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Lorenz Hirsch–Smale–Devaney Chapter 11–14 MATLAB 2026 5 25 3 3 1 4 2 SIR 4 2.1 SI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 SIRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 van der Pol Hopf 9 3.1 Lienard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.4 Poincare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.5 Hopf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 Kepler 14 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.4 W = 1/ r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1

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Chapter 11–14 Hirsch–Smale–Devaney 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5 Lorenz 18 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.2 Lorenz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.5 Poincare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 6 24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 SIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 van der Pol / Hopf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Kepler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Lorenz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 7 MATLAB 26 27 2

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Chapter 11–14 Hirsch–Smale–Devaney Chapter 11–14 SIR van der Pol Hopf Kepler Lorenz . 20 1 SIR 4 van der Pol/Hopf 5 Kepler 4 Lorenz 5 1 30 SIRS Poincare Kepler Lorenz . SIR SIRS ν / β I + S − ( ν / β ) log S van der Pol Hopf f μ ( x ) = x 3 − μx Kepler W = 1/ r Lorenz Chapter 11–14 (1) SIRS (2) Poincare–Bendixson ω - Lorenz (3) Kepler ℓ ̸ = 0 (4) Lorenz Lorenz Poincare Lorenz SIR van der Pol Kepler Lorenz MATLAB 19 SIRS van der Pol Hopf Kepler Lorenz Poincare 3

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Chapter 11–14 Hirsch–Smale–Devaney 1 11–14 11 12 13 14 SIR Chapter 11.1 van der Pol / Hopf Chapter 12.3–12.4 Kepler Chapter 13.3–13.6 Lorenz Chapter 14.1–14.5 = ⇒ = ⇒ = ⇒ . . 2 SIR 2.1 SI S ( t ) I ( t ) R ( t ) t β > 0 ν > 0 SIR SI − βSI + βSI νI S ′ = − βSI, I ′ = βSI − νI, R ′ = νI. (1) 2.1 ( ) . τ = S + I + R (1) ( S + I + R ) ′ = − βSI + ( βSI − νI ) + νI = 0 . τ 4

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Chapter 11–14 Hirsch–Smale–Devaney R = τ − S − I S, I S ′ = − βSI, I ′ = I ( βS − ν ) . (2) . − βSI 2.2 2.2 ( ) . S ′ = F ( S, I ) I ′ = G ( S, I ) F ( S, I ) = 0 S - G ( S, I ) = 0 I - SIR (2) S ′ = 0 ⇐⇒ S = 0 I = 0 , I ′ = 0 ⇐⇒ I = 0 S = ν β . S = ν β (3) ν / β threshold level 2.1 ( ) . I ( t ) > 0 S ( t ) > ν / β I ′ ( t ) > 0 S ( t ) < ν / β I ′ ( t ) < 0 S ( t ) = ν β . . (2) I ′ = I ( βS − ν ) I > 0 I ′ βS − ν . ν / β S I S I 1: SIR S = ν / β 5

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Chapter 11–14 Hirsch–Smale–Devaney 2: S 0 . S ν / β 2.3 SIR S ( t ) I ( t ) I S S > 0 I > 0 d I d S = I ′ S ′ = βSI − νI − βSI = − 1 + ν βS . S I = − S + ν β log S + C. H ( S, I ) = I + S − ν β log S (4) 2.2 (SIR ) . S > 0 I > 0 SIR (4) S ′ = − βSI < 0 S > ν / β S < ν / β 6

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Chapter 11–14 Hirsch–Smale–Devaney 3: SIR S = ν / β . 2.4 SIRS SIR SIRS S ′ = − βSI + μR, I ′ = βSI − νI, R ′ = νI − μR, (5) μ > 0 τ = S + I + R R = τ − S − I S ′ = − βSI + μ ( τ − S − I ) , I ′ = βSI − νI. (6) ( τ, 0) , ( S ∗ , I ∗ ) = ( ν β , μ ( τ − ν / β ) ν + μ ) . (7) τ ≥ ν / β τ = ν / β 7

