Numerical Continuation and Normal Form Analysis of Limit Cycle Bifurcations without Computing Poincaré Maps

Numerical Continuation and Normal Form Analysis of Limit Cycle Bifurcations without Computing Poincaré Maps Yuri A. Kuznetsov joint work with W. Govaerts, A. Dhooge(Gent), and E. Doedel (Montreal) LCBIF p.1/47

Contents References Limit cycles of ODEs and their local bifurcations Standard approach using Poincaré maps Continuation and normal form analysis using BVPs Open problems LCBIF p.2/47

1. References Kuznetsov, Yu. A., Govaerts, W., Doedel, E. J., and Dhooge, A. Numerical periodic normalization for codim 1 bifurcations of limit cycles. SIAM J. Numer. Analysis 43 (2005), 1407-1435 Govaerts, W., Kuznetsov, Yu.A., and Dhooge, A. Numerical continuation of bifurcations of limit cycles in MATLAB. SIAM J. Sci. Comput. 27 (2005), 231-252 Doedel, E.J., Govaerts, W., Kuznetsov, Yu.A., and Dhooge, A. Numerical continuation of branch points of equilibria and periodic orbits. Int. J. Bifurcation & Chaos 15 (2005), 841-860 Doedel, E.J., Govaerts, W., and Kuznetsov, Yu.A. Computation of periodic solution bifurcations in ODEs using bordered systems. SIAM J. Numer. Analysis 41 (2003), 401-435 LCBIF p.3/47

2. Limit cycles of ODEs and their local bifurcations dx dt = f(x, α), x Rn, α R m. A limit cycle C 0 corresponds to a periodic solution x 0 (t + T 0 ) = x 0 (t) and has Floquet multipliers {µ 1, µ 2,..., µ n 1, µ n = 1} = σ(m(t 0 )), where Ṁ(t) f x (x 0 (t), α 0 )M(t) = 0, M(0) = I n. Im µ Re µ LCBIF p.4/47

2. Limit cycles of ODEs and their local bifurcations dx dt = f(x, α), x Rn, α R m. A limit cycle C 0 corresponds to a periodic solution x 0 (t + T 0 ) = x 0 (t) and has Floquet multipliers {µ 1, µ 2,..., µ n 1, µ n = 1} = σ(m(t 0 )), where Ṁ(t) f x (x 0 (t), α 0 )M(t) = 0, M(0) = I n. Im µ Fold (LPC): µ 1 = 1; Re µ LCBIF p.4/47

2. Limit cycles of ODEs and their local bifurcations dx dt = f(x, α), x Rn, α R m. A limit cycle C 0 corresponds to a periodic solution x 0 (t + T 0 ) = x 0 (t) and has Floquet multipliers {µ 1, µ 2,..., µ n 1, µ n = 1} = σ(m(t 0 )), where Ṁ(t) f x (x 0 (t), α 0 )M(t) = 0, M(0) = I n. Im µ Fold (LPC): µ 1 = 1; Flip (PD): µ 1 = 1; Re µ LCBIF p.4/47

2. Limit cycles of ODEs and their local bifurcations dx dt = f(x, α), x Rn, α R m. A limit cycle C 0 corresponds to a periodic solution x 0 (t + T 0 ) = x 0 (t) and has Floquet multipliers {µ 1, µ 2,..., µ n 1, µ n = 1} = σ(m(t 0 )), where Ṁ(t) f x (x 0 (t), α 0 )M(t) = 0, M(0) = I n. Im µ Fold (LPC): µ 1 = 1; Flip (PD): µ 1 = 1; Torus (NS): µ 1,2 = e ±iθ 0. θ 0 Re µ LCBIF p.4/47

