Continuation of cycle-to-cycle connections in 3D ODEs

HET p. 1/2 Continuation of cycle-to-cycle connections in 3D ODEs Yuri A. Kuznetsov joint work with E.J. Doedel, B.W. Kooi, and G.A.K. van Voorn

HET p. 2/2 Contents Previous works Truncated BVP s with projection BC s The defining BVP in 3D Finding starting solutions with homopoty Implementation in AUTO Example Open questions

HET p. 3/2 Previous works W.-J. Beyn [1994] On well-posed problems for connecting orbits in dynamical systems. In Chaotic Numerics (Geelong, 1993), volume 172 of Contemp. Math. Amer. Math. Soc., Providence, RI, 131 168. T. Pampel [2001] Numerical approximation of connecting orbits with asymptotic rate, Numer. Math. 90, 309 348. L. Dieci and J. Rebaza [2004] Point-to-periodic and periodic-to-periodic connections, BIT Numerical Mathematics 44, 41 62. L. Dieci and J. Rebaza [2004] Erratum: Point-to-periodic and periodic-to-periodic connections, BIT Numerical Mathematics 44, 617 618.

HET p. 4/2 2. Truncated BVP s with projection BC s Some notations Isolated families of connecting orbits Truncated BVP Error estimate

HET p. 5/2 Some notations Consider the (local) flow ϕ t generated by a smooth ODE du dt = f(u, α), f : Rn R p R n.

HET p. 5/2 Some notations Consider the (local) flow ϕ t generated by a smooth ODE du dt = f(u, α), f : Rn R p R n. Let O be a hyperbolic limit cycle with dim W u = m u.

HET p. 5/2 Some notations Consider the (local) flow ϕ t generated by a smooth ODE du dt = f(u, α), f : Rn R p R n. Let O be a hyperbolic limit cycle with dim W u = m u. Let O + be a hyperbolic limit cycle with dim W s + = m + s.

HET p. 5/2 Some notations Consider the (local) flow ϕ t generated by a smooth ODE du dt = f(u, α), f : Rn R p R n. Let O be a hyperbolic limit cycle with dim W u = m u. Let O + be a hyperbolic limit cycle with dim W s + = m + s. Let x ± (t) be periodic solutions (with minimal periods T ± ) corresponding to O ± and M ± = D x ϕ T ± (x) x=x ± (0) (monodromy matrices).

HET p. 5/2 Some notations Consider the (local) flow ϕ t generated by a smooth ODE du dt = f(u, α), f : Rn R p R n. Let O be a hyperbolic limit cycle with dim W u = m u. Let O + be a hyperbolic limit cycle with dim W s + = m + s. Let x ± (t) be periodic solutions (with minimal periods T ± ) corresponding to O ± and M ± = D x ϕ T ± (x) x=x ± (0) (monodromy matrices). Then m + s = n + s + 1 and m u = n u + 1, where n + s and n u are the numbers of eigenvalues of M ± satisfying µ < 1 and µ > 1, resp.

HET p. 6/2 Isolated families of connecting orbits Necessary condition: p = n m + s m u + 2 (Beyn, 1994).

HET p. 6/2 Isolated families of connecting orbits Necessary condition: p = n m + s m u + 2 (Beyn, 1994). The cycle-to-cycle connections in R 3 : W u W s W s + W s ± O O + W u + O ± W u ± (a) heteroclinic (b) homoclinic

HET p. 6/2 Isolated families of connecting orbits Necessary condition: p = n m + s m u + 2 (Beyn, 1994). The cycle-to-cycle connections in R 3 : W u W s W s + W s ± O O + W u + O ± W u ± (a) heteroclinic (b) homoclinic L.P. Shilnikov [1967] On a Poincaré-Birkhoff problem," Math. USSR-Sb. 3, 353-371.

HET p. 7/2 Truncated BVP The connecting solution u(t) is truncated to an interval [τ, τ + ].

HET p. 7/2 Truncated BVP The connecting solution u(t) is truncated to an interval [τ, τ + ]. The points u(τ + ) and u(τ ) are required to belong to the linear subspaces that are tangent to the stable and unstable invariant manifolds of O + and O, respectively: L + (u(τ + ) x + (0)) = 0, L (u(τ ) x (0)) = 0.

