Reduction of dimension for nonlinear dynamical systems

TL;DR

This paper explores methods for reducing the dimensionality of nonlinear dynamical systems, enabling simpler analysis and computation of solutions, especially for chaotic systems, using differential elimination and symbolic software.

Contribution

It introduces an algorithmic approach for reducing nonlinear systems to single equations or integro-differential forms, facilitating analysis and solution approximation.

Findings

Reduction to single equations is possible in some cases.

Reduced equations improve efficiency in chaos diagnostics.

The approach is implemented in symbolic software like MAPLE and SageMath.

Abstract

We consider reduction of dimension for nonlinear dynamical systems. We demonstrate that in some cases, one can reduce a nonlinear system of equations into a single equation for one of the state variables, and this can be useful for computing the solution when using a variety of analytical approaches. In the case where this reduction is possible, we employ differential elimination to obtain the reduced system. While analytical, the approach is algorithmic, and is implemented in symbolic software such as {\sc MAPLE} or {\sc SageMath}. In other cases, the reduction cannot be performed strictly in terms of differential operators, and one obtains integro-differential operators, which may still be useful. In either case, one can use the reduced equation to both approximate solutions for the state variables and perform chaos diagnostics more efficiently than could be done for the original higher-dimensional system, as well as to construct Lyapunov functions which help in the large-time study of the state variables. A number of chaotic and hyperchaotic dynamical systems are used as examples in order to motivate the approach.