2.1. Generic LPC bifurcation: µ 1 = 1 C 1 C 2 C 0 W c β W c 0 W c β β(α) < 0 β(α) = 0 β(α) > 0 LCBIF p.5/47

2.2. Generic PD bifurcation: µ 1 = 1 W c β C 0 W0 c Wβ c C 1 β(α) < 0 β(α) = 0 β(α) > 0 LCBIF p.6/47

2.3. Generic NS bifurcation: µ 1,2 = e ±iθ 0 C 0 T 2 β(α) < 0 β(α) = 0 β(α) > 0 LCBIF p.7/47

3. Standard approach using Poincaré maps Poincaré map and its fixed points Continuation of PD and LP bifurcations of maps Continuation of NS bifurcation of maps Smooth normal forms of maps on center manifolds Difficulties using Poincaré maps Reference on finite-dimensional bordering techniques: Govaerts, W. Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM, 2000. LCBIF p.8/47

3.1. Poincaré map and its fixed points Poincaré map P : R n 1 R m R n 1 A limit cycle C 0 corresponds to a fixed point: W c α y y 0 P(y, α) P(y 0, α) y 0 = 0. C 0 Σ Write P(y 0 + η, α 0 ) = y 0 + Aη + 1 2 B(η, η) + 1 6 C(η, η, η) + O( η 4 ), where A = P y (y 0, α 0 ) and, for i = 1, 2,..., n 1, B i (η, ζ) = C i (η, ζ, ω) = n 1 j,k=1 n 1 j,k,l=1 2 P i (y, α 0 ) y j y k 3 P i (y, α 0 ) y j y k y l η j ζ k, y=y0 η j ζ k ω l. y=y0 LCBIF p.9/47

3.2. Continuation of PD and LC bifurcations of maps Defining system: (y, α) R n 1 R 2 P(y, α) y = 0, g(y, α) = 0, where g is defined by solving P y(y, α) ± I n 1 w 1 v1 T 0 v g = 0 1 Vectors v 1, w 1 R n 1 are adapted to make the linear system nonsingular. (g y, g α ) can be computed efficiently using the adjoint linear system. LCBIF p.10/47

3.3. Continuation of NS bifurcation of maps Defining system: (y, α, κ) R n 1 R 2 R P(y, α) y = 0, g 11 (y, α, κ) = 0, g 22 (y, α, κ) = 0, where κ = cos θ 0 and g ij are defined by solving I n 1 2κP y (y, α) + [P y (y, α)] 2 w 1 w 2 r s v1 T 0 0 g 11 g 12 v2 T 0 0 g 21 g 22 = 0 0 1 0 0 1 Vectors v 1,2, w 1,2 R n 1 are adapted to ensure unique solvability. Efficient computation of derivatives of g ij. LCBIF p.11/47

3.4. Smooth normal forms of maps on center manifolds Eigenvectors Normal form Critical coefficients LP Aq = q A T p = p p, q = 1 ξ β + ξ + bξ 2 +O(ξ 3 ), ξ R b = 1 2 p, B(q, q) PD Aq = q A T p = p p, q = 1 ξ (1 + β)ξ + cξ 3 +O(ξ 4 ), ξ R c = 1 6 p, C(q, q, q) + 3B(q, h 2) h 2 = (I n A) 1 B(q, q) NS Aq = e iθ 0 q A T p = e iθ 0 p e iνθ0 1 ν = 1, 2, 3, 4 ξ ξe iθ ( 1 + β + d ξ 2 ) + O( ξ 4 ), ξ C d = 1 2 e iθ 0 p, C(q, q, q) + 2B(q, h 11 ) + B( q, h 20 ) h 11 = (I n A) 1 B(q, q) p, q = 1 h 20 = (e 2iθ 0 I n A) 1 B(q, q) LCBIF p.12/47

LP-coefficient b Assume y 0 = 0 and write P(H) = AH + 1 2 B(H, H) + O( H 3 ), and locally represent the center manifold W0 c H : R R n 1, as the graph of a function u = H(ξ) = ξq + 1 2 h 2ξ 2 + O(ξ 3 ), ξ R, h 2 R n 1. The restriction of the Poincaré map to W0 c is ξ G(ξ) = ξ + bξ 2 + O(ξ 3 ). The invariance of the center manifold W0 c means P(H(ξ)) = H(G(ξ)). LCBIF p.13/47

