modelling the chaotic waterwheel

Modelling the Chaotic Waterwheel

G14DIS

Mathematics 4th Year Dissertation

2014/15

School of Mathematical Sciences

University of Nottingham

Edward Pode

Supervisor: Dr. P. C. Matthews

Research group: Industrial and Applied Mathematics

Project code: PCM D1

Assessment type: Investigation

I have read and understood the School and University guidelines on plagiarism. I confirm

that this work is my own, apart from the acknowledged references.

Abstract

This paper examines a basic model of the Malkus chaotic Waterwheel, a mechanical

analogy of the Lorenz equations. Governing equations are derived by balancing the effects

of mass and torque. Contrary to most previous works this paper uses analytical mechanics

for derivations of the discrete and continuous versions of the Waterwheel, reducing the

scope for error. A Centre of Mass representation is presented and the merits of each frame

of reference are discussed.We derive a refined model of the waterwheel by introducing the

apparently simple refinement of water flow between buckets. This takes the form of the

capture coefficient (cij) leading to new, more physically accurate governing equations.

Comparison to the basic system shows that this has a drastic impact on the dynamics of

the Waterwheel.

Contents

1 Introduction 8

2 Background to Chaos 10

2.1 Lorenz’s Work on Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 The Mathematics of the Waterwheel 18

3.1 The Chaotic Waterwheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Mass Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Torque Balances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Evaluating the Discrete and Continuous Cases 27

4.1 Simplifying equations, equivalence to Lorenz . . . . . . . . . . . . . . . . . 27

4.2 The Stability of the Waterwheel . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Refining the Model 41

5.1 Bucket positioning and assumptions . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Examples of the Basic and Refined systems . . . . . . . . . . . . . . . . . . 43

6 Effect of Refinement on the System 53

6.1 2 Bucket systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.2 3 Bucket Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7 Conclusions and Scope for Future Work 57

A List of Nomenclature, subdivided by type of model 59

B Matlab Code 61

B.1 Basic Lorenz system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3

B.2 Comparison plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

B.3 Lorenz attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

B.4 Basic 2 Bucket system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

B.5 Refined 2 Bucket system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

B.6 Basic 3 Bucket system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B.7 Refined 3 Bucket system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

C Website 70

D References 86

4

List of Figures

1 Simple Convection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Convection in Coffee. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Comparison of Lorenz systems. . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Lorenz Attractor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Constructed Waterwheel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

6 Strogatz’s Original Waterwheel. . . . . . . . . . . . . . . . . . . . . . . . . 18

7 Discrete Waterwheel, four buckets. . . . . . . . . . . . . . . . . . . . . . . 19

8 Strogatz’s Continuous Waterwheel. . . . . . . . . . . . . . . . . . . . . . . 20

9 Mass focuses, Discrete and Continuous. . . . . . . . . . . . . . . . . . . . . 21

10 Hopf Boundary plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

11 Bucket Coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

12 Continuum boundaries, 2 Bucket Refined model. . . . . . . . . . . . . . . . 47

13 Bucket Arrangements, 3 Bucket Waterwheel. . . . . . . . . . . . . . . . . . 51

14 Example plots of the 2 Bucket Basic/Refined systems. . . . . . . . . . . . . 53

15 Example plots of the 3 Bucket Basic/Refined systems. . . . . . . . . . . . . 55

5

List of Key Equations

2.1 Lorenz System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1 Mass Balance, Discrete system . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.8 Mass Balance, Continuous system . . . . . . . . . . . . . . . . . . . . . . . . 24

3.11 Torque Balance, Discrete system . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.14 Torque Balance, Continuous system . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Normalised Centres of Mass, Centre of Mass system . . . . . . . . . . . . . . 28

4.4 ω governing equation, Centre of Mass system . . . . . . . . . . . . . . . . . . 28

4.8 ycm governing equation, Centre of Mass system . . . . . . . . . . . . . . . . . 29

4.9 xcm governing equation, Centre of Mass system . . . . . . . . . . . . . . . . . 29

4.11 Change of Variable from Centre of Mass system to Lorenz system . . . . . . . 29

4.17 Governing equations, Continuous system . . . . . . . . . . . . . . . . . . . . . 31

4.20 Change of Variable from Continuous system to Lorenz system . . . . . . . . . 32

4.23 Fixed points of the Lorenz system . . . . . . . . . . . . . . . . . . . . . . . . 33

4.24 General Jacobian for the Lorenz System . . . . . . . . . . . . . . . . . . . . . 33

4.25 (0, 0, 0) Jacobian for the Lorenz System . . . . . . . . . . . . . . . . . . . . . 34

