Chaos in a non-autonomous nonlinear system describing asymmetric water wheels

TL;DR

This paper models an asymmetric water wheel with unsteady water inflow as a non-autonomous nonlinear system, analyzes conditions for chaos using a generalized competitive modes method, and demonstrates the prevalence of chaotic behavior.

Contribution

It introduces a generalized competitive modes analysis for non-autonomous systems and applies it to reveal chaos in an asymmetric water wheel model with unsteady inflow.

Findings

Chaotic regimes are identified in the water wheel model.

Asymmetric unsteady water wheels exhibit more disorder than steady or symmetric ones.

Chaos is likely common in asymmetric water wheels with unsteady inflow.

Abstract

We use physical principles to derive a water wheel model under the assumption of an asymmetric water wheel for which the water inflow rate is in general unsteady (modeled by an arbitrary function of time). Our model allows one to recover the asymmetric water wheel with steady flow rate, as well as the symmetric water wheel, as special cases. Under physically reasonable assumptions we then reduce the underlying model into a non-autonomous nonlinear system. In order to determine parameter regimes giving chaotic dynamics in this non-autonomous nonlinear system, we consider an application of competitive modes analysis. In order to apply this method to a non-autonomous system, we are required to generalize the competitive modes analysis so that it is applicable to non-autonomous systems. The non-autonomous nonlinear water wheel model is shown to satisfy competitive modes conditions for chaos in certain parameter regimes, and we employ the obtained parameter regimes to construct the chaotic attractors. As anticipated, the asymmetric unsteady water wheel exhibits more disorder than does the asymmetric steady water wheel, which in turn is less regular than the symmetric steady state water wheel. Our results suggest that chaos should be fairly ubiquitous in the asymmetric water wheel model with unsteady inflow of water.