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

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## Abstract

We derive a water wheel model from first principles 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.

## Keywords

Asymmetric water wheel Unsteady water inflow rate Non-autonomous dynamical systems Chaotic attractors Competitive modes analysis## 1 Introduction

A water wheel has several porous water containers attached along the rim of a wheel which rotates in a tilted plane. The angle \(\theta \in [0,2\pi )\) is measured around the wheel in the counterclockwise direction. When the water is poured into the wheel at a certain angle, the containers start filling up and the wheel starts moving due to gravity. The water inflow and the loss of said water due to leakage create a seemingly random revolving motion where the wheel starts rotating in either direction at unpredictable instants of time; see Fig. 1 for a schematic.

*t*),

*R*is the radius of the wheel,

*K*is the leakage rate, \(\nu \) is the rotational damping rate,

*I*is the moment of inertia of the wheel, and

*g*is the gravitational constant.

*m*and

*q*in (1)–(2):

*q*is symmetric, no sine terms appear in the Fourier expansion of

*q*. The presence of sine terms is a manifestation of

*asymmetric*inflow of water. As far as we are aware, all previous works have considered steady inflow of water [1, 2, 3, 4, 5], while we have assumed an unsteady inflow instead in the present work, for the sake of greater generality. This unsteady inflow results in the time-dependent harmonics \(p_n\) and \(q_n\) in the Fourier expansion of

*q*.

Many other seemingly disparate physical phenomena have been shown to have dynamical formulations which are slight variations of (6). We mention some of them here without being exhaustive. It was shown in [3] that the dynamics of thermal convection in a circular tube, with heat sources and sinks spread along its length, is equivalent to the Lorenz system. While thermal convection is a different physical phenomena than what we study in this paper, this example is similar in setup and geometry as the heat sources and sinks are conceptually equivalent to the water inflow and leakage, and the circular tube analogous to the rim of the wheel. In [9], the author has shown Maxwell-Block equations representing a laser model to be equivalent to the Lorenz system as well. In [10], the author has shown a model of disk dynamo to be equivalent to a special case of (6). More recently, authors of [11] have shown a star-shaped gas turbine to have a dynamical model similar to (6). This turbine model is referred to as an augmented Lorenz model which consists of several Lorenz subsystems that share the angular velocity of the turbine as the central node.

The remainder of the paper is organized as follows. In Sect. 2, we analyze the dynamical systems (4) and (6) qualitatively. Then we give an introduction to the theory of competitive modes and extend it to non-autonomous dynamical systems in Sect. 3. This theory is then applied to the system (6) to estimate chaotic parameter regimes in Sect. 4. We discuss interesting findings in Sect. 5.

## 2 Qualitative analysis of the water wheel dynamical system

*n*to simplify the notation. Assuming \(a^{(0)} = b^{(0)} =0\) and solving the integral equation (9) iteratively, we find

*steady asymmetric*water wheel model. In contrast to previously studied steady water wheel models, here we consider an unsteady water wheel model: system (6) represents a water wheel where water inflow is asymmetric and unsteady. As a consequence of this latter attribute of the water inflow, the ODE system (6) is non-autonomous. If we assume a symmetric inflow [no sine terms in the Fourier series of \(q(\theta ,t)\), (3), and \(\mu =0\) in (6)], we obtain an

*unsteady symmetric*water wheel model, instead.

*x*,

*y*,

*z*) in (6) results in a different system. While Malkus’ symmetric water wheel is equivalent to the Lorenz system [1, 2], (6) cannot be reduced to the Lorenz system [5]. Nonetheless, like the Lorenz system, the dynamical system (6) contracts volume:

## 3 General competitive modes analysis

Because of its proximity and similar characteristics to the Lorenz system, we expect chaos in certain parameter regimes for the system (6). Traditionally, one would search the parameter space \((\sigma , r(\tau ), \mu (\tau ))\) for a set of parameters which result in a chaotic attractor. Competitive modes analysis makes this process of searching for chaotic parameter regimes more methodical and organized. The theory helps delineate chaotic regimes in nonlinear autonomous dynamical systems, and can be used to predict chaos in the parameter regimes where mode frequencies are competitive.