4.28 (±√β(ρ− 1),±

√β(ρ− 1), (ρ− 1)) Jacobian for the Lorenz System . . . . . 34

4.35 Limit to trajectories of zero Volume . . . . . . . . . . . . . . . . . . . . . . . 36

4.36 Fixed points of the Truncated system . . . . . . . . . . . . . . . . . . . . . . 37

4.37 General Jacobian for the Truncated system . . . . . . . . . . . . . . . . . . . 38

4.38 (0, 0, r) Jacobian for the Truncated system . . . . . . . . . . . . . . . . . . . . 38

4.43 (±√

gqrν− k2,±

√νk2

gq(r − νk2

gq), νk

2

gq)) Jacobian for the Truncated system . . . . 39

5.1 Governing equations, Refined system . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 Definition of the Capture Coefficient . . . . . . . . . . . . . . . . . . . . . . . 42

5.4 Governing equations, 2 Bucket Basic system . . . . . . . . . . . . . . . . . . . 44

5.5 Fixed points of the 2 Bucket Basic system . . . . . . . . . . . . . . . . . . . . 44

5.6 General Jacobian for the 2 Bucket Basic system . . . . . . . . . . . . . . . . . 45

6

5.7 (q

k, 0, 0, 0) Jacobian for the 2 Bucket Basic system . . . . . . . . . . . . . . . 45

5.11 Governing equations, 2 Bucket Refined system . . . . . . . . . . . . . . . . . . 46

5.12 Fixed points of the 2 Bucket Refined system . . . . . . . . . . . . . . . . . . . 46

5.13 General Jacobian for the 2 Bucket Refined system . . . . . . . . . . . . . . . 47

5.14 (q

k,q

k, 0, 0) Jacobian for the 2 Bucket Refined system . . . . . . . . . . . . . . 47

5.16 (q

k,q

k, 0,±sin−1(1

2)) Jacobians for the 2 Bucket Refined system . . . . . . . . 48

5.17 Governing equations, 3 Bucket Basic system . . . . . . . . . . . . . . . . . . . 49

5.18 Fixed point of the 3 Bucket Basic system . . . . . . . . . . . . . . . . . . . . 49

5.19 General Jacobian for the 3 Bucket Basic system . . . . . . . . . . . . . . . . . 50

5.20 (q

k, 0, 0, 0) Jacobian for the 3 Bucket Basic system . . . . . . . . . . . . . . . 50

5.22 Governing equations, 3 Bucket Refined system . . . . . . . . . . . . . . . . . . 51

5.23 Fixed coordinates of the 3 Bucket Refined system . . . . . . . . . . . . . . . . 52

5.24 Values of θ allowing for ω = 0, 3 Bucket Refined system . . . . . . . . . . . . 52

7

1 Introduction

According to the Oxford English Dictionary, chaos in its current mathematical guise can

be defined as the: “behaviour of a system which is governed by deterministic laws but

is so unpredictable as to appear random, owing to its extreme sensitivity to changes in

parameters or its dependence on a large number of independent variables”1.

Despite initial insights in the late 19th and early 20th centuries by academics such as

Poincare and Kolmogorov2, Edward Lorenz can truly be described as ‘the father of chaos

theory’. His equations and the mechanical analogue waterwheel developed by Willem

Malkus in the early 1970s, see Section 2, brought chaotic systems to light and convinced

the scientific community that deterministic chaos is a reality.

Today, chaos theory is an important field of study in mathematics, with applications in

many other disciplines including meteorology, sociology, physics, engineering, economics,

biology, and philosophy. A better understanding of the mechanics of chaos theory makes

it applicable to ever more sensitive systems where they appear in real life. As Chris

Budd [2] shows, chaos theory can help maintain car tyres, encrypt systems, prevent ships

capsizing, aid in the design of lasers and help restore a fibrillating heart to its normal

regularity. However, the sensitivity of chaotic systems means they can be hard to model.

For this reason, the aim of this project is to refine the governing equations for the chaotic

Waterwheel model by considering the effect of water flow between buckets.