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Chapter 11–14 Hirsch–Smale–Devaney 4: SIRS S ≥ 0 , I ≥ 0 , S + I ≤ τ ( − βI − μ − βS − μ βI βS − ν ) . ( τ, 0) − μ βτ − ν τ > ν / β . 2.5 . SIR βSI νI ( S, I ) I ′ = I ( βS − ν ) S ν / β S = ν / β I = 0 SIRS 8

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Chapter 11–14 Hirsch–Smale–Devaney 3 van der Pol Hopf 3.1 Lienard 12 RLC x ′ = y − f ( x ) , y ′ = − x. (8) Lienard f ( x ) = x 3 − x, van der Pol x ′ = y − x 3 + x, y ′ = − x. (9) 5: van der Pol x - y = x 3 − x y - x = 0 . f ( x ) f ( x ) = x 3 − x van der Pol 3.2 W ( x, y ) = 1 2 ( x 2 + y 2 ) . Lienard (8) ˙ W = x ( y − f ( x )) + y ( − x ) = − xf ( x ) . 9

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Chapter 11–14 Hirsch–Smale–Devaney xf ( x ) > 0 x ̸ = 0 ˙ W < 0 van der Pol f ( x ) = x 3 − x = x ( x 2 − 1) ˙ W = − x ( x 3 − x ) = x 2 (1 − x 2 ) . (10) | x | < 1 ⇒ ˙ W > 0 , | x | > 1 ⇒ ˙ W < 0 . 6: van der Pol . van der Pol 3.3 (9) y − x 3 + x = 0 , x = 0 , A = ( 1 1 − 1 0 ) . λ 2 − λ + 1 = 0 1/2 > 0 3.1 (van der Pol ) . van der Pol (9) 10

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Chapter 11–14 Hirsch–Smale–Devaney 7: van der Pol (1) (2) (3) Poincare 3.4 Poincare y - v + = { (0 , y ) : y > 0 } . (0 , y 0 ) v + P ( y 0 ) = v + y . Poincare P ( y 0 ) = y 0 . → . Lorenz 11

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Chapter 11–14 Hirsch–Smale–Devaney 3.5 Hopf μ x ′ = y − x 3 + μx, y ′ = − x. (11) f μ ( x ) = x 3 − μx μ = 1 van der Pol A μ = ( μ 1 − 1 0 ) , λ ± = 1 2 ( μ ± √ μ 2 − 4 ) . − 1 ≤ μ < 0 0 < μ ≤ 1 (11) ˙ W = x ( y − x 3 + μx ) + y ( − x ) = x 2 ( μ − x 2 ) . μ < 0 ˙ W < 0 x ̸ = 0 μ > 0 3.2 (Hopf ) . (11) μ 0 γ μ μ → 0 + γ μ 12

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Chapter 11–14 Hirsch–Smale–Devaney 8: Hopf μ < 0 μ > 0 9: Hopf μ 13

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Chapter 11–14 Hirsch–Smale–Devaney . μ > 0 γ μ → 0 μ → 0 + 3.6 . van der Pol W = ( x 2 + y 2 )/2 ˙ W = x 2 (1 − x 2 ) μ μ < 0 μ > 0 Hopf 4 Kepler 4.1 R n X ( t ) V ( t ) = X ′ ( t ) mX ′′ = F ( X ) . (12) U F ( X ) = − grad U ( X ) , E = 1 2 m | V | 2 + U ( X ) . 4.1 ( ) . E . d d t ( 1 2 m | V | 2 + U ( X ) ) = mV · V ′ + grad U ( X ) · X ′ . mV ′ = − grad U ( X ) X ′ = V V · ( − grad U ( X )) + grad U ( X ) · V = 0 . 4.2 L = m ( X × V ) . L X = ( r cos θ, r sin θ ) . m = 1 ℓ = r 2 θ ′ . (13) 14