A(ξq + 1 2 h 2ξ 2 ) + 1 2 B(ξq, ξq) + O( ξ 3 ) = (ξ + bξ 2 )q + 1 2 h 2ξ 2 + O( ξ 3 ) The ξ-terms give the identity: Aq = q. The ξ 2 -terms give the equation for h 2 : (A I n 1 )h 2 = B(q, q) + 2 bq. It is singular and its Fredholm solvability implies where A T p = p, p, q = 1. 1 b = p, B(q, q), 2 LCBIF p.14/47

3.5. Difficulties using Poincaré maps The computation of P can be unstable numerically near saddle or repelling limit cycles. Finite differences give low accuracy for B and C even if the cycle C 0 is stable. Simultaneous solving variational equations for the components of A, B, and C is possible but costly and difficult to implement. LCBIF p.15/47

4. Continuation and normal form analysis using BVPs Continuation of limit cycles in one parameter Simple bifurcation points Continuation of bifurcations in two parameters Periodic normalization on center manifolds LCBIF p.16/47

4.1. Continuation of limit cycles in one parameter Defining system (BVP with IC) [Keller & Doedel, 1981]: u(τ) T f(u(τ), α) = 0, τ [0, 1], u(0) u(1) = 0, 1 0 ũ(τ), u(τ) dτ = 0, where ũ is a reference periodic solution. Linearization w.r.t. (u, T, α): D T f x (u, α) f(u, α) T f α (u, α) δ 0 δ 1 0 0 Int ũ 0 0 LCBIF p.17/47

Discretization via orthogonal collocation Mesh points: 0 = τ 0 < τ 1 < < τ N = 1. Basis points: τ i,j = τ i + j m (τ i+1 τ i ), i = 0, 1,..., N 1, j = 0, 1,..., m. Approximation: u (i) (τ) = m j=0 ui,j l i,j (τ), τ [τ i, τ i+1 ], where l i,j (τ) are the Lagrange basis polynomials and u i,m = u i+1,0. Collocation [de Boor & Swartz, 1973]: ( m j=0 ui,j l i,j i,k)) (ζ T f( m j=0 ui,j l i,j (ζ i,k ), α) = 0, u 0,0 u N 1,m = 0, m j=0 σ i,j ũ i,j, u i,j = 0, N 1 i=0 where ζ i,k, k = 1, 2,..., m, are the Gauss points. LCBIF p.18/47

Sparse Jacobian matrix u 0,0 u 0,1 u 1,0 u 1,1 u 2,0 u 2,1 u 3,0 T α LCBIF p.19/47

4.2. Simple bifurcation points Φ(τ) T f x (u(τ), α 0 )Φ(τ) = 0, Φ(0) = I n, Ψ(τ) + T fx T (u(τ), α 0 )Ψ(τ) = 0, Ψ(0) = I n. LPC: (Φ(1) I n )q 0 = 0, (Φ(1) I n )q 1 = q 0, (Ψ(1) I n )p 0 = 0, (Ψ(1) I n )p 1 = p 0. PD: (Φ(1) + I n )q 2 = 0, (Ψ(1) + I n )p 2 = 0. NS: κ = cos θ 0 (Φ(1) e iθ 0 I n )(q 3 + iq 4 ) = 0, (Ψ(1) e iθ 0 I n )(p 3 + ip 4 ) = 0. We have (I n 2κΦ(1) + Φ 2 (1))q 3,4 = 0. LCBIF p.20/47

4.3. Continuation of bifurcations in two parameters PD and LPC: (u, T, α) C 1 ([0, 1], R n ) R R 2 u(τ) T f(u(τ), α) = 0, τ [0, 1], u(0) u(1) = 0, 1 0 ũ(τ), u(τ) dτ = 0, G[u, T, α] = 0. NS: (u, T, α, κ) C 1 ([0, 1], R n ) R R 2 R u(τ) T f(u(τ), α) = 0, τ [0, 1], u(0) u(1) = 0, 1 0 ũ(τ), u(τ) dτ = 0, G 11 [u, T, α, κ] = 0, G 22 [u, T, α, κ] = 0. LCBIF p.21/47

