Interval Constraint-Solving Strategies for Solving Dynamical Systems

There are abundant phenomena that humans can describe through mathematical models. Dynamical systems are one such type of model, describing the behavior of phenomena that change over time. For example, a scientist can measure and analyze an insect\u27s wing parameters and movements to create a dynamical system of that behavior. We can then use this model in different applications, such as creating a nano vehicle with insect-like propulsion. For many real life problems, there exist analytical solutions. These can represent a full description of the state of a dynamical system at any moment in continuous time. However, in most practical cases, there are no such analytical solutions. Instead, we use numerical methods: to find an approximation of the state of a system at a specific (future) time, we select a series of discrete moments in time between the start and end times, at which we will compute intermediate state approximations. This discretization process can turn even a dynamical system with a single equation into a system of dozens if not hundreds or thousands of equations and artificial variables representing these intermediate states over time. With the aid of computing algorithms and tools, it is possible to solve these complex systems in feasible time. The tools commonly used to solve dynamical systems are efficient and useful, but are not well-suited to solve certain types of problems; for example, identifying changes in the parameters of an unfolding event using on-the-fly observations, or finding the parameters that minimize the power consumption of our nano vehicle. Existing tools can find one solution, but they depend on an initial guess; if multiple solutions exist, they cannot guarantee to have found all of them. On occasion, these methods will not find a solution, and cannot guarantee that they could not find it because no such solution exists. In our work, we use interval methods, which are guaranteed to find solutions if any exists, and guarantee that if none is found then the system has no solution. There are two categories of methods used to solve dynamical systems: step-based methods that compute each state by using explicit discretizations in which each state equation uses previously-computed states; and constraint-solving techniques that work on the entire system of state equations generated using implicit discretization, whose state equations involve previous and future states. The research we present in this document concerns interval constraint-solving techniques used to solve dynamical systems. There are two main drawbacks for using constraint-solving techniques: the first is the amount of time it takes to solve an implicitly discretized dynamical system, which can be impossibly long or just unfeasible. The second is the need for a discretization that is guaranteed to be accurate. Here we present the Sliding Windows algorithm, a promising strategy that uses interval constraint-solving techniques in a novel way to solve a discretized dynamical system in a more feasible amount of time compared to traditional strategies that employ constraint-solvers. We also use this algorithm to compare discretization techniques of various levels of complexity, to identify the ones that provide better accuracy without excessively extending computation time