BIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs

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1 BIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs Yuri A. Kuznetsov August, 2010

2 Contents 1. Solutions and orbits: equilibria cycles connecting orbits compact invariant manifolds strange invariant sets 2. Bifurcations of n-dimensional ODEs local equilibrium bifurcations local bifrcations of cycles bifurcations of homoclinic orbits to equilibria some other cases

3 Literature 1. L.P. Shilnikov, A.L. Shilnikov, D.V. Turaev, and L.O. Chua Methods of Qualitative Theory in Nonlinear Dynamics, Part I, World Scientific, Singapore, L.P. Shilnikov, A.L. Shilnikov, D.V. Turaev, and L.O. Chua Methods of Qualitative Theory in Nonlinear Dynamics, Part II, World Scientific, Singapore, V.I. Arnol d, V.S. Afraimovich, Yu.S. Il yashenko, and L.P. Shil nikov Bifurcation theory, In: V.I. Arnol d (ed), Dynamical Systems V. Encyclopaedia of Mathematical Sciences, Springer-Verlag, New York, Yu.A. Kuznetsov Elements of Applied Bifurcation Theory, 3rd ed. Applied Mathematical Sciences 112, Springer-Verlag, New York, Yu.A. Kuznetsov, O. De Feo, and S. Rinaldi (2001), Belyakov homoclinic bifurcations in a tritrophic food chain model, SIAM J. Appl. Math. 62,

4 1. SOLUTIONS AND ORBITS Consider a smooth system X 3 Ẋ = f(x), X R n. Orbits, phase portraits, and topological equivalence are defined as in the case n = 2 X 2 Equilibria: f(x 0 ) = 0 X 1 Definition 1 An equilibrium is called hyperbolic if R(λ) 0 for all eigenvalues of its Jacobian matrix A = f X (X 0 ). Theorem 1 (Grobman-Hartman) If equilibrium X 0 = 0 is hyperbolic, Ẋ = f(x) is locally topologically equivalent near the origin to Ẏ = AY.

5 Stable and unstable invariant manifolds of equilibria: If a hyperbolic equilibrium X 0 has n s eigenvalues with R(λ) < 0 and n u eigenvalues with R(λ) > 0, it has the n s -dimensional smooth invariant manifold W s composed of all orbits approaching X 0 as t, and the n u -dimensional smooth invariant manifold W u composed of all orbits approaching X 0 as t v 1 W u 1 W ss λ 3 λ 2 λ 1 X 0 Av j = λ j v j stable unstable W s v 2 v 3 W u 2

6 Periodic orbits (cycles) The Poincaré map ξ ξ = P(ξ) is defined on a smooth (n 1)-dimensional crossection: ξ P(ξ) 0 Σ P : Σ Σ. If C 0 coresponds to ξ = 0 then P(0) = 0 and P(ξ) = Mξ + O(2) µ 1 µ 2 µ n 1 = exp ( T0 0 (div f)(x0 (t))dt ) > 0 C 0 Definition 2 A cycle is called hyperbolic if µ = 1 for all eigenvalues (multipliers) of the matrix M = P ξ (0). Theorem 2 (Grobman-Hartman for maps) The Poincaré map ξ P(ξ) of a hyperbolic cycle is locally topologically equivalent near the origin to ξ Mξ.

7 Stable and unstable invariant manifolds of cycles: If a hyperbolic cycle C 0 has m s multipliers with µ < 1 and m u multipliers with µ > 1, it has the (m s +1)-dimensional smooth invariant manifold W s composed of all orbits approaching C 0 as t, and the (m u + 1)-dimensional smooth invariant manifold W u composed of all orbits approaching C 0 as t C 0 W u (C 0 ) W u (C 0 ) W s (C 0 ) P(ξ) Σ ξ W s (C 0 ) P(ξ) P 2 (ξ) ξ Σ C 0

