  Hybrid systems are traditionally difficult to identify and analyze using
classical dynamical systems theory. Moreover, recently developed model
identification methodologies largely focus on identifying a single set of
governing equations solely from measurement data. In this article, we develop a
new methodology, Hybrid-Sparse Identification of Nonlinear Dynamics
(Hybrid-SINDy), which identifies separate nonlinear dynamical regimes, employs
information theory to manage uncertainty, and characterizes switching behavior.
Specifically, we utilize the nonlinear geometry of data collected from a
complex system to construct a set of coordinates based on measurement data and
augmented variables. Clustering the data in these measurement-based coordinates
enables the identification of nonlinear hybrid systems. This methodology
broadly empowers nonlinear system identification without constraining the data
locally in time and has direct connections to hybrid systems theory. We
demonstrate the success of this method on numerical examples including a
mass-spring hopping model and an infectious disease model. Characterizing
complex systems that switch between dynamic behaviors is integral to overcoming
modern challenges such as eradication of infectious diseases, the design of
efficient legged robots, and the protection of cyber infrastructures.