PD-continuation There exist v 01, w 01 C 0 ([0, 1], R n ), and w 02 R n, such that N 1 : C 1 ([0, 1], R n ) R C 0 ([0, 1], R n ) R n R, D T f x (u, α) w 01 N 1 = δ 0 δ 1 w 02, Int v01 0 is one-to-one and onto near a simple PD bifurcation point. Define G by solving N 1 v 0 = 0 G. 1 LCBIF p.22/47

The BVP for (v, G) can be written in the classical form v(τ) T f x (u(τ), α)v(τ) + Gw 01 (τ) = 0, τ [0, 1], v(0) + v(1) + Gw 02 = 0, 1 0 v 01(τ), v(τ) dτ 1 = 0. One can take and w 02 = 0 w 01 (τ) = Ψ(τ)p 2, v 01 (τ) = Φ(τ)q 2. LCBIF p.23/47

LPC-continuation There exist v 01, w 01 C 0 ([0, 1], R n ), w 02 R n, and v 02, w 03 R such that N 2 : C 1 ([0, 1], R n ) R 2 C 0 ([0, 1], R n ) R n R 2, D T f x (u, α) f(u, α) w 01 δ 0 δ 1 0 w 02 N 2 =, Int f(u,α) 0 w 03 Int v01 v 02 0 is one-to-one and onto near a simple LPC bifurcation point. 0 v Define G by solving N 2 S = 0. 0 G 1 LCBIF p.24/47

NS-continuation There exist v 01, v 02, w 11, w 12 C 0 ([0, 2], R n ), and w 21, w 22 R n, such that N 3 : C 1 ([0, 2], R n ) R 2 C 0 ([0, 2], R n ) R n R 2, D T f x (u, α) w 11 w 12 δ 0 2κδ 1 + δ 2 w 21 w 22 N 3 =, Int v01 0 0 Int v02 0 0 is one-to-one and onto near a simple NS bifurcation point. 0 0 r s Define G jk by solving N 3 G 11 G 12 = 0 0. 1 0 G 21 G 22 0 1 LCBIF p.25/47

Remarks on continuation of bifurcations After discretization via orthogonal collocation, all linear BVPs for G s have sparsity structure that is identical to that of the linearization of the BVP for limit cycles. For each defining system holds: Simplicity of the bifurcation + Transversality Regularity of the defining BVP. Jacobian matrix of each (discretized) defining BVP can be efficiently computed using adjoint linear BVP. Border adaptation using solutions of the adjoint linear BVPs. Actually implemented in MATCONT, also with compiled C-codes for the Jacobian matrices. LCBIF p.26/47

4.4. Periodic normalization on center manifolds Parameter-dependent periodic normal forms for LPC, PD, and NS Critical normal form coefficients References on periodic normal forms: Iooss, G. Global characterization of the normal form for a vector field near a closed orbit. J. Diff. Equations 76 (1988), 47-76. Iooss, G. and Adelmeyer, M. Topics in Bifurcation Theory and Applications. World Scientific, 1992. LCBIF p.27/47

LPC bifurcation T 0 -periodic normal form on Wα: c dτ dt = 1 + ν(α) ξ + a(α)ξ 2 + O(ξ 3 ), dξ dt = β(α) + b(α)ξ 2 + O(ξ 3 ), where a, b R, sign(b) = sign( b). C 1 C 2 C 0 W c β W c 0 W c β β(α) < 0 β(α) = 0 β(α) > 0 LCBIF p.28/47

