proposition 32.26 The Hopf normal form

Rests on

• depends_on theorem 32.25 Hopf bifurcation, quoted ¶
  • depends_on definition 32.22 Bifurcation ¶
    • depends_on definition 32.3 Dynamical system, phase space, flow ¶
      • depends_on definition 13.82 Vector field ¶
        • depends_on definition 13.81 Vector on a manifold ¶
      • depends_on definition 9.1 Ordinary differential equation ¶
    • depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
      • depends_on definition 32.12 Linearization ¶
        • depends_on definition 32.10 Fixed point ¶
        • depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
      • depends_on definition 6.7 Homeomorphism ¶
        • depends_on definition 3.47 Bijective map ¶
        • depends_on definition 3.51 Inverse map ¶
        • depends_on definition 6.6 Continuous map ¶
      • depends_on theorem 9.32 Hartman–Grobman; quoted ¶
        • depends_on definition 9.30 Hyperbolic equilibrium ¶
        • depends_on definition 6.7 Homeomorphism ¶ ↺
        • depends_on theorem 9.23 Solution of a constant-coefficient system ¶
  • depends_on proposition 32.17 Classification of planar fixed points ¶
    • depends_on definition 32.12 Linearization ¶ ↺
    • depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
      • depends_on definition 5.70 Characteristic polynomial ¶
        • depends_on definition 5.56 Endomorphism ¶
        • depends_on equation 5.19 eq:lin-leibniz-det ¶
        • depends_on equation 5.74 eq:lin-matrix-rep ¶
        • proves proof ch:03-linear-algebra-representations@proof-25 ¶
      • depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
        • depends_on definition 5.56 Endomorphism ¶ ↺
        • depends_on definition 5.39 Kernel, image, nullity, rank ¶
      • depends_on proposition 5.49 Injective, surjective, invertible ¶
        • depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
        • depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
        • depends_on theorem 5.40 Rank–nullity ¶
        • proves proof ch:03-linear-algebra-representations@proof-18 ¶
      • proves proof ch:03-linear-algebra-representations@proof-26 ¶
    • proves proof ch:15-nonlinear-dynamics-chaos@proof-5 ¶
• proves proof ch:15-nonlinear-dynamics-chaos@proof-9 ¶

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