## Differential equations and dynamical systemsThis is a text in applied mathematics concerning differential equations and dynamical systems. |

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asymptotically stable behavior bifurcation diagram bifurcation value center manifold Chapter Cl(E compute Corollary Definition determine differential equation dynamical system eigenvalues eigenvectors equilibrium point xo family of limit family of periodic family of rotated field f finite number fundamental matrix global phase portrait Hamiltonian system homeomorphism homoclinic loop homoclinic orbit Hopf bifurcation initial value problem interval of existence invariant Lemma Lienard system lim x(t limit cycle linear system Lorenz system maximal interval multiplicity neighborhood node nonhyperbolic periodic orbit nonlinear system one-parameter family open subset parameter periodic orbit planar systems Poincare map Poincare sphere Poincare-Bendixson Theorem Problem Set proof result rotated vector fields satisfies Section 3.4 separatrix cycle shown in Figure solution curves solution x(t stable and unstable stable limit cycle Stable Manifold Theorem structurally stable successive approximations sufficiently small system in Example system x tangent topologically equivalent trajectories two-dimensional unstable manifolds w-limit set zero