proposition 32.24 Transcritical and pitchfork bifurcations

Rests on

• 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 13.71 Differentiable curve ¶
        • depends_on definition 13.45 Differentiable map on a topological space ¶
        • depends_on definition 13.48 Differentiable 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 definition 32.3 Dynamical system, phase space, flow ¶ ↺
      • depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
        • depends_on proposition 7.30 Leibniz rule ¶
        • depends_on theorem 7.34 Rolle ¶
        • proves proof ch:05-real-analysis@proof-22 ¶
    • depends_on definition 6.7 Homeomorphism ¶
      • depends_on definition 3.47 Bijective map ¶
      • depends_on definition 3.51 Inverse map ¶
        • depends_on definition 3.47 Bijective map ¶ ↺
        • depends_on definition 3.49 Composition map ¶
        • depends_on definition 3.50 Identity map ¶
      • depends_on definition 6.6 Continuous map ¶
        • depends_on definition 3.53 Preimage ¶
        • depends_on definition 6.2 Open set ¶
        • depends_on definition 6.1 Topological space ¶
    • depends_on theorem 9.32 Hartman–Grobman; quoted ¶
      • depends_on definition 9.30 Hyperbolic equilibrium ¶
        • depends_on definition 7.98 Functions of class $C^{1}$ ¶
        • depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
      • depends_on definition 6.7 Homeomorphism ¶ ↺
      • depends_on theorem 9.23 Solution of a constant-coefficient system ¶
        • depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
        • depends_on proposition 9.22 The exponential and its derivative ¶
        • proves proof ch:07-odes-sturm-liouville@proof-12 ¶
• depends_on theorem 32.13 Linear stability ¶
  • depends_on definition 32.12 Linearization ¶ ↺
  • depends_on theorem 5.77 Criterion for diagonalizability ¶
    • depends_on definition 5.76 Diagonalizable operator ¶
      • depends_on definition 5.56 Endomorphism ¶
        • depends_on definition 5.37 Linear transformation ¶
      • depends_on equation 5.74 eq:lin-matrix-rep ¶
    • depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶ ↺
    • depends_on proposition 5.75 Eigenvectors for distinct eigenvalues are independent ¶
      • depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶ ↺
      • depends_on definition 5.14 Linear independence ¶
        • depends_on definition 5.5 Linear combination ¶
      • proves proof ch:03-linear-algebra-representations@proof-29 ¶
    • proves proof ch:03-linear-algebra-representations@proof-30 ¶
  • proves proof ch:15-nonlinear-dynamics-chaos@proof-4 ¶
• proves proof ch:15-nonlinear-dynamics-chaos@proof-8 ¶

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