PD bifurcation 2T 0 -periodic normal form on Wα: c dτ dt = 1 + ν(α) + a(α)ξ 2 + O(ξ 4 ), dξ dt = β(α)ξ + c(α)ξ 3 + O(ξ 4 ), where a, c R, sign(c) = sign( c). W c β C 0 W0 c Wβ c C 1 β(α) < 0 β(α) = 0 β(α) > 0 LCBIF p.29/47

NS bifurcation T 0 -periodic normal form on Wα: c dτ dt dξ dt = 1 + ν(α) + a(α) ξ 2 + O( ξ 4 ), ( = β(α) + iθ(α) T (α) where a R, d C, sign(re(d)) = sign(re( d)). ) ξ + d(α)ξ ξ 2 + O( ξ 4 ), C 0 T 2 β(α) < 0 β(α) = 0 β(α) > 0 LCBIF p.30/47

Critical normal form coefficients At a codimension-one point write f(x 0 (t)+v, α 0 ) = f(x 0 (t), α 0 )+A(t)v+ 1 2 B(t; v, v)+1 6 C(t; v, v, v)+o( v 4 ), where A(t) = f x (x 0 (t), α 0 ) and the components of the multilinear functions B and C are given by and B i (t; u, v) = C i (t; u, v, w) = n j,k=1 n j,k,l=1 2 f i (x, α 0 ) x j x k 3 f i (x, α 0 ) x j x k x l for i = 1, 2,..., n. These are T 0 -periodic in t. u j v k x=x0 (t) u j v k w l, x=x0 (t) LCBIF p.31/47

Fold (LPC): µ 1 = 1 Critical center manifold W c 0 : τ [0, T 0], ξ R x = x 0 (τ) + ξv(τ) + H(τ, ξ), where H(T 0, ξ) = H(0, ξ), τ C 0 H(τ, ξ) = 1 2 h 2(τ)ξ 2 + O(ξ 3 ) Critical periodic normal form on W c 0 : dτ dt dξ dt = 1 ξ + aξ 2 + O(ξ 3 ), = bξ 2 + O(ξ 3 ), where a, b R, while the O(ξ 3 )-terms are T 0 -periodic in τ. ξ W c 0 LCBIF p.32/47

LPC: Eigenfunctions implying where ϕ satisfies v(τ) A(τ)v(τ) f(x 0 (τ), α 0 ) = 0, τ [0, T 0 ], v(0) v(t 0 ) = 0, T0 0 v(τ), f(x 0 (τ), α 0 ) dτ = 0, T0 0 ϕ (τ), f(x 0 (τ), α 0 ) dτ = 0, ϕ (τ) + A T (τ)ϕ (τ) = 0, τ [0, T 0 ], ϕ (0) ϕ (T 0 ) = 0, T0 0 ϕ (τ), v(τ) dτ 1 = 0. LCBIF p.33/47

LPC: Computation of b Substitute into Collect dx dt = x ξ dξ dt + x τ dτ dt ξ 0 : ẋ 0 = f(x 0, α 0 ), ξ 1 : v A(τ)v = ẋ 0, ξ 2 : ḣ 2 A(τ)h 2 = B(τ; v, v) 2af(x 0, α 0 ) + 2 v 2bv. Fredholm solvability condition b = 1 2 T0 0 ϕ (τ), B(τ; v(τ), v(τ)) + 2A(τ)v(τ) dτ. LCBIF p.34/47

LPC: Example in MATCONT (ABC-reaction model) u 1 = u 1 + p 1 (1 u 1 )e u 3, u 2 = u 2 + p 1 (1 u 1 p 5 u 2 )e u 3, u 3 = u 3 p 3 u 3 + p 1 p 4 (1 u 1 + p 2 p 5 u 2 )e u 3. p 3 = 1.5 p 4 = 8.0 p 5 = 0.04 1.25 1.2 1.15 CPC 1.1 p 2 1.05 1 b = 0 at CPC (cusp) 0.95 0.9 0.19 0.192 0.194 0.196 0.198 0.2 0.202 0.204 0.206 p 1 LCBIF p.35/47