HET p. 7/2 Truncated BVP The connecting solution u(t) is truncated to an interval [τ, τ + ]. The points u(τ + ) and u(τ ) are required to belong to the linear subspaces that are tangent to the stable and unstable invariant manifolds of O + and O, respectively: L + (u(τ + ) x + (0)) = 0, L (u(τ ) x (0)) = 0. Generically, the truncated BVP composed of the ODE, the above projection BC s, and a phase condition on u, has a unique solution family (û, ˆα), provided that the ODE has a connecting solution family satisfying the pahase condition and Beyn s equality.

HET p. 8/2 Error estimate If u is a generic connecting solution to the ODE at parameter value α, then the following estimate holds: (u [τ,τ + ], α) (û, ˆα) Ce 2 min(µ τ,µ + τ + ), where is an appropriate norm in the space C 1 ([τ, τ + ], R n ) R p, u [τ,τ + ] is the restriction of u to the truncation interval, µ ± are determined by the eigenvalues of the monodromy matrices M ±. (Pampel, 2001; Dieci and Rebaza, 2004)

HET p. 9/2 3. The defining BVP in 3D u f 0 w + x + u(0) O x 0 w (0) x w w + (0) O + f + 0 x + 0 u(1) It has cycle- and connection-related parts.

HET p. 10/2 Cycle-related equations Periodic solutions: ẋ ± f(x ±, α) = 0, x ± (0) x ± (T ± ) = 0.

HET p. 10/2 Cycle-related equations Periodic solutions: ẋ ± f(x ±, α) = 0, x ± (0) x ± (T ± ) = 0. Adjoint eigenfunctions: µ + = 1 µ + u, µ = 1 µ s ẇ ± + f T u (x ±, α)w ± = 0, w ± (T ± ) µ ± w ± (0) = 0, w ± (0), w ± (0) 1 = 0,.

HET p. 10/2 Cycle-related equations Periodic solutions: ẋ ± f(x ±, α) = 0, x ± (0) x ± (T ± ) = 0. Adjoint eigenfunctions: µ + = 1 µ + u, µ = 1 µ s ẇ ± + f T u (x ±, α)w ± = 0, w ± (T ± ) µ ± w ± (0) = 0, w ± (0), w ± (0) 1 = 0, Projection BC: w ± (0), u(τ ± ) x ± (0) = 0..

HET p. 11/2 Connection-related equations The equation for the connection: u f(u, α) = 0.

HET p. 11/2 Connection-related equations The equation for the connection: u f(u, α) = 0. We need the base points x ± (0) to move freely and independently upon each other along the corresponding cycles O ±.

HET p. 11/2 Connection-related equations The equation for the connection: u f(u, α) = 0. We need the base points x ± (0) to move freely and independently upon each other along the corresponding cycles O ±. We require the end-point of the connection to belong to a plane orthogonal to the vector f(x + (0), α), and the starting point of the connection to belong to a plane orthogonal to the vector f(x (0), α): f(x ± (0), α), u(τ ± ) x ± (0) = 0.

HET p. 12/2 The defining BVP in 3D: λ ± = ln µ ±, s ± = signµ ± ẋ ± T ± f(x ±, α) = 0, x ± (0) x ± (1) = 0, ẇ ± + T ± fu T (x ±, α)w ± + λ ± w ± = 0, w ± (1) s ± w ± (0) = 0, w ± (0), w ± (0) 1 = 0, u T f(u, α) = 0, f(x + (0), α), u(1) x + (0) = 0, f(x (0), α), u(0) x (0) = 0, w + (0), u(1) x + (0) = 0, w (0), u(0) x (0) = 0, u(0) x (0) 2 ε 2 = 0.

HET p. 13/2 4. Finding starting solutions with homopoty Adjoint scaled eigenfunctions. Homotopy to connection. References to homotopy techniques for point-to-point connections: E.J. Doedel, M.J. Friedman, and A.C. Monteiro [1994] On locating connecting orbits", Appl. Math. Comput. 65, 231 239. E.J. Doedel, M.J. Friedman, and B.I. Kunin [1997] Successive continuation for locating connecting orbits", Numer. Algorithms 14, 103 124.

