theorem 9.23 Solution of a constant-coefficient system

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:778 · p. 281

Rests on

depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
- depends_on definition 9.2 Linear equation; homogeneity ¶
  - depends_on definition 9.1 Ordinary differential equation ¶
- depends_on remark 9.3 Normal form and first-order systems ¶
- depends_on theorem 9.8 Picard–Lindelöf ¶
  - depends_on definition 9.4 Lipschitz condition ¶
  - depends_on equation 9.3 eq:slt-ode-system ¶
  - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - depends_on definition 7.45 Series ¶
      - depends_on theorem 7.8 Cauchy criterion ¶
      - proves proof ch:05-real-analysis@proof-28 ¶
    - proves proof ch:07-odes-sturm-liouville@proof-1 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-2 ¶
- 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 definition 5.19 Norm ¶
    - depends_on definition 4.33 Vector space ¶
      - depends_on definition 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
  - depends_on proposition 5.125 prop:lin-matrix-algebra ¶
    - depends_on definition 5.121 Algebra ¶
      - depends_on definition 4.4 Internal binary operation; magma ¶
      - depends_on definition 4.33 Vector space ¶ ↺
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶ ↺
    - depends_on equation 5.150 eq:lin-ctesesc2 ¶
    - depends_on equation 5.74 eq:lin-matrix-rep ¶
    - depends_on proposition 5.45 prop:lin-matrix-unique ¶
      - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - depends_on equation 5.29 eq:lin-indepli2 ¶
      - depends_on equation 5.74 eq:lin-matrix-rep ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-16 ¶
    - proves proof ch:03-linear-algebra-representations@proof-55 ¶
- depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶ ↺
- depends_on theorem 7.51 Termwise differentiation ¶
  - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
  - depends_on proposition 7.27 Differentiable implies continuous ¶
    - depends_on definition 7.20 Continuity at a point ¶ ↺
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-11 ¶
  - depends_on proposition 7.48 Ratio test ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
    - depends_on proposition 7.46 Geometric series ¶
      - depends_on corollary A.46 Monotone convergence ¶ ↺
      - depends_on definition 7.45 Series ¶ ↺
      - proves proof ch:05-real-analysis@proof-27 ¶
    - depends_on theorem 7.8 Cauchy criterion ¶ ↺
    - proves proof ch:05-real-analysis@proof-29 ¶
  - depends_on proposition 7.6 Algebra of limits ¶ ↺
  - depends_on theorem 7.50 Power series; radius of convergence ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
    - depends_on proposition 7.46 Geometric series ¶ ↺
    - proves proof ch:05-real-analysis@proof-31 ¶
  - proves proof ch:05-real-analysis@proof-32 ¶
- proves proof ch:07-odes-sturm-liouville@proof-11 ¶
proves proof ch:07-odes-sturm-liouville@proof-12 ¶

Supports

depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
- depends_on corollary 9.27 Linear stability ¶
  - depends_on corollary 9.41 Hill's equation ¶
    - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
  - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
  - depends_on proposition 9.29 The planar classification ¶
  - depends_on theorem 9.39 Floquet ¶
    - depends_on corollary 9.41 Hill's equation ¶ ↺
- depends_on proposition 9.29 The planar classification ¶ ↺
depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
- depends_on proposition 9.88 The model problem, and the error of the composite ¶
depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶ ↺
depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶ ↺
depends_on theorem 9.32 Hartman–Grobman; quoted ¶
- depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
  - depends_on definition 32.22 Bifurcation ¶
    - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶
    - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
    - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
      - depends_on proposition 32.26 The Hopf normal form ¶
  - depends_on example 32.18 A linear centre that is really a stable focus ¶
depends_on theorem 9.26 Structure of the solutions ¶
- depends_on corollary 9.27 Linear stability ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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Edges

type	direction	node	provenance	where
`depends_on`	→	Linear equations: existence on the whole interval	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:795
`depends_on`	→	The exponential and its derivative	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:795
`depends_on`	←	Normal modes of a diagonalizable system	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:855
`depends_on`	←	The distinguished limit, and which end carries the layer	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:3319
`depends_on`	←	Lyapunov's first method: asymptotic stability	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1170
`depends_on`	←	The unmodulated line, and the feet of the tongues	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1585
`depends_on`	←	Hartman–Grobman; quoted	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1223
`depends_on`	←	Structure of the solutions	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:891
`proves`	←	ch:07-odes-sturm-liouville@proof-12	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:798