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Chapter 11–14 Hirsch–Smale–Devaney 4.1 ( ) . A ( t ) t A ′ ( t ) = 1 2 r ( t ) 2 θ ′ ( t ) = 1 2 ℓ. 10: . Kepler Kepler 4.3 X ′′ = − X | X | 3 . (14) U ( X ) = − 1 | X | = − 1 r . E = 1 2 ( ( r ′ ) 2 + ( rθ ′ ) 2 ) − 1 r . (15) ℓ = r 2 θ ′ . ℓ = 0 ℓ ̸ = 0 15

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Chapter 11–14 Hirsch–Smale–Devaney 11: 4.4 W = 1/ r ℓ = r 2 θ ′ ̸ = 0 θ r θ W ( θ ) = 1 r ( θ ) . K = ℓ 2 2 [ ( d W d θ ) 2 + W 2 ] . (16) U = − W K = E − U = E + W (16) ( d W d θ ) 2 + W 2 = 2 ℓ 2 ( E + W ) . (17) θ d 2 W d θ 2 + W = 1 ℓ 2 . (18) . Kepler W = 1/ r θ 16

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Chapter 11–14 Hirsch–Smale–Devaney 12: Kepler 4.5 (18) W ( θ ) = 1 ℓ 2 + A cos θ + B sin θ. W ( θ ) = 1 ℓ 2 + C cos θ. C = 1 ℓ 2 √ 1 + 2 Eℓ 2 . 1 r = 1 ℓ 2 ( 1 + √ 1 + 2 Eℓ 2 cos θ ) . (19) 1 r = 1 κ (1 + e cos θ ) , e = √ 1 + 2 Eℓ 2 . (20) 4.2 (Kepler ) . (14) ℓ ̸ = 0 E < 0 E = 0 E > 0 17

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Chapter 11–14 Hirsch–Smale–Devaney 13: Kepler . ℓ = 0 4.6 . Kepler X ′′ = − X / | X | 3 W = 1/ r W ′′ + W = 1/ ℓ 2 1/ r 1 + e cos θ 5 Lorenz 5.1 Poincare–Bendixson . Poincare–Bendixson ω - Lorenz Lorenz 18

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Chapter 11–14 Hirsch–Smale–Devaney 5.2 Lorenz Lorenz x ′ = σ ( y − x ) , y ′ = rx − y − xz, z ′ = xy − bz, (21) σ > 0 r > 0 b > 0 σ = 10 , b = 8 3 , r = 28 . x, y, z 14: Lorenz . Lorenz 5.3 σ ( y − x ) = 0 , rx − y − xz = 0 , xy − bz = 0 . 19

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Chapter 11–14 Hirsch–Smale–Devaney (0 , 0 , 0) r > 1 Q ± = ( ± √ b ( r − 1) , ± √ b ( r − 1) , r − 1 ) . (22) r = 1 15: Lorenz r r = 1 Lorenz S ( x, y, z ) = ( − x, − y, z ) . ( x ( t ) , y ( t ) , z ( t )) ( − x ( t ) , − y ( t ) , z ( t )) 5.4 P 1 = (0 , 2 , 0) , P 2 = (0 , 2 . 01 , 0) . 20

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Chapter 11–14 Hirsch–Smale–Devaney 16: Lorenz 17: Lorenz 21

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Chapter 11–14 Hirsch–Smale–Devaney 18: Lorenz 5.1 ( ) . . Lorenz 5.5 Poincare Σ Poincare Φ : Σ → Σ . 22

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Chapter 11–14 Hirsch–Smale–Devaney 19: Lorenz Poincare x = 0 x ′ > 0 Lorenz . Lorenz 5.6 Lorenz 5.1 (Lorenz ) . Lorenz Poincare A (1) (2) A (3) A chaotic (1) (2) 23