Flip (PD): µ 1 = 1 Critical center manifold W c 0 : τ [0, 2T 0], ξ R x = x 0 (τ) + ξw(τ) + H(τ, ξ), where H(2T 0, ξ) = H(0, ξ), τ C 0 H(τ, ξ) = 1 2 h 2(τ)ξ 2 + 1 6 h 3(τ)ξ 3 + O(ξ 4 ) ξ Critical periodic normal form on W c 0 : dτ dt dξ dt = 1 + aξ 2 + O(ξ 4 ), = cξ 3 + O(ξ 4 ), where a, c R, while the O(ξ 4 )-terms are 2T 0 -periodic in τ. W c 0 LCBIF p.36/47

PD: Eigenfunctions w(τ) = v(τ), τ [0, T 0 ], v(τ T 0 ), τ [T 0, 2T 0 ], w (τ) = v (τ), τ [0, T 0 ], v (τ T 0 ), τ [T 0, 2T 0 ], with v(τ) A(τ)v(τ) = 0, τ [0, T 0 ], v(0) + v(t 0 ) = 0, T0 0 v(τ), v(τ) dτ 1 = 0, v (τ) + A T (τ)v (τ) = 0, τ [0, T 0 ], v (0) + v (T 0 ) = 0, T0 0 v (τ), v(τ) dτ 1/2 = 0. LCBIF p.37/47

PD: Quadratic terms ξ 2 : ḣ 2 A(τ)h 2 = B(τ; w, w) 2aẋ 0, τ [0, 2T 0 ]. Since Ker ( d dτ A(τ)) = span{w, ψ = ẋ 0 }, we must have where ψ satisfies 2T0 0 w (τ), B(τ; w(τ), w(τ)) 2aẋ 0 (τ) dτ = 0, 2T0 0 ψ (τ), B(τ; w(τ), w(τ)) 2aẋ 0 (τ) dτ = 0, ψ (τ) + A T (τ)ψ (τ) = 0, τ [0, T 0 ], ψ (0) ψ (T 0 ) = 0, T0 0 ψ (τ), f(x 0 (τ), α 0 ) dτ 1/2 = 0, and is extended to [T 0, 2T 0 ] by periodicity. LCBIF p.38/47

PD: Computation of a and h 2 a = 1 2 The first Fredholm condition holds identically for all a, while the second gives 2T0 0 ψ (τ), B(τ; w(τ), w(τ) dτ = T0 0 ψ (τ), B(τ; v(τ), v(τ) dτ. Define h 2 on [0, T 0 ] as the unique solution to ḣ 2 (τ) A(τ)h 2 (τ) B(τ; v(τ), v(τ)) + 2af(x 0 (τ), α 0 ) = 0, h 2 (0) h 2 (T 0 ) = 0, T0 0 ψ (τ), h 2 (τ) dτ = 0, and extend it by periodicity to [T 0, 2T 0 ]. LCBIF p.39/47

PD: Computation of c Cubic terms: ξ 3 : ḣ 3 A(τ)h 3 = C(τ; w, w, w) + 3B(τ; w, h 2 ) 6aẇ 6cw. The Fredholm solvability condition implies 6c = or 2T0 0 2T0 0 w (τ), C(τ; w(τ), w(τ), w(τ)) + 3B(τ; w(τ), h 2 (τ)) dτ w (τ), 6aA(τ)w(τ) dτ c = 1 3 T0 0 v (τ), C(τ; v(τ), v(τ), v(τ))+3b(τ; v(τ), h 2 (τ)) 6aA(τ)v(τ) dτ LCBIF p.40/47

