Computation of homoclinic and heteroclinic orbits for flows

Transcription

1 Computation of homoclinic and heteroclinic orbits for flows Jean-Philippe Lessard BCAM BCAM Mini-symposium on Computational Math February 1st, 2011

2 Rigorous Computations Connecting Orbits Compute a set of equilibria.

3 Rigorous Computations Connecting Orbits Compute a set of equilibria. Local representation of the invariant manifolds. Parameterization method

4 Rigorous Computations Connecting Orbits Compute a set of equilibria. Local representation of the invariant manifolds. Parameterization method Connecting orbits between the equilibria? Boundary value problem Fixed point method

5 Local representation of the invariant manifolds Parameterization method Differential equation: u = g(u). Local parameterization of an invariant manifold at an equilibrium solution: P (θ) = α N n a α θ α θ R n : parameter n: dimension of the manifold

6 Local representation of the invariant manifolds Parameterization method Differential equation: u = g(u). Local parameterization of an invariant manifold at an equilibrium solution: P (θ) = α N n a α θ α θ R n : parameter n: dimension of the manifold Invariance Equation } g P (θ) =P L u (θ) P (0) = u 0 DP(0) = ξ u 0 : equilibrium solution Solve for the {a α } α N n Get the first terms exactly Asymptotic exponential decay ξ: tangent vectors to the manifold.

7 Formulation: projected boundary value problem

8 Formulation: projected boundary value problem u = g(u), lim t ± u(t) =u ± u u +

9 Formulation: projected boundary value problem u = g(u), lim t ± u(t) =u ± P 1 (θ 1 ) P 2 (θ 2 ) W u (u ) W s (u + ) (n 1 -dimensional) (n 2 -dimensional) u u +

10 Formulation: projected boundary value problem u = g(u), lim t ± u(t) =u ± P 1 (θ 1 )= P 2 (θ 2 )= a (1) 1 α N n a (2) α N n 2 α θ α 1 : local parameterization of W u (u ). α θ α 2 : local parameterization of W s (u + ). P 1 (θ 1 ) P 2 (θ 2 ) W u (u ) W s (u + ) (n 1 -dimensional) (n 2 -dimensional) u u +

11 Formulation: projected boundary value problem u = g(u), lim t ± u(t) =u ± P 1 (θ 1 )= P 2 (θ 2 )= a (1) 1 α N n a (2) α N n 2 α θ α 1 : local parameterization of W u (u ). α θ α 2 : local parameterization of W s (u + ). P 1 (θ 1 ) P 2 (θ 2 ) W u (u ) W s (u + ) (n 1 -dimensional) (n 2 -dimensional) u u +

12 Formulation: projected boundary value problem u = g(u), lim t ± u(t) =u ± P 1 (θ 1 )= α θ1 α : local parameterization of W u (u ). P 2 (θ 2 )= a (1) α N n 1 a (2) α N n 2 α θ α 2 : local parameterization of W s (u + ). u( L) = a (1) α θ α 1 u(l) = a (2) α θ α 2 α N n 1 u = g(u) t [ L, L] α N n 2

13 Formulation: projected boundary value problem u = g(u), lim t ± u(t) =u ± P 1 (θ 1 )= α θ1 α : local parameterization of W u (u ). P 2 (θ 2 )= a (1) α N n 1 a (2) α N n 2 α θ α 2 : local parameterization of W s (u + ). u( L) = a (1) α θ α 1 u(l) = a (2) α θ α 2 α N n 1 u = g(u) t [ L, L] α N n 2 F(θ, u) =0 θ =(θ 1,θ 2 ) R n1+n2 u C[ L, L]

14 Derivation of F(θ, u) =0 Symmetric connecting orbits for the system of second order equations d 2 u dτ 2 = Ψ(u), Ψ:Rn R n lim u(t) t ± =u± R n.

15 Derivation of F(θ, u) =0 Symmetric connecting orbits for the system of second order equations (BVP) d 2 u dτ 2 = Ψ(u), Ψ:Rn R n lim u(t) t ± =u± R n. d 2 u dt 2 = L2 Ψ(u), in [0, 1], u (0) = 0, (even homoclinics) u(1) = P (0) (θ), u (1) = P (1) (θ).

16 Derivation of F(θ, u) =0 Symmetric connecting orbits for the system of second order equations (BVP) d 2 u dτ 2 = Ψ(u), Ψ:Rn R n lim u(t) t ± =u± R n. d 2 u dt 2 = L2 Ψ(u), in [0, 1], u (0) = 0, (even homoclinics) u(1) = P (0) (θ), u (1) = P (1) (θ). F(θ, u) = Define [ F : R n C[0, 1] n R n C[0, 1] n P (1) (θ) L Ψ(u(s))ds P (0) (θ)+(t 1)L 2 t 0 Ψ(u(s))ds + L2 1 (s 1)Ψ(u(s))ds u t by ]