In Section 2 background reading on the subject is explored and some sample calcu-

lations regarding the Lorenz system are evaluated. Key details to focus on from this

section are bifurcation theory, moments of inertia in rotating systems and torque. These

are needed to grasp the mechanics of the waterwheel. Then in Section 3 we familiarise

ourselves with the equations of motion for the two varieties of the Waterwheel, Discrete

and Continuous. Section 4 explains how we simplify the equations we have and what

properties are needed for Chaos to arise. Then in Section 5 we derive a Refined model of

1There has been a great deal of debate about the definition of chaos see for example Gleick [6, pg.306]2For a full description of the history of chaos theory see Oestreicher [16]

8

the chaotic Waterwheel by introducing a factor, the capture coefficient, to represent water

being captured by one bucket as it falls from another. Example calculations of the Basic

and Refined models are included to illustrate the point and to provide talking points for

the next section. In Section 6 we evaluate the differences between models caused by our

refinements. We then have a conclusion summarising the main points of the report.

Appendix A provides a list of nomenclature for the Waterwheels. Appendix B is the

Matlab Code used for our example calculations and Appendix C illustrates the website

we have designed and the supporting java application that supports this work.

9

2 Background to Chaos

2.1 Lorenz’s Work on Chaos

The idea of chaos theory and chaotic systems, which are bounded, aperiodic and ex-

tremely sensitive to initial conditions, originates with Edward Lorenz. He showed that

chaotic systems exist in his 1963 paper, ‘Deterministic Nonperiodic Flow’ [10], which has

subsequently been cited over 14,000 times according to Google Scholar.

Lorenz, a “mathematician in meteorologist’s clothing” [6, pg.22], was attempting to

model the motion of convection in fluid as part of his study of the weather. He saw that

when a liquid or gas is heated from below, the fluid tends to circulate in a steady pattern

of rolls, as shown in Figure 1. Hot fluid rises on one side of a cylinder, loses heat and

descends on the other side. This is the process of convection. When the difference in

temperature is slightly larger, an instability sets in and the rolls develop a wobble that

moves back and forth along the length of the cylinders. This is the region of deterministic

chaos. At even higher temperatures, the flow becomes wild and turbulent and is very

difficult to describe [6, pg.25].

Figure 1: Simple Convection [13].

The example that Lorenz preferred, according to Gleick, was based on convection

in a cup of hot coffee. If the coffee is just warm, its heat will dissipate without any

hydrodynamic motion at all. The coffee remains in a steady state. However, if it is hot

enough, a convective overturning will bring hot coffee from the bottom of the cup up to

the cooler surface. The chaotic convection in coffee becomes visible to the eye when milk

10

is dribbled into the cup [6, pg.24].

Figure 2: Convection in a cup of coffee. The milk is rising ‘chaotically’ to thetop [4].

As Gleick explains, “Lorenz took a set of equations for convection and stripped it to the

bone, throwing out everything that could possibly be extraneous, making it unrealistically

simple” [6, pg.25]. Lorenz arrived at the set of equations that now bear his name :

dx

dt= σ(y − x) (2.1a)

dy

dt= x(ρ− z)− y (2.1b)

dz

dt= xy − βz . (2.1c)

11

The significance of the variables in the equation shown are as follows:

• x is convective overturning,

• y is horizontal temperature variation,

• z is the vertical temperature variation,

• σ, ρ and β correspond to physical properties and are positive constants.

As Gleick explains, “Lorenz’s discovery was an accident, one more in a line stretching

back to Archimedes and his bathtub” [6, pg.21]. Using an early computer (the Royal

McBee LGP-30 electronic computing machine), Lorenz ran multiple simulations of a hy-

drodynamical system in an attempt to spot trends that would carry through into real

weather forecasting. In the words of Gleick:

“One day in the winter of 1961, wanting to examine one sequence at greater

length, Lorenz took a shortcut. Instead of starting the whole run over, he

started midway through. To give the machine its initial conditions, he typed

the numbers straight from the earlier printout. Then he walked down the hall

to get away from the noise and drink a cup of coffee. When he returned an

hour later, he saw something unexpected, something that planted a seed for

a new science”.

The seed that was planted was that minor changes result in wildly different results

after sufficient time has passed. The values were bounded, not obviously periodic/regular

and a minor change to initial conditions resulted in very different values after an hour,

showing them to be sensitive to initial conditions therefore fulfilling all the requirements

of a chaotic system.