*j*, one can get the following second-order system of differential equations by differentiating (12) with respect to

*t*:

### Conjecture 1

- 1.
there exist at least two

*g*’s in the system; - 2.
at least two

*g*’s are competitive or nearly competitive, that is, there are \(g_i \backsimeq g_j > 0\) at some*t*; - 3.
at least one of

*g*’s is the function of evolution variables such as*t*; and - 4.
at least one of

*h*’s is the function of the system variables.

The \(x_i\)’s for which all four conditions of the conjecture are satisfied are called *competitive modes*.

*j*, and \(f_{n+1} = 1\). With these definitions we get from (14) that the necessary second-order oscillator system corresponding to a non-autonomous first-order system is given by

*j*. This system is equivalent to (13) with one additional system variable. Therefore, making use of this modification, one can use the theory of competitive modes to analyze non-autonomous systems.

Competitive modes analysis has been used to predict chaotic regimes and custom design chaotic systems. Competitive modes were shown to result in a chaotic solution on a slow time scale in parametrically driven surface waves [12]. In [13], a competitive modes analysis was used to predict chaos in nonlinear dynamical systems and was employed to construct custom design chaotic systems. Competitive modes analysis has been used to obtain parameter regimes giving chaos in a variety of nonlinear systems, such as the T system [14], a generalized Lorenz system [15], a general chaotic bilinear system of Lorenz type [16], and the canonical form of the blue sky catastrophe [17], to name a few application. In [7], a competitive modes analysis was used to find general chaotic regimes for a quadratic dynamical system by relaxing some conditions of Conjecture 1.

## 4 Competitive modes analysis of the water wheel system

*x*(0) as a free parameter.

Let us rationalize our choice for the function \(r(\tau )\). First, let us remember that the function \(r(\tau )\) is a scalar multiple of the first harmonic \(q_1\) in the Fourier expansion of the water inflow rate \(q(\theta ,t)\). A smooth approximation of step function for \(q_1\) corresponds to changing the water inflow rate abruptly. Secondly, we make the inflow more realistic by including small sinusoidal oscillations to model small fluctuations in the inflow. The function \(r(\tau )\) thus represents small fluctuations and large changes in the water inflow rate. Similar arguments can be used to justify the choice of \(p_2(\tau )\) and \(q_2(\tau )\).

*g*functions to satisfy condition 2 of the conjecture, solutions of the dynamical system (6) must satisfy at least one of the following conditions:

*n*will in general have

*n*additional manifolds compared to an autonomous system of the same dimension. Therefore, the non-autonomous system has more

*g*’s to meet the conditions of Conjecture 1 than the autonomous system, potentially enlarging the chaotic parameter regime.

Figure 2 shows a typical representation of the manifolds listed in (23)–(28), while Fig. 3 gives corresponding time series plots. Figure 2 shows a snapshot of the evolving parameter-dependent manifolds. The maximum amplitude of the expressions \(g_i - g_j\) increases with increasing value of the parameter \(r(\tau )\) in Fig. 3.

Conjecture 1 requires that the competitive *g*’s be positive. In order to verify this, we plot the \(x-z\) region where none of the competitive \((g_i,g_j),i\ne j\) pair is positive with the same parameters as in the previous plot. This region is shown in Fig. 2. Clearly, there are regions of the phase space where competitive *g*’s are positive for the chosen parameters. When the solution (*x*, *y*, *z*) of the system (6) resides in these regions, positivity condition of Conjecture 1 is satisfied. Clearly, manifolds \(g_2=0\) and \(g_3=0\) delineate the positive competitive region of the space from non-competitive region.

It is necessary for the solution \((x(\tau ),y(\tau ),z(\tau ))\) to lie on at least one of the manifolds of Fig. 2 intermittently for the system to be chaotic. Solving (6) numerically (taking \(x(0)=1\)) and evaluating \(g_i -g_j\), for all \(i \ne j\), along the numerical solution, we obtain Fig. 3. For the chosen parameter values, the pairs \(\{g_2,g_4\}, \{g_3,g_4\}\), and \(\{g_2,g_3\}\) are competitive according to the time series plotted. No pair is competitive around the time interval \(\tau \in [0,20]\), as expected, because \(r(\tau ) \approx 0\) in that interval.