17 Finite dimensional reduction X = R n C[0, 1] n : Infinite dimensional Banach space

18 Finite dimensional reduction X = R n C[0, 1] n : Infinite dimensional Banach space h :0=t 0 <t 1 <t 2 < <t m =1 h = } max {t k t k 1 }. k=1,...,m Mesh on [0,1] S h : Piecewise linear functions on h Π h : C[0, 1] S h : Interpolation at the mesh points

19 Finite dimensional reduction X = R n C[0, 1] n : Infinite dimensional Banach space h :0=t 0 <t 1 <t 2 < <t m =1 h = } max {t k t k 1 }. k=1,...,m Mesh on [0,1] S h : Piecewise linear functions on h Π h : C[0, 1] S h : Interpolation at the mesh points X = R n C[0, 1] n Π m = I (Π h ) n Π =0 (I Π h ) n X = X m X X m = R n (S h ) n X = {0} n {(I Π h )C[0, 1]} n

20 Finite dimensional reduction X = R n C[0, 1] n : Infinite dimensional Banach space h :0=t 0 <t 1 <t 2 < <t m =1 h = } max {t k t k 1 }. k=1,...,m Mesh on [0,1] S h : Piecewise linear functions on h Π h : C[0, 1] S h : Interpolation at the mesh points X = R n C[0, 1] n Π m = I (Π h ) n Π =0 (I Π h ) n X = X m X X m = R n (S h ) n X = {0} n {(I Π h )C[0, 1]} n Π m F : X m X m

21 Fixed point operator T (θ, u) =(θ, u) X m = R n(m+2) Π mf : R n(m+2) R n(m+2)

22 Fixed point operator T (θ, u) =(θ, u) X m = R n(m+2) Π mf : R n(m+2) R n(m+2) Numerical ingredients Π m F(ˆθ, û h ) 0; DΠ m F(ˆθ, û h ). DΠ m F(ˆθ, û h ). An approximation solution The Jacobian matrix The approximate inverse A m is injective. A m of

23 Fixed point operator T (θ, u) =(θ, u) X m = R n(m+2) Π mf : R n(m+2) R n(m+2) Numerical ingredients Π m F(ˆθ, û h ) 0; DΠ m F(ˆθ, û h ). DΠ m F(ˆθ, û h ). An approximation solution The Jacobian matrix The approximate inverse A m is injective. Define the Newton-like operator A m T (θ, u) def = (Π m A mπ m F)(θ, u)+π (F(θ, u)+(θ, u)). of T : X X by

24 Fixed point operator T (θ, u) =(θ, u) X m = R n(m+2) Π mf : R n(m+2) R n(m+2) Numerical ingredients Π m F(ˆθ, û h ) 0; DΠ m F(ˆθ, û h ). DΠ m F(ˆθ, û h ). An approximation solution The Jacobian matrix The approximate inverse A m is injective. Define the Newton-like operator A m T (θ, u) def = (Π m A mπ m F)(θ, u)+π (F(θ, u)+(θ, u)). of T : X X Lemma: T (θ, u) =(θ, u) F (θ, u) =0. by

25 Fixed point operator T (θ, u) =(θ, u) X m = R n(m+2) Π mf : R n(m+2) R n(m+2) Numerical ingredients Π m F(ˆθ, û h ) 0; DΠ m F(ˆθ, û h ). DΠ m F(ˆθ, û h ). An approximation solution The Jacobian matrix The approximate inverse A m is injective. Define the Newton-like operator A m T (θ, u) def = (Π m A mπ m F)(θ, u)+π (F(θ, u)+(θ, u)). of T : X X Lemma: T (θ, u) =(θ, u) F (θ, u) =0. proof. Since the matrix is injective, then Π m T (θ, u) = (Π m A mπ m F)(θ, u) =Π m (θ, u) Π m F(θ, u) =0. Also, one has that A m Π T (θ, u) =Π (F(θ, u)+(θ, u)) = Π (θ, u) Π F(θ, u) =0. by

26 Q: How to find a ball such that B x (r) T : B x (r) B x (r) is a contraction? ˆx x X B x (r)

27 Q: How to find a ball such that B x (r) T : B x (r) B x (r) is a contraction? B x (r) = x + B(r) x =(ˆθ, û h ) ˆx x X B x (r)

28 Q: How to find a ball such that B x (r) T : B x (r) B x (r) is a contraction? B x (r) = x + B(r) x =(ˆθ, û h ) Ball of radius r centered at 0 in the space X ˆx x X B x (r)

29 Q: How to find a ball such that B x (r) T : B x (r) B x (r) is a contraction? B x (r) = x + B(r) x =(ˆθ, û h ) Ball of radius r centered at 0 in the space X ˆx x X B x (r) A: Radii polynomials {p k (r)} k : upper bounds satisfying [T ( x) x]k + sup b,c B(r) [DT( x + b)c]k r pk (r)

30 Q: How to find a ball such that B x (r) T : B x (r) B x (r) is a contraction? B x (r) = x + B(r) x =(ˆθ, û h ) Ball of radius r centered at 0 in the space X ˆx x X B x (r) A: Radii polynomials {p k (r)} k : upper bounds satisfying [T ( x) x]k + sup b,c B(r) [DT( x + b)c]k r pk (r) Lemma: If there exists r>0such that p k (r) < 0 for all k, then there is a unique ˆx B x (r) s.t.. proof. Banach fixed point theorem. F(ˆx) =0

