COMPUTING CONNECTING ORBITS IN DELAY ...

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connecting˙orbits

COMPUTING CONNECTING ORBITS IN DELAY DIFFERENTIAL EQUATIONS USING DDE-BIFTOOL

G. SAMAEY∗ AND D. ROOSE Department of Computer Science, Katholieke Universiteit Leuven, Celestijnenlaan 200 A, B-3001 Leuven

Connecting orbits in delay differential equations (DDEs) are approximated using projection boundary conditions, which involve the stable and unstable manifolds of a steady state solution. The stable manifold of a steady state of a DDE is infinitedimensional. We circumvent this problem by reformulating the end conditions using a special bilinear form. The solution of the resulting boundary value problem using a collocation method is implemented in the Matlab package DDE-BIFTOOL. In a model system for intracellular signalling, we show how the method can be used to obtain travelling wave solutions of a delay partial differential equation.

1. Introduction We discuss an algorithm to compute connecting orbits in systems of autonomous delay differential equations (DDEs), x(t) ˙ = f (x(t), x(t − τ ), η),

(1)

where τ > 0 is a (constant) delay, η ∈ Rp represents a number of physical parameters, and f : Rn × Rn × Rp → Rn . Computation of connecting orbits allows to study global bifurcations and travelling waves in delay PDEs. We call a solution x∗ (t) of (1) at η = η ∗ a connecting orbit if there exist steady state solutions x± , such that lim x∗ (t) = x+ .

lim x∗ (t) = x− ,

t→−∞

t→+∞

(2)

Computation of such orbits for ODEs is usually done with a collocation method with projection boundary conditions 1 , e.g. in AUTO 2 and its extension HomCont. We extended this approach to the DDE case 6 and included it in DDE-BIFTOOL, a publicly available Matlab package for bifurcation analysis of DDEs 3 . ∗ Giovanni

Samaey is a Research Assistant of the Fund for Scientific Research - Flanders. 202

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2. Delay differential equations (DDEs) The state of a system, described by (1) is given by a function segment x∗s ∈ Cτ =C([−τ, 0], Rn ). We call Cτ the (infinite-dimensional) state space. When we define the n × n-dimensional matrix ∆ as ∆(x∗ , η, λ) := λI − A0 − A1 e−λτ , (3) ∂f , the characteristic equation where, using f ≡ f (x0 , x1 , η), Ai := ∂x i ∗ (x ,η)

can be written as, 

∆(x∗ , η, λ)v = 0 cH v − 1 = 0

or



∆H (x∗ , η, λ)w = 0 dH w − 1 = 0,

where c and d are normalizing vectors, and a superscript H denotes the complex-conjugate and transpose. This equation has an infinite number of roots λ ∈ C, called the characteristic roots, which form a discrete, countable set in the complex plane. We assume that there are no multiple roots and no roots on the imaginary axis. The associated right and left eigenfunctions are then given by φ(θ) = veλθ , θ ∈ [−τ, 0], resp. ψ(θ) = w H e−λθ , θ ∈ [0, τ ], with φ(θ) ∈ C([−τ, 0]; Cn ) and ψ(θ) ∈ C([0, τ ]; C1×n ) 4 . The steady state solution x∗ is (asymptotically) stable if all characteristic roots have negative real part; otherwise it is unstable. Since the number of characteristic roots in any half plane γ, γ ∈ R, is finite 4 , the stability is always determined by a finite number of roots. 3. Computing connecting orbits in DDEs To approximate a connecting orbit on a finite domain of length T , we apply projection boundary conditions, which approximate the stable and unstable manifolds of the steady states by their eigenspaces. Because of the delay term, these need to be written in terms of solution segments. The initial function segment can be written as a linear combination of the unstable eigenfunctions. However, we cannot write the end conditions for the final function segment in a similar way, because there are infinitely many stable eigenfuctions. Therefore, we use a special bilinear form 4 to require that the final function segment is in the complement of the unstable eigenspace of x+ . This leads to extra conditions of the form Z 0
+ H H wk+ (x(T ) − x+ ) + wk+ e−λk (θ+τ ) A1 (x+ , η) x(T + θ) − x+ dθ = 0 . −τ

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connecting˙orbits

204 + + where (λ+ k ,wk ) are the left eigenvectors and eigenvalues of x . The complete determining system, including additional equations and a numerical convergence analysis is given in Ref. 6 .

4. Example Consider the following reaction-diffusion equation 5 for the intracellular signal transmission in autocrine relays,   ∂r s(x, t − τ ) − s0 (x, t) = −r(x, t) + σ ∂t δ 2 ∂ s ∂s (x, t) − s(x, t) + r(x, t), τs (x, t) = ∂t ∂x2 where τs , s0 and δ are physical parameters, and σ is a sigmoidal function. A continuation in τ of the travelling waves for δ = 0.1, s0 = 0.6 and τs = 0.1 is shown in the figure. The wave speed decreases with increasing delay. The travelling wave is shown for τ ≈ 2.195 · 10−2 , where c ≈ 2.46 · 10−1 . 1 0.9 0.25 0.8 0.7

(r(ξ),s(ξ))

c − velocity

0.2

0.15

0.1

0.6 0.5 0.4 0.3 0.2

0.05 0.1 0 0

0

2

τ

4

6

0

0.2

0.4

ξ

0.6

0.8

1

References 1. W. J. Beyn. The numerical computation of connecting orbits in dynamical systems. IMA Journal of Numerical Analysis, 9:379–405, 1990. 2. E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, and X. Wang. AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), March 1998. 3. K. Engelborghs, T. Luzyanina, and G. Samaey. DDE-BIFTOOL v2.00: a Matlab package for bifurcation analysis of delay differential equations. Technical Report TW-330, Dept. of Computer Science, KULeuven, October 2001. 4. J. K. Hale. Theory of Functional Differential Equations, volume 3 of Applied Mathematical Sciences. Springer-Verlag, 1977. 5. M. Pribyl, C. B. Muratov, S. Y. Shvartsman. Long-range signal transmission in autocrine relays. Biophysical Journal, 84:883-896,2003. 6. G. Samaey, K. Engelborghs, and D. Roose. Numerical computation of connecting orbits in delay differential equations. Numerical Algorithms,30(3-4):335– 352, 2002.