8 Connecting orbits Homoclinic orbits are intersections of W u and W s of an equilibrium/cycle. Heteroclinic orbits are intersections of W u and W s of two different equilibria/cycles. Γ 0 Γ 0 W u W u 1 W u 2 X 0 X 1 X 0 W s W s 1 W s 2

9 Generically, the closure of the 2D invariant manifold near a homoclinic orbit Γ 0 to an equilibriun with real eigenvalues (saddle) in R 3 is either simple (orientable) or twisted (non-orientable): Γ 0 Γ 0 W s W s

10 Compact invariant manifolds 1. tori Example: 2D-torus T 2 with periodic or quasi-periodic orbits 2. spheres 3. Klein bottles

11 Strange (chaotic) invariant sets - have fractal structure (not a manifold); - contain infinite number of hyperbolic cycles; - demonstrate sensitive dependence of solutions on initial conditions; - can be attracting (strange attractors); - orbits can be coded by sequences of symbols (symbolic dynamics).

12 2. BIFURCATIONS OF N-DIMENSIONAL ODES Ẋ = f(x, α) Local (equilibrium) bifurcations Center manifold reduction: Let X 0 = 0 be non-hyperbolic with stable, unstable, and critical eigenvalues: I(λ) R(λ) n c n s n u Theorem 3 For all sufficiently small α, there exists a local invariant center manifold W c (α) of dimension n c that is locally attracting if n u = 0, repelling if n s = 0, and of saddle type if n s n u > 0. Moreover W c (0) is tangent to the critical eigenspace of A = f X (0,0).

13 v 3 λ 2 λ 1 v 2 λ 3 λ 1 λ 2 W c x 0 v 1 x 0 Im v 1 W c Re v 1 Remark: W c (0) is not unique; however, all W c (0) have the same Taylor expansion. Theorem 4 If ξ = g(ξ, α) is the restriction of Ẋ = f(x, α) to W c (α), then this system is locally topologically equivalent to ξ = g(ξ, α), ξ R n c, α R m, ẋ = x, x R n s, ẏ = y, y R n u.

14 Codim 1 equilibrium bifurcations: α R f(x,0) = AX B(X, X) + 1 C(X, X, X) + O(4) 6 Fold (saddle-node): λ 1 = 0 and g(ξ,0) = aξ 2 + O(3) If a 0 then the restriction of Ẋ = f(x, α) to its 1D W c (α) is locally topologically equivalent to ξ = β(α) + aξ 2. a > 0, λ 2 < 0 W c W c W c O 1 O 2 0 β < 0 β = 0 β > 0 Computational formula: a = 1 2 q, B(q, q) where Aq = AT p = 0 with p, q = q, q = 1.

15 Andronov-Hopf: λ 1,2 = ±iω (ω > 0) and ξ = g(ξ,0) is locally smoothly equivalent to ż = iωz + c 1 z z 2 + O(5), z C. If l 1 = R(c 1) ω 0 then the restriction of Ẋ = f(x, α) to its 2D W c (α) is { ρ = ρ(β(α) + l1 ρ locally topologically equivalent to 2 ), ϕ = 1. l 1 < 0, λ 3 < 0 C β β < 0 β = 0 β > 0 l 1 = 1 2ω R [ p, C(q, q, q) 2 p, B(q, A 1 B(q, q)) where Aq = iωq, A T p = iωp, p, q = q, q = 1. + p, B( q,(2iωe n A) 1 B(q, q)) ],

16 Codim 2 equilibrium bifurcations: α R 2 1. Cusp: λ 1 = 0, a = 0 (n c = 1) If c 0, then the restriction of Ẋ = f(x, α) to W c (α) is locally topologically equivalent to ξ = β 1 (α) + β 2 (α)ξ + sξ 3, where s = sign(c) = ±1. 2. Bogdanov-Takens: λ 1 = λ 2 = 0 (n c = 2) If ab 0, then the restriction of Ẋ = f(x, α) to W c (α) is locally topologically equivalent to ẋ = y, ẏ = β 1 (α) + β 2 (α)x + x 2 + sxy, where s = sign(ab) = ±1. 3. Bautin: λ 1,2 = ±iω, ω > 0 (n c = 2) If l 2 0, then the restriction of Ẋ = f(x, α) to W c (α) is locally topologically equivalent to ρ = ρ(β 1 (α) + β 2 (α)ρ 2 + sρ 4 ), ϕ = 1, where s = sign(l 2 ) = ±1.

