(194e) Data-Driven Local Bifurcation Analysis Via Homeomorphism Learning

Bifurcation is a commonly encountered phenomenon in nonlinear dynamical systems, which refers to the dependence of their qualitative properties (such as contraction, cycling, and chaos) on parameters. Since bifurcations are often associated with severe consequences (e.g., when the chemical reaction is exothermic in a reactor, insufficient cooling may result in runaway), they need to be well predicted by engineers. Plenty of literature in process systems engineering and chemical reaction engineering have revealed the bifurcative nature of reactor dynamics and proposed methods to characterize different types of bifurcations [1, 2]. However, since first-principles models often need to be largely simplified to be amenable for such bifurcation analyses, the applicability can be restrictive. For real-world processes, it may also be difficult to establish highly accurate models. Therefore, in this work, the problem of detecting bifurcations in a data-driven manner is considered.

This work focuses on local bifurcations instead of global ones (i.e., the effect of parametric variations on the global behavior of dynamics such as homoclinic and heteroclinic orbits), leaving the latter category to be investigated in future studies. To detect local bifurcation, the problem is to determine the local topological equivalence between the system under different parameter values. Such a topological equivalence is a continuous and continuously invertible one-to-one mapping, namely a homeomorphism, between the two dynamics. The problem of learning the homeomorphism from data is formulated under two different settings:

• In the first setting, a linear reference model is a priori known, which is the local linearization of the nonlinear model under a reference parameter value. Thus, the problem is to find a transformation of the states under the new system parameter, so that the transformed states evolve according to the given linear dynamics.
• In the second setting, a linearized model is not given. Assuming that there exist n eigenfunctions of the Koopman operator that form a basis of the state space [3], these eigenfunctions can be statistically consistently learned via extended dynamic mode decomposition (EDMD) [4], which also finds the linear dynamics on these eigenfunctions. Hence, the problem is equipped with the same form as the first setting.

The learning of homeomorphism boils down to a convex optimization problem, if the transformation to be learned is considered to be in the span of a set of basis functions. The loss term that captures the discrepancy with the linearized dynamics is quadratic in the coefficients to be optimized. To ensure the invertibility of this mapping to be learned, barrier terms are added to restrict the Jacobian at sample points to be positive definite. Using standard arguments from statistical learning theory (e.g., Hoeffding’s inequality), a probabilistic bound on the generalization error of predicting the system evolution with the learned homeomorphism and linear dynamics, as an evidence of the absence of bifurcation, is proved. The proposed approach is motivated by [5], where matching of Koopman functions was proposed to verify the equivalence of two dynamical systems. A numerical study on a system with pitchfork bifurcation is used to demonstrate the performance of the bifurcation detection approach proposed.

It should be noted that, despite often being implicit, bifurcation analysis is considered as a pre-requisite of process design and controller synthesis of nonlinear systems. As recent research in process control is witnessing a transition towards data-driven methods and algorithms, bifurcation analysis in a data-driven setting is a worth studying problem by the process control community that should be combined with state observation [6] and data-driven control [7] methods.

References

[1] Jensen, K. F., & Ray, W. H. (1982). Chem. Eng. Sci., 37, 199-222.

[2] Balakotaiah, V., Dommeti, S. M., & Gupta, N. (1999). Chaos, 9, 13-35.

[3] Mezić, I. (2020). J. Nonlin. Sci., 30, 2091-2145.

[4] Williams, M. O., Kevrekidis, I. G., & Rowley, C. W. (2015). J. Nonlin. Sci., 25, 1307-1346.

[5] Bollt, E. M., Li, Q., Dietrich, F., & Kevrekidis, I. (2018). SIAM J. Appl. Dyn. Syst., 17, 1925-1960.

[6] Tang, W. (2023). AIChE J., 69, e18224.

[7] Tang, W., & Daoutidis, P. (2022, June). In 2022 American Control Conference (ACC) (pp. 1048-1064).