Regularization of Neutral Delay Differential Equations with Several Delays
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Regularization of Neutral Delay Differential Equations with Several Delays Nicola Guglielmi · Ernst Hairer
Received: 11 September 2012 / Published online: 29 January 2013 © Springer Science+Business Media New York 2013
Abstract For neutral delay differential equations the right-hand side can be multi-valued, when one or several delayed arguments cross a breaking point. This article studies a regularization via a singularly perturbed problem, which smooths the vector field and removes the discontinuities in the derivative of the solution. A low-dimensional dynamical system is presented, which characterizes the kind of generalized solution that is approximated. For the case that the solution of the regularized problem has high frequency oscillations around a codimension-2 weak solution of the original problem, a new stabilizing regularization is proposed and analyzed. Keywords Neutral delay differential equation · Regularization · Singularly perturbed problem · Generalized solution · Codimension-2 weak solution Mathematics Subject Classification (2010)
34K40 · 34K26 · 34E05
1 Introduction We consider systems of neutral delay differential equations y˙ (t) = f y(t), y˙ α1 (y(t)) , . . . , y˙ αm (y(t)) for t > 0 y(t) = ϕ(t) for t ≤ 0
(1)
N. Guglielmi (B) Dipartimento di Matematica Pura e Applicata, Università dell’Aquila, via Vetoio (Coppito) 67010, L’Aquila, Italy e-mail: [email protected] E. Hairer Section de Mathématiques, Université de Genève, 1211, Geneva 24, Switzerland e-mail: [email protected]
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J Dyn Diff Equat (2013) 25:173–192
with smooth functions f (y, z 1 , . . . , z m ), ϕ(t) and α j (y). More general situations, where f also depends on t and on y(α j (y(t))), can be treated as well without any further difficulties. We consider time intervals where the solution satisfies α j (y(t)) < t for all j. The solution y(t) is continuous at t = 0, but its derivative has a jump discontinuity at t = 0 if ϕ(0) ˙ = f ϕ(0), ϕ˙ α1 (ϕ(0)) , . . . , ϕ˙ αm (ϕ(0)) . (2) This article focusses on this situation, and we use the notation ˙ y˙0+ = f ϕ(0), ϕ˙ α1 (ϕ(0)) , . . . , ϕ˙ αm (ϕ(0)) . y˙0− = ϕ(0), As long as α j (y(t)) < 0 for all j we are concerned with the ordinary differential equation (3) y˙ (t) = f y(t), ϕ˙ α1 (y(t)) , . . . , ϕ˙ αm (y(t)) and the classical theory can be applied. An interesting situation arises when for the first time one of the lag terms becomes zero, for example, α1 (y(t1 )) = 0. Such a time instant is called breaking point. Because of (2) the vector field has a jump discontinuity along the manifold M1 = {y; α1 (y) = 0} and we have to distinguish two situations. Either, a classical solution continues to exist in the region α1 (y) > 0, or the vector field points towards the manifold M1 from both sides. In the second case it is possible to define a weak solution evolving in the manifold (see Sect. 2). Consider such a weak solution and assume that at some time t2 the second lag term becomes zero: α2 (y(t2 )) = 0. We then encounter se
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