17 4. Fold-Hopf: λ 1 = 0, λ 2,3 = ±iω, ω > 0 (n c = 3) Generically, the restriction of Ẋ = f(x, α) to W c (α) is smoothly orbitally equivalent to ξ = β 1 (α) + ξ 2 + sρ 2 + P(ξ, ρ, ϕ, α), ρ = ρ(β 2 (α) + θ(α)ξ + ξ 2 ) + Q(ξ, ρ, ϕ, α), ϕ = ω 1 (α) + θ 1 (α)ξ + R(ξ, ρ, ϕ, α), where s = ±1, θ(0) 0, ω 1 (0) > 0, P, Q, R = O( (ξ, ρ) 4 ). The bifurcation diagrams depend on O(4)-terms. Big picture is determined by the truncated normal form without the O(4)-terms. There exist invariant tori and homoclinic orbits near the fold-hopf bifurcation.

18 5. Hopf-Hopf: λ 1,2 = ±ω 1, λ 3,4 = ±iω 2, ω j > 0 (n c = 4) Generically, the restriction of Ẋ = f(x, α) to W c (α) is smoothly orbitally equivalent to ṙ 1 = r 1 (β 1 (α) + p 11 (α)r p 12(α)r s 1(α)r 4 2 ) + Φ 1(r, ϕ, α), ṙ 2 = r 2 (β 2 (α) + p 21 (α)r p 22(α)r s 2(α)r 4 1 ) + Φ 2(r, ϕ, α), ϕ 1 = ω 1 (α) + Ψ 1 (r, ϕ, α), ϕ 2 = ω 2 (α) + Ψ 2 (r, ϕ, α) where Φ j = O( r 6 ), Ψ j = O( r ). The bifurcation diagrams depend on Φ j - and Ψ j -terms. Big picture is determined by the truncated normal form without these terms. There exist invariant tori and homoclinic orbits near the Hopf- Hopf bifurcation.

19 Local bifurcations of cycles I(µ) dim W c = m c Critical cases of codim 1: m s m c R(µ) cyclic fold (saddle-node): µ 1 = 1 m u period-doubling: µ 1 = 1 Neimark-Sacker (torus): µ 1,2 = e ±iθ 0, 0 < θ 0 < π

20 Fold bifurcation of cycles: µ 1 = 1 (m c = 1) The restriction of ξ P(ξ,0) to its W c (0) is ξ ξ + bξ 2 + O(3). If b 0 then the restriction of ξ P(ξ, α) to its W c (α) is locally topologically equivalent to ξ ξ + β(α) + bξ 2. C 1 C 2 C0 β < 0 β = 0 β > 0

21 Period-doubling: µ 1 = 1 (m c = 1) The restriction of ξ P(ξ,0) to its W c (0) is locally smoothly equivalent to ξ ξ + cξ 3 + O(5). If c 0 then the restriction of ξ P(ξ, α) to its W c (α) is locally topologically equivalent to ξ (1 + β(α))ξ + cξ 3. C 0 C0 C 0 β < 0 β = 0 β > 0 C 1

22 Torus: µ 1 = 1 (m c = 1) If e ikθ 0 1 for k = 1,2,3,4, then the restriction of the Poincaré map to its W c (α) is locally smoothly equivalent to ( ) ( ρ ρ(1 + β(α) + d(α)ρ 2 ) ( ) ) R(ρ, ϕ, α) +, ϕ ϕ + θ(α) S(ρ, ϕ, α) where θ(0) = θ 0 and R = O(ρ 4 ), S = O(ρ 2 ) C 0 C 0 C 0 β > 0 β = 0 β > 0 T 2

23 Codim 1 bifurcations of homoclinic orbits to equilibria Homoclinic orbit to a hyperbolic equilibrium: central nonleading stable λ 1 µ 1 0 λ 2 µ 2 λ 3 nonleading unstable leading stable leading unstable µ 0 λ0 Definition 3 Saddle quantity σ = R(µ 1 ) + R(λ 1 ). Theorem 5 (Homoclinic Center Manifold) Generically, there exists an invariant finitely-smooth manifold W h (α) that is tangent to the central eigenspace at the homoclinic bifurcation.

