Numerical Methods for Parameter Estimation in Dynamical Systems Using Measurements at Bifurcation Points

Mathematical models play a crucial role in understanding the complex dynamics of real-world systems. One of the main challenges in mathematical modeling is the estimation of model parameters ensuring that model predictions align well with observed data. Traditional approaches to parameter estimation typically rely on measurements of observables over multiple points in time. However, obtaining such data can be challenging or even infeasible for many experimental systems, particularly those that operate on a fast timescale. For such experimental systems, we propose an alternative approach to parameter estimation that uses the values of applied external controls at bifurcation points, instead of time-series data to calibrate the mathematical models. Bifurcations are, in fact, a valuable source of information, particularly in systems that exhibit bistability, hysteresis, and oscillations. Therefore, in this thesis, we formulate a constrained nonlinear least-squares problem to estimate model parameters by fitting the measured control values at bifurcation points to the corresponding theoretical predictions of the model. We solve this optimization problem using the generalized Gauss-Newton method with efficient structure-exploitation. To ensure reliable convergence, we combine optimization solvers with numerical continuation methods to create a robust numerical strategy for generating initial guesses for the optimization variables. Furthermore, we implement this parameter estimation framework in an open-source software package called bifit. This software enables researchers to easily apply, adapt, and extend the methods we develop in this thesis. Using bifit, we also demonstrate the effectiveness of our approach through four case studies across various fields. Finally, we adapt the standard optimal experimental design approach to our bifurcation-based framework, enabling researchers to strategically select new measurement points to minimize parameter uncertainty. Overall, this thesis provides a new perspective on the kind of data that can be used for model calibration and lays the groundwork for further advancements in parameter estimation and experimental design using bifurcation points.