The graph in Figure 3, made using MATLAB’s ODE45 function, demonstrates these

properties. It is a plot of an example of the Lorenz system. Details of the code used can be

found in Appendix B. The blue and green lines are initially very similar (indistinguishable)

12

Figure 3: a) Blue line has initial conditions x = 1, y = 1, z = 1 at t = 0b) Green line has initial conditions x = 1 + 10−3, y = 1, z = 1 at t = 0.

but as time progresses they deviate until eventually they bear no similarity at all. This

sensitivity is ‘Chaos’. You can also see that the values in each case are bounded (|x| < 20)

and aperiodic. This example is the Lorenz equation where both of the cases have ρ = 28,

σ = 10 and β = 83. As indicated by the caption their initial conditions are (very slightly)

different, the graph clearly show the chaotic nature of the Lorenz system.

Chaos is also notable for exhibiting patterns. A phase space plot of the three Lorenz

equations produces an image known as a ‘Strange Attractor’, which resembles a figure

eight or pair of butterfly wings (Figure 4).

Because the Lorenz equations are so co-dependent, their trajectories orbit back and

forth between two centres but never cross. Such properties (combined with the sensitivity

to initial conditions) are what make the system chaotic.

At the time Lorenz described the process to Willem Malkus. A professor of applied

mathematics at the Massachusetts Institute of Technology(MIT). Malkus laughed and

said, “Ed, we know we know very well that fluid convection doesn’t do that at all.” Malkus

told him that the complexity would surely be damped out, with the system settling down

to steady, regular motion. Malkus later acknowledged, “Of course, we completely missed

13

the point ... Ed wasn’t thinking in terms of our physics at all. He was thinking in terms of

some sort of generalized or abstracted model which exhibited behavior that he intuitively

felt was characteristic of some aspects of the external world” [6, pg.31].

Figure 4: A plot of Lorenz’s strange attractor for values ρ = 28, σ = 10,β = 8/3 see Appendix B for details.

The idea of a perfectly predictable deterministic system was so widespread in the

1960s that a real world example was needed to demonstrate that chaos, and sensitivity to

initial conditions (often referred to as ‘the butterfly effect’), were realities and not mere

mathematical artifacts. To this end a ‘chaotic Waterwheel’, a mechanical analog of the

Lorenz equations, was built at MIT by Willem Malkus and Lou Howard in the early 1970s

and this, coupled with Malkus’s paper, ‘Non-periodic convection at high and low Prandtl

number’ [11], in the words of Gassman, “helped to convince the scientific community that

deterministic chaos is a reality in physical systems” [5, pg.2216].

14

2.2 Literature Review

The importance of the Malkus chaotic Waterwheel is dependant on certain key character-

istics. It is a system simple enough to be properly reduced to a low dimensional system

of differential equations; yet it is complex enough to present chaotic dynamics. It can be

properly isolated and tested in a laboratory; but, unlike Haken’s lasers [7] which can also

be used to model the Lorenz equations, Malkus’s Waterwheel is a macroscopic system

visible to the naked eye.

Since the 1970s, numerous authors have built upon the work of Malkus. A non-

exhaustive sample of these papers includes Kol and Gumbs [9]. In 1992 they presented

the derivation of a state of the art (Continuous) Waterwheel. Ideal for laboratory mea-

surement and verification. Gassmann not only described, ‘Noise-induced chaos-order

transitions’ [5] but also gave a brief introduction to the work of Malkus and devel-

oped a website that demonstrates, with a Java applet, the Chaotic Waterwheel. We

have developed with his permission, see Appendix C for full details or directly visit

https://sites.google.com/site/gassmannlab/home). Clauset, Grigg, Lim, and Miller, in

‘Chaos You Can Play In’ [3], designed a chaotic Waterwheel using a very basic experi-

mental setup (illustrated in Figure 5) and adopted more realistic assumptions regarding

the Waterwheel in their derivation i.e. finite cup volume. In 2006 Mishra and Sanghi

[14] studied an asymmetric version of the Chaotic Waterwheel. They introduced extra

parameters that caused a favouritism for one ‘wing’ of the Lorenz attractor. In 2007,

Matson made a centre of mass derivation [12].

David Becerra Alonso [1] is the only author we have identified who dealt with the

equations of motion for a Discrete case (N buckets) Waterwheel. His doctoral thesis

discussed many aspects of the Waterwheel, including a brief history of weighted wheels,

the development of the Waterwheel and a toroidal convection model developed by Malkus

that presents the same dynamics.

15

Figure 5: Clauset, Grigg, Lim, and Miller’s experimental version of chaoticWaterwheel [3].

Alonