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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...
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- Fall '10
- Hamiltonian mechanics, Lagrangian mechanics, Lagrangian, Euler–Lagrange equation, invariant manifolds, corresponding invariant manifolds