24 Saddle homoclinic orbit: σ = µ 1 + λ 1 Assume that Γ 0 approaches X 0 along the leading eigenvectors. W h (0) Γ 0 Γ 0 W h (0) The Poincaré map near Γ 0 : ξ ξ = β + Aξ µ 1 λ where generically A 0, so that a unique hyperbolic cycle bifurcates from Γ 0 (stable in W h if σ < 0 and unstable in W h if σ > 0).

25 3D saddle homoclinic bifurcation with σ < 0: Assume that µ 2 < µ 1 < 0 < λ 1 (otherwise reverse time: t t). W u W u Γ 0 W u C β W s W s W s β < 0 β = 0 β > 0

26 3D saddle homoclinic bifurcation with σ > 0: Assume that µ 2 < µ 1 < 0 < λ 1 (otherwise reverse time: t t). Γ 0 W u C β W u W u A > 0 W s W s W s β < 0 β = 0 β > 0 Γ 0 W u W u W u A < 0 C β W s W s W s β < 0 β = 0 β > 0

27 Saddle-focus homoclinic orbit: σ = R(µ 1 ) + λ 1 3D saddle-focus homoclinic bifurcation with σ < 0: Assume that R(µ 2 ) = R(µ 1 ) < 0 < λ 1 (otherwise reverse time: t t). W u W u W u Γ 0 C β W s W s W s β < 0 β = 0 β > 0

28 3D saddle-focus homoclinic bifurcation with σ > 0: x x x CHAOTIC INVARIANT SET Focus-focus homoclinic orbit: σ = R(µ 1 ) + R(λ 1 ) CHAOTIC INVARIANT SET

29 Homoclinic orbit(s) to a non-hyperbolic equilibrium W ss W u W s Γ 0 Γ 0 One homoclinic orbit: a unique hyperbolic cycle W u W s Γ 1 Γ 0 Several homoclinic orbits: CHAOTIC INVARIANT SET

30 Some other cases C β C 0 C β β < 0 Homoclinic tangency of a hyperbolic cycle: CHAOS β = 0 β > 0 W ss (C 0 ) W u (C 0 ) W ss (C 0 ) W u (C 0 ) W ss (C 0 ) W u (C 0 ) C 0 C 0 C 0 (a) (b) (c) Homoclinics to nonhyperbolic cycle: torus/chaos/cycle

31 Example: Bifurcations in a food chain model The tri-trophic food chain model by Hogeweg & Hesper (1978): ( ẋ 1 = rx 1 1 x ) 1 a 1x 1 x 2, K 1 + b 1 x 1 where ẋ 2 = e 1 a 1 x 1 x b 1 x 1 a 2x 2 x b 2 x 2 d 1 x 2, ẋ 3 = e 2 a 2 x 2 x b 2 x 2 d 2 x 3, x 1 prey biomass x 2 predator biomass x 3 super predator biomass

32 À Ì Ì Ì Ì ½ Ì Ì À Ô Local bifurcations ¼º¼¾ Å ¼º¼½ ¾ ¼º¼½¾ À ¼º¼¼ ¼º ¼º ¼º ¼º ¼º½

33 Ì Ì Ì Ì Local and key global bifurcations ¼º¼½ ¼º¼½ À ¾ ¼º¼½¾ ½ ¼º¼½ ¾ ¼º¼¼ ¼º ¼º ¼º ¼º¾ ¼º½ ½