Dimensionality Reduction for Dynamical Systems with Parameters

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Abstract

Dimensionality reduction methods allow for the study of high-dimensional systems by producing low-dimensional descriptions that preserve the relevant structure and features of interest. For dynamical systems, attractors are particularly important examples of such features, as they govern the long-term dynamics of the system, and are typically low-dimensional even if the state space is high- or infinite-dimensional. Methods for reduction need to be able to determine a suitable reduced state space in which to describe the attractor, and to produce a reduced description of the corresponding dynamics. In the presence of a parameter space, a system can possess a family of attractors. Parameters are important quantities that represent aspects of the physical system not directly modelled in the dynamics, and may take different values in different instances of the system. Therefore, including the parameter dependence in the reduced system is desirable, in order to capture the model's full range of behaviour.Existing methods typically involve algebraically manipulating the original differential equation, either by applying a projection, or by making local approximations around a fixed-point. In this work, we take more of a geometric approach, both for the reduction process and for determining the dynamics in the reduced space. For the reduction, we make use of an existing secant-based projection method, which has properties that make it well-suited to the reduction of attractors. We also regard the system to be a manifold and vector field, consider the attractor's normal and tangent spaces, and the derivatives of the vector field, in order to determine the desired properties of the reduced system.We introduce a secant culling procedure that allows for the number of secants to be greatly reduced in the case that the generating set explores a low-dimensional space. This reduces the computational cost of the secant-based method without sacrificing the detail captured in the data set. This makes it feasible to use secant-based methods with larger examples.We investigate a geometric formulation of the problem of dimensionality reduction of attractors, and identify and resolve the complications that arise. The benefit of this approach is that it is compatible with a wider range of examples than conventional approaches, particularly those with angular state variables. In turn this allows for application to non-autonomous systems with periodic time-dependence. We also adapt secant-based projection for use in this more general setting, which provides a concrete method of reduction.We then extend the geometric approach to include a parameter space, resulting in a family of vector fields and a corresponding family of attractors. Both the secant-based projection and the reproduction of dynamics are extended to produce a reduced model that correctly responds to the parameter dependence. The method is compatible with multiple parameters within a given region of parameter space. This is illustrated by a variety of examples.

Keyword(s)

Grassmannian; dimensionality reduction; dynamical systems; secant

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