Solving Discontinuous Initial Value Problems with Unique Solutions Is Equivalent to Computing over the Transfinite

We study a precise class of dynamical systems that we call solvable ordinary differential equations. We prove that analog systems mathematically ruled by solvable ordinary differential equations can be used for transfinite computation, solving tasks such as the halting problem for Turing machines and any Turing jump of the halting problem in the hyperarithmetical hierarchy. We prove that the computational power of such analog systems is exactly the one of transfinite computations of the hyperarithmetical hierarchy. It has been proved recently that polynomial ordinary differential equations correspond unexpectedly naturally to Turing machines. Our results show that the more general exhibited class of solvable ordinary differential equations corresponds, even unexpectedly, naturally to transfinite computations. From a wide philosophical point of view, our results contribute to state that the question of whether such analog systems can be used to solve untractable problems (both for complexity for polynomial systems and for computability for solvable systems) is provably related to the question of the relations between mathematical models, models of physics and our real world. More technically, we study a precise class of dynamical systems: bounded initial value problems involving ordinary differential equations with a unique solution. We show that the solution of these systems can still be obtained analytically even in the presence of discontinuous dynamics once we carefully select the conditions that describe how discontinuities are distributed in the domain. We call the class of right-hand terms respecting these natural and simple conditions the class of solvable ordinary differential equations. We prove that there is a method for obtaining the solution of such systems based on transfinite recursion and taking at most a countable number of steps. We explain the relevance of these systems by providing several natural examples and showcasing the fact that these solutions can be used to perform limit computations and solve tasks such as the halting problem for Turing machines and any Turing jump of the halting problem in the hyperarithmetical hierarchy.