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Solving neutral delay differential equations with state-dependent delays. (English) Zbl 1166.65034

Summary: We consider a class of neutral delay differential equations with state dependent delays. For such equations the possible discontinuity in the derivative of the solution at the initial point may propagate along the integration interval giving rise to subsequent points, called “breaking points”, where the solution derivative is still discontinuous. As a consequence, in a right neighbourhood of each such point we have to face a Cauchy problem where the equation has a discontinuous right-hand side. In this case the existence and the uniqueness of the solution is no longer guaranteed to the right of such points and hence the solution of the neutral equation may either cease to exist or bifurcate.
After illustrating why uniqueness and existence of the solution is no longer guaranteed for general state-dependent problems and showing a possible way to detect these occurrences automatically, we explain how to generalize/regularize the problem in order to suitably extend the solution beyond the breaking point. This is important, for example, when exploring numerically the presence of possible periodic orbits.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34K13 Periodic solutions to functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
34K40 Neutral functional-differential equations

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RODAS; RADAR5
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References:

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