Undecidable event detection problems for ODEs of dimension one and two

The ability of dynamical systems of various kinds to simulate Turing machines and thus manifest a universal computation power (and beyond) has gathered a lot of interest interest lately see e.g. [16], [5], [6] and [4]. A similar line of investigation for ordinary differential equations was started in [111] and continued in [12] and [13]. In this context the minimum dimension required for universal computation is of interest. The dynamical systems in [5] and [6] are of small dimension and the topic of [4] is to find the smallest dimension for certain types of dynamical systems. The results in this paper show that for ODEs dimension two can be reached and allowing some what complicated events, even dimension one.