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Chapter 11–14 Hirsch–Smale–Devaney (3) . Lorenz Lorenz 5.7 . Poincare–Bendixson Lorenz x ′ = σ ( y − x ) y ′ = rx − y − xz z ′ = xy − bz Poincare Lorenz 6 Hirsch–Smale–Devaney 11 14 SIR van der Pol Hopf Kepler Lorenz SIR SIR S I R βSI νI S ′ = − βSI, I ′ = βSI − νI, R ′ = νI. ( S, I ) I ′ = I ( βS − ν ) . I > 0 S ν / β S > ν / β S < ν / β S = ν / β S ′ = − βSI < 0 I + S − ν β log S = constant . 24

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Chapter 11–14 Hirsch–Smale–Devaney S ( t ) I ( t ) SIRS ν / β van der Pol / Hopf Lienard x ′ = y − f ( x ) , y ′ = − x. f ( x ) = x 3 − x van der Pol x ′ = y − x 3 + x, y ′ = − x. W = ( x 2 + y 2 )/2 ˙ W = x 2 (1 − x 2 ) . | x | < 1 ˙ W > 0 | x | > 1 ˙ W < 0 μ x ′ = y − x 3 + μx, y ′ = − x. ( μ 1 − 1 0 ) . μ < 0 μ > 0 Hopf μ < 0 μ > 0 Kepler mX ′′ = F ( X ) . U F = − grad U E = 1 2 m | V | 2 + U ( X ) L = m ( X × V ) ℓ = r 2 θ ′ A ′ = 1 2 r 2 θ ′ X ′′ = − X | X | 3 . 25

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Chapter 11–14 Hirsch–Smale–Devaney − 1/ r E = 1 2 (( r ′ ) 2 + ( rθ ′ ) 2 ) − 1 r . W = 1/ r ℓ = r 2 θ ′ ̸ = 0 r θ W ′′ + W = 1 ℓ 2 . W ( θ ) = 1 ℓ 2 + C cos θ. 1 r = 1 ℓ 2 ( 1 + √ 1 + 2 Eℓ 2 cos θ ) . e = √ 1 + 2 Eℓ 2 Kepler Lorenz Lorenz Poincare–Bendixson Lorenz Lorenz x ′ = σ ( y − x ) , y ′ = rx − y − xz, z ′ = xy − bz. σ = 10 b = 8/3 r = 28 Lorenz Lorenz (0 , 2 , 0) (0 , 2 . 01 , 0) Poincare chaotic SIR van der Pol Kepler Lorenz 7 MATLAB MATLAB selected_ch11_14_figures_revised.m revised_report_figures 26

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Chapter 11–14 Hirsch–Smale–Devaney sir_time_series_v2.png SIR S ( t ) , I ( t ) , R ( t ) sir_phase_v2.png SIR sir_threshold_cases_v2.png sirs_phase_v2.png SIRS vdp_nullclines_v2.png van der Pol vdp_limit_cycle_v2.png van der Pol vdp_energy_balance_v2.png van der Pol hopf_phase_portraits_v2.png μ Hopf hopf_bifurcation_diagram_v2.png Hopf kepler_equal_area_v2.png kepler_effective_potential_v2.png Newton kepler_conics_v2.png Kepler kepler_logic_chain_v2.png Kepler lorenz_attractor_v2.png Lorenz lorenz_time_series_v2.png Lorenz lorenz_sensitivity_traces_v2.png x ( t ) lorenz_sensitivity_distance_v2.png Lorenz lorenz_poincare_section_v2.png Lorenz Poincare lorenz_equilibria_bifurcation_v2.png Lorenz r Morris W. Hirsch, Stephen Smale, Robert L. Devaney, Differential Equations, Dynamical Sys- tems, and an Introduction to Chaos , Third Edition, Academic Press, 2013. Chapter 11–14 27