They are both stable and unstable manifolds of the

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Unformatted text preview: tangent to ! ! . 5.2 Homoclinic and Heteroclinic Orbits An orbit Γ is heteroclinic if each ! ∈ Γ is backward asymptotic to an invariant set ! and forward asymptotic to an invariant set !. That is, Γ ⊂ ! ! ! ∩ ! ! (!). 3 Ch5Lecs An orbit Γ is homoclinic if each ! ∈ Γ is both forward and backward asymptotic to the same invariant set !. That is Γ ⊂ ! ! ! ∩ ! ! (!). A saddle connection is a heteroclinic orbit between two saddle equilibria. Example (continued). The stable and unstable manifolds of the origin are heteroclinic orbits. They are also saddle connections. Alternately, we could consider the domain of the system to be !×ℝ. 〈Sketch the phase portrait on the cylinder〉 Now the curves !± are homoclinic orbits. They are both stable and unstable manifolds of the saddle. ■ ! Example. The fish. Meiss discusses the Hamiltonian system with ! ! , ! = ! ! ! − ! ! + !! ! , where ! > 0. The equations of motion are != !" !" = ! !" ! = − !" = ! − 3!! ! Linearize about the origin by throwing away the nonlinear terms. The origin is again a saddle, now corresponding to energy ! = 0. Solve for ! from ! ! , ! = 0 to obtain ! = ±! 1 − 2!" . There are two intersection with the !  ­axis: ! = 0 and ! = 2! !! . 〈sketch the phase portrait〉 The homoclinic loop in the right half plane is part of both the stable and unstable manifolds of the origin. In the left half plane these manifolds are separate and go to ∞. ■ Homoclinic and heteroclinic orbits of Hamiltonian systems are easy to discuss because they are level sets of !(! , !). We can generate non ­Hamiltonian examples with homoclinic and heteroclinic orbits by modifying the Hamilton...
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