Figures 4, 5, and 6 show phase spaces and time series of the system variables obtained by solving (6) and (4) numerically for the unsteady symmetric, steady asymmetric, and unsteady asymmetric water wheel models, respectively, taking \(x(0)=1\). In the phase space plots, the black curve is the solution for \(t\in [0,20]\), the blue curve is the solution for \(t\in [20,40]\), and red curve is the solution for \(t\in [40,60]\). In the top right panels, the green curve represents the evolving circle (8).

When the parameter \(r(\tau )\) is nonzero, one expects chaos, as predicted by the competitive modes analysis. Since the parameters \(r(\tau )\) and \(\mu (\tau )\) are time dependent, the system adjusts its response and settles into chaotic motion in different phase space regions. For \(t>20\), the *x* and *y* time series switch signs unpredictably and the solution seems to settle on two different butterfly attractors. This indicates chaotic motion of the water wheel for large time. An examination of the phase portraits in Figs. 4, 5, 6 and Fig. 2 reveals that a large part of the butterfly attractor lies outside of the red area in Fig. 2. Also, the time series plot of the *z* variable for all the water wheels satisfies equation of the manifold \(g_2-g_3=0\) intermittently, thus satisfying condition 2 of the competitive modes Conjecture 1.

Figures 4, 5 and 6 also show phase space trajectories of higher-order modes (\(n=2\), although one can similarly plot modes for \(n\ge 3\)) in a chaotic regime and the evolving circle given in (8) (the latter depicted by a green curve). As discussed in Sect. 2, all trajectories are attracted toward the boundary of the evolving circle (8) for large time. Therefore, as the radius of the circle grows with time, so does the amplitude of the modes \(a_n, b_n\). Also, since the phase space volume contracts (see (7)), the modes remain bounded in a region of the phase space.

We change the initial angular velocity to \(x(0) =0\), and give corresponding results for the symmetric, steady asymmetric, and unsteady asymmetric water wheels in Figs. 7, 8 and 9. For this modified condition, we do not expect the symmetric water wheel to spin at all (as discussed in Sect. 2), and this is the behavior which we observe in Fig. 7. On the other hand, the other two asymmetric water wheels can still move, and we observe again chaotic motion of these two water wheels in Figs. 8 and 9. In the absence of rotation in the symmetric water wheel case, modes corresponding to \(n=2\) also get confined to a very small area of the phase space. This suggests that the modes occupy a larger proportion of perimeter of the circle in a chaotic regime compared to non-chaotic regimes.

## 5 Conclusions

We have derived a non-autonomous nonlinear dynamical system governing an asymmetric water wheel with unsteady water inflow rate. In relevant limits, this reduces to a dynamical system for an asymmetric water wheel with steady water inflow rate, as well as a dynamical system for the symmetric water wheel. In order to determine parameter regimes likely to give chaotic dynamics in this model, we have applied the competitive modes analysis. As our nonlinear dynamical system is non-autonomous, we have modified the standard competitive modes approach by introducing an auxiliary function which accounts for the non-autonomous contribution, casting the system as a higher-order autonomous dynamical system for sake of the competitive modes analysis. The caveat here is that one must first convert the non-autonomous system into an autonomous system, through a change of variable. While this results in a higher-order ODE system, the extra term is simple enough so that it does not contribute an extra non-trivial mode frequency in the competitive modes analysis, and hence it does not complicate the analysis.

As was true for the symmetric and asymmetric steady water wheel systems, the asymmetric unsteady water wheel model permits chaotic dynamics for some parameter regimes. Owing to the non-autonomous contributions, we find that the dynamics are less regular in general, and an enlarged chaotic parameter regime is found when compared to the symmetric and asymmetric steady cases. Physically, our results suggest that chaos should be fairly ubiquitous in the asymmetric water wheel model with unsteady inflow of water, and can occur even in parameter regimes where the corresponding symmetric and asymmetric steady water wheel systems give regular, non-chaotic dynamics.

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