Marginalization on Bifurcation Diagrams: A New Perspective on Infinite-Horizon Prediction

This work addresses the problem of accurate infinite-horizon forecasting of dynamical systems with uncertain parameters. We introduce an intermediate representation of the probability distribution though the marginalization onto the sparse bifurcation structure of an ordinary differential equation (ODE) through integration over regions of attraction. With operations on this representation naturally completed in the space of parameter and state, metrics for the space of limiting behavior can be directly applied. Both limit points and limit cycles are investigated using the Hausdorff distance to treat them in a unified manner. The technique is further applied to stability detection, resulting in a likelihood ratio test.