example 32.18 A linear centre that is really a stable focus

Rests on

• depends_on proposition 32.17 Classification of planar fixed points ¶
  • 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 definition 13.82 Vector field ¶
        • depends_on definition 9.1 Ordinary differential equation ¶
    • depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
      • depends_on proposition 7.30 Leibniz rule ¶
        • depends_on definition 7.12 Product of functions ¶
        • depends_on equation 7.14 eq:ana-derivh ¶
        • depends_on proposition 7.27 Differentiable implies continuous ¶
        • depends_on proposition 7.6 Algebra of limits ¶
        • proves proof ch:05-real-analysis@proof-14 ¶
      • depends_on theorem 7.34 Rolle ¶
        • depends_on lemma 7.33 Fermat: interior extremum ¶
        • depends_on theorem 7.24 Extreme value theorem ¶
        • proves proof ch:05-real-analysis@proof-18 ¶
      • proves proof ch:05-real-analysis@proof-22 ¶
  • 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 definition 5.37 Linear transformation ¶
      • 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 definition 5.37 Linear transformation ¶ ↺
        • depends_on definition 5.7 Vector subspace ¶
        • proves proof ch:03-linear-algebra-representations@proof-12 ¶
    • 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 definition 5.47 Inverse of a linear transformation ¶
        • depends_on definition 5.37 Linear transformation ¶ ↺
        • proves proof ch:03-linear-algebra-representations@proof-17 ¶
      • depends_on theorem 5.40 Rank–nullity ¶
        • depends_on definition 5.15 Basis ¶
        • depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
        • depends_on lemma 5.38 Exchange and completion ¶
        • proves proof ch:03-linear-algebra-representations@proof-13 ¶
      • 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 ¶
• depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
  • depends_on definition 32.12 Linearization ¶ ↺
  • 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 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 7.20 Continuity at a point ¶
        • depends_on definition 7.97 Partial derivative; gradient ¶
      • 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 definition 9.2 Linear equation; homogeneity ¶
        • depends_on remark 9.3 Normal form and first-order systems ¶
        • depends_on theorem 9.8 Picard–Lindelöf ¶
        • proves proof ch:07-odes-sturm-liouville@proof-3 ¶
      • depends_on proposition 9.22 The exponential and its derivative ¶
        • depends_on definition 9.21 Matrix exponential ¶
        • depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
        • depends_on theorem 7.51 Termwise differentiation ¶
        • proves proof ch:07-odes-sturm-liouville@proof-11 ¶
      • proves proof ch:07-odes-sturm-liouville@proof-12 ¶

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