Torus (NS): µ 1,2 = e ±iθ 0 with e iνθ0 1, ν = 1, 2, 3, 4 Critical center manifold W c 0 : τ [0, T 0], ξ C x = x 0 (τ) + ξv(τ) + ξ v(τ) + H(τ, ξ, ξ), H(T 0, ξ, ξ) = H(0, ξ, ξ), H(τ, ξ, ξ) = 1 2 h 20(τ)ξ 2 + h 11 (τ)ξ ξ + 1 2 h 02(τ) ξ 2 + 1 6 h 30(τ)ξ 3 + 1 2 h 21(τ)ξ 2 ξ + 1 2 h 12(τ)ξ ξ 2 + 1 6 h 03(τ) ξ 3 + O( ξ 4 ). Critical periodic normal form on W0 c: dτ = 1 + a ξ 2 + O( ξ 4 ), dt dξ = iθ 0 ξ + dξ ξ 2 + O( ξ 4 ), dt T 0 where a R, d C, and the O( ξ 4 )-terms are T 0 -periodic in τ. LCBIF p.41/47

NS: Complex eigenfunctions and v(τ) A(τ)v(τ) + iθ 0 T 0 v(τ) = 0, τ [0, T 0 ], v(0) v(t 0 ) = 0, T0 0 v(τ), v(τ) dτ 1 = 0. v (τ) + A T (τ)v (τ) iθ 0 T 0 v (τ) = 0, τ [0, T 0 ], v (0) v (T 0 ) = 0, T0 0 v (τ), v(τ) dτ 1 = 0. LCBIF p.42/47

NS: Quadratic terms ξ 2 ξ0 : ḣ 20 A(τ)h 20 + 2iθ 0 T 0 h 20 = B(τ; v, v) Since e 2iθ 0 is not a multiplier of the critical cycle, the BVP ḣ 20 (τ) A(τ)h 20 (τ) + 2iθ 0 T 0 h 20 (τ) B(τ; v(τ), v(τ)) = 0, has a unique solution on [0, T 0 ]. ξ 2 : ḣ 11 A(τ)h 11 = B(τ; v, v) aẋ 0 Here Ker ( ) d dτ A(τ) h 20 (0) h 20 (T 0 ) = 0. = span(ϕ = ẋ 0 ). LCBIF p.43/47

NS: Computation of a and h 11 Define ϕ as the unique solution of ϕ (τ) + A T (τ)ϕ (τ) = 0, τ [0, T 0 ], ϕ (0) ϕ (T 0 ) = 0, T0 0 ϕ (τ), f(x 0 (τ), α 0 ) dτ 1 = 0. Fredholm solvability: a = T 0 0 ϕ (τ), B(τ; v(τ), v(τ) dτ. Then find h 11 on [0, T 0 ] from the BVP ḣ 11 (τ) A(τ)h 11 (τ) B(τ; v(τ), v(τ)) + af(x 0 (τ), α 0 ) = 0, h 11 (0) h 11 (T 0 ) = 0, T0 0 ϕ (τ), h 11 (τ) dτ = 0. LCBIF p.44/47

NS: Computation of d Cubic terms: ξ 2 ξ : ḣ21 Ah 21 + iθ 0 T 0 h 21 = 2B(τ; h 11, v) + B(τ; h 20, v) Fredholm solvability condition: + C(τ; v, v, v) 2a v 2dv. d = 1 2 T0 0 + 1 2 v (τ), C(τ; v(τ), v(τ), v(τ)) dτ T0 0 a v (τ), B(τ; h 11 (τ), v(τ)) + B(τ; h 20 (τ), v(τ)) dτ T0 0 v (τ), A(τ)v(τ) dτ + iaθ 0 T 0. LCBIF p.45/47

Remarks on numerical periodic normalization Only the derivatives of f(x, α 0 ) are used, not those of the Poincaré map P(y, α 0 ). Detection of codim 2 points is easy. After discretization via orthogonal collocation, all linear BVPs involved have the standard sparsity structure. One can re-use solutions to linear BVPs appearing in the continuation to compute the normal form coefficients. Actually implemented in MATCONT. LCBIF p.46/47

5. Open problems Automatic differentiation of the Poincaré map vs. BVPs. Periodic normal forms for codim 2 bifurcations of limit cycles. Branch switching at codim 2 points to limit cycle codim 1 continuation. LCBIF p.47/47