31 An application: the Gray-Scott model { a t = D A 2 a x k 2 1 ab 2 + k f (a 0 a), b t = D B 2 b x + k 2 1 ab 2 k 2 b,

32 An application: the Gray-Scott model { a t = D A 2 a x k 2 1 ab 2 + k f (a 0 a), b t = D B 2 b x + k 2 1 ab 2 k 2 b, Stationary profiles: Pulses Homoclinic orbits of u 1 = u 1 u 2 2 λ + λu 1 u 2 = 1 γ (u 2 u 1 u 2 2)

33 An application: the Gray-Scott model { a t = D A 2 a x k 2 1 ab 2 + k f (a 0 a), b t = D B 2 b x + k 2 1 ab 2 k 2 b, Stationary profiles: Pulses Homoclinic orbits of u 1 = u 1 u 2 2 λ + λu 1 u 2 = 1 γ (u 2 u 1 u 2 2) ū 1 (x) def = 1 P { } 3γ 1 + Q cosh(x/ γ), ū 2(x) def = Q cosh(x/ γ) v u tric homoclinic solutions of (77) for the steady-state ( ) = (1 Q = 1 9γ x 4 P = {(λ, γ) λ =1/γ and 0 <γ<2/9}, γ =.15 Hale, Peletier, Troy. Exact homoclinic and heteroclinic solutions of the Gray-Scott model for autocatalysis. SIAM Journal of Math. Analysis, 2000.

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35 W u (U 0 ) W s (U 0 ) U 0 U 0 =(u, u,v,v )=(1, 0, 0, 0)

36 U 0

37 U 0

38 U 0

39 U 0

40 U 0

41 U 0

42 U 0

43 U 0

44 U 0

45 U 0

46 U 0

47 Ingredients for the construction of the radii polynomials Explicit decay of the coefficients of the parameterization of the manifolds Estimates of integrals on each mesh intervals: computations & analysis tk+1 tk+1 t k v (1) (s) ds t k (s 1)v (1) (s) ds tk+1 tk+1 t k v (2) (s) ds t k (s 1)v (2) (s) ds tk+1 tk+1 t k v (3) (s) ds t k (s 1)v (3) (s) ds Rigorous computations using interval arithmetic (Intlab) Bounds on the truncation error terms (I Π h )F i ( x) max k=0,...,m 1 max û 2 2 (s) + λ + 2 û 1 (s)û 2 (s) s [t k,t k+1 ] û2 (1/γ) max 2 (s) + 1 2û 1 (s)û 2 (s) (t k+1 t k )ω s [t k,t k+1 ] max (s 1)(û 2 2 (s) + λ) + 2 (s 1)û 1 (s)û 2 (s) s [t k,t k+1 ] (1/γ) max (s 1)û 2 2 (s) + (s 1)(1 2û 1 (s)û 2 (s)) (t k+1 t k )ω s [t k,t k+1 ] 2 max 2/γ { û 1(s) + 2 û 2 (s) } (t k+1 t k )(ω + 1) 2 s [t k,t k+1 ] 2 max 2/γ { (s 1)û 1(s) + 2 (s 1)û 2 (s) } (t k+1 t k )(ω + 1) 2 s [t k,t k+1 ] 3 (t 3/γ k+1 t k )(ω + 1) 3 3 t 3/γ k+1 (1 t k+1 2 ) t k(1 t k 2 ) (ω + 1) 3, = max k=0,...,m 1 { (t k+1 t k ) 2 { L 2 8 sup s [t k,t k+1 ] 8 (t k+1 t k ) 2 sup s [t k,t k+1 ] } d 2 d 2 t F i( x)(s) } Ψ i (û h (s)).

48 Theorem u2 u1 t/l proof. Computation of the radii polynomials and verification of the existence of a positive radius which makes them simultaneously negative.

49 References X. Cabré, E. Fontich & R. de la Llave. The parameterization method for invariant manifolds III. Overview and Applications. Journal of Differential Equations, N. Yamamoto. A numerical verification method for solutions of boundary value problems with local uniqueness by Banach s fixed point theorem. SIAM Journal of Numerical Analysis, M.T. Nakao. Numerical verification methods for solutions of ordinary and partial differential equations. Numer. Funct. Anal. Optim., S. Day, J.-P. L. and K. Mischaikow. Validated continuation for equilibria of PDEs. SIAM Journal of Numerical Analysis, J.B. van den Berg, J.-P. L., J. Mireles James and K. Mischaikow. Rigorous computation of connecting orbits of dynamical systems. In preparation, 2010.

50 Future & ongoing work Continuation of the connecting orbits Connecting orbits between periodic orbits Organizing center for chaos Traveling waves for PDEs The Navier-Stokes equations

51 Collaborators Jan Bouwe van den Berg (VU University Amsterdam) Jason Mireles James (Rutgers University) Konstantin Mischaikow (Rutgers University)

52 Thank you!