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Undecidability of event detection for ODEs. (English) Zbl 0771.65035

The paper is concerned with the event detection and event location for ordinary differential equations (ODEs) given by \(y'=f(x,y)\), \(y(0)=y_ 0\). The numerical solution of computable ordinary differential equations is not effective. On the other hand, it follows from certain undecidabilities in symbolical computation that symbolical solution of ODEs is also noneffective.
It is shown here that multidimensional ODEs \(Y'=F(X,Y)\) of simple structure are capable of simulating dynamically arbitrary deterministic Turing machines, even with a smooth \(F\). This adds noneffectiveness for ODEs and difficulties for detection of events.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
68W30 Symbolic computation and algebraic computation
03D35 Undecidability and degrees of sets of sentences
34A34 Nonlinear ordinary differential equations and systems
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
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