The Dynamics of a Magnetic Field Around a Heteroclinic Structure |
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A₂ angle arctan basin of attraction belong boundary circle closed orbit collection common boundary computation cone configuration of wires connected Consequence Consider containing corresponding crossing curve defined Denote depend direct sum domains dynamics eigenvalues associated eigenvectors equilibrium cone equilibrium points p+ equivalent family of periodic field H field lines field surface Figure Fix(R Fix(S fixed flow given hence heteroclinic orbit hyperbolic equilibrium points implies intersect invariant manifolds involution limex linear vector field magnetic field magnetic-type vector field MICHIGAN Observe parameters periodic orbits phase portrait phase space plane polycycle position Proof properties Proposition radius relation respectively rot(H saddle type equilibrium semiline sets stable and unstable stable submanifold stable/unstable submanifold symmetric Taking tangent Theorem topological type transversal intersections two-dimensional type equilibrium point unbounded unique University of Timişoara unstable sets zero