HET p. 14/2 Adjoint scaled eigenfunctions For fixed α and any λ, x ± (τ) = x ± old (τ), w± (τ) 0, and h ± = 0 satisfy ẋ ± T ± f(x ±, α) = 0, x ± (0) x ± (0) = 0, 1 0 ẋ± old (τ), x± (τ) = 0, ẇ ± + T ± f T u (x ±, α)w ± + λw ± = 0, w ± (1) s ± w ± (0) = 0, w ± (0), w ± (0) h ± = 0,

HET p. 14/2 Adjoint scaled eigenfunctions For fixed α and any λ, x ± (τ) = x ± old (τ), w± (τ) 0, and h ± = 0 satisfy ẋ ± T ± f(x ±, α) = 0, x ± (0) x ± (0) = 0, 1 0 ẋ± old (τ), x± (τ) = 0, ẇ ± + T ± f T u (x ±, α)w ± + λw ± = 0, w ± (1) s ± w ± (0) = 0, w ± (0), w ± (0) h ± = 0, A branch point at λ ± 1 corresponds to the adjoint multiplier µ ± = s ± e λ± 1. Branch switching and continuation towards h ± = 1 gives the eigenfunction w ±.

HET p. 15/2 Homotopy to connection in (T, h jk ) ẋ ± T ± f(x ±, α) = 0, x ± (0) x ± (1) = 0, Φ ± [x ± ] = 0, ẇ ± + T ± fu T (x ±, α)w ± + λ ± w ± = 0, w ± (1) s ± w ± (0) = 0, w ± (0), w ± (0) 1 = 0, u T f(u, α) = 0, f(x + (0), α), u(1) x + (0) h 11 = 0, f(x (0), α), u(0) x (0) h 12 = 0, w + (0), u(1) x + (0) h 21 = 0, w (0), u(0) x (0) h 22 = 0.

HET p. 16/2 5. Implementation in AUTO U(τ) F (U(τ), β) = 0, τ [0, 1], where b(u(0), U(1), β) = 0, 1 0 q(u(τ), β)dτ = 0, U( ), F (, ) R n d, b(, ) R n bc, q(, ) R n ic, β R n fp, The number n fp of free parameters β is n fp = n bc + n ic n d + 1. In our primary BVPs: n d = 15, n ic = 0, and n bc = 19 so that n fp = 5.

HET p. 17/2 6. Example: Poincaré homoclinic structure in ecology The standard tri-trophic food chain model: ẋ 1 = x 1 (1 x 1 ) a 1x 1 x 2, 1 + b 1 x 1 ẋ 2 = a 1x 1 x 2 a 2x 2 x 3 d 1 x 2, 1 + b 1 x 1 1 + b 1 x 2 ẋ 3 = a 2x 2 x 3 d 2 x 3, 1 + b 1 x 2 with a 1 = 5, a 2 = 0.1, b 1 = 3, and b 2 = 2.

HET p. 17/2 6. Example: Poincaré homoclinic structure in ecology The standard tri-trophic food chain model: ẋ 1 = x 1 (1 x 1 ) a 1x 1 x 2, 1 + b 1 x 1 ẋ 2 = a 1x 1 x 2 a 2x 2 x 3 d 1 x 2, 1 + b 1 x 1 1 + b 1 x 2 ẋ 3 = a 2x 2 x 3 d 2 x 3, 1 + b 1 x 2 with a 1 = 5, a 2 = 0.1, b 1 = 3, and b 2 = 2. M.P. Boer, B.W. Kooi, and S.A.L.M. Kooijman [1999] Homoclinic and heteroclinic orbits to a cycle in a tri-trophic food chain, J. Math. Biol. 39, 19 38. Yu.A. Kuznetsov, O. De Feo, and S. Rinaldi [2001] Belayakov homoclinic bifurcations in a tritrophic food chain model, SIAM J. Appl. Math. 62, 462 487.

HET p. 18/2 Continuation of cycle-to-cycle connections Homoclinic orbit to the cycle at (d1, d2) = (0.25, 0.0125):

HET p. 18/2 Continuation of cycle-to-cycle connections Homoclinic orbit to the cycle at (d 1, d 2 ) = (0.25, 0.0125): Limit points: d 1 = 0.2809078 and d 1 = 0.2305987.

HET p. 19/2 Homoclinic tangency curve

Detailed bifurcation diagram HET p. 20/2

HET p. 21/2 Open questions n > 3?

HET p. 21/2 Open questions n > 3? Should all this be integrated in AUTO?