Computation of homoclinic and heteroclinic orbits for flows

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

Rigorous Computations Connecting Orbits Compute a set of equilibria.

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

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

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.

Formulation: projected boundary value problem

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

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 +

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 +

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

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.

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 2 1 0 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 ]

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

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

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

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

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)

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)

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)

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

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,

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)

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) 2.5 2 ū 1 (x) def = 1 P { } 3γ 1 + Q cosh(x/ γ), ū 2(x) def = 3 1 + Q cosh(x/ γ) 1.5 1 v u tric homoclinic solutions of (77) for the steady-state ( ) = (1 Q = 1 9γ 2 0.5-4 -2 00 2 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.

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

U 0

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)).

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.

References X. Cabré, E. Fontich & R. de la Llave. The parameterization method for invariant manifolds III. Overview and Applications. Journal of Differential Equations, 2005. 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, 1998. M.T. Nakao. Numerical verification methods for solutions of ordinary and partial differential equations. Numer. Funct. Anal. Optim., 2001. S. Day, J.-P. L. and K. Mischaikow. Validated continuation for equilibria of PDEs. SIAM Journal of Numerical Analysis, 2007. 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.

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

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

Thank you!