Homework Set 1 Due by the end of the...

This preview shows pages 1–2. Sign up to view the full content.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 CDS 140b: Homework Set 1 Due by the end of the first week after the first unit Problems. 1. Consider the following planar, autonomous vector field: ˙ x =- x + y 2 , ˙ y =- 2 x 2 + 2 xy 2 , ( x,y ) ∈ R 2 . (a) Prove that y = x 2 is an invariant manifold for this vector field. (b) Prove that there exists a heteroclinic connection between the equilibrium points ( x,y ) = (0 , 0) and ( x,y ) = (1 , 1). 2. Consider the system of equations in the plane R 2 given by ˙ x = y ˙ y =- 4 x- μy + 6 x 3- 2 x 7 (a) Show that for μ = 0, the system is Hamiltonian and find a Hamiltonian function. (b) Still assuming that μ = 0, write this system as Euler-Lagrange equations and find a Lagrangian function. (c) Assume that μ ≥ 0. Show that solutions exist for all positive time for any initial data. (d) Assume that μ ≥ 0. Show that (0 , 1) is an equilibrium point and find the eigenvalues of the linearized system at this equilibrium. Explain how your answer is consistent with the possible eigenvalue configurations for...
View Full Document

• Fall '10
• list
• Hamiltonian mechanics, Lagrangian mechanics, Lagrangian, Euler–Lagrange equation, invariant manifolds, corresponding invariant manifolds

{[ snackBarMessage ]}

### Page1 / 2

140b_hw1 - 1 CDS 140b Homework Set 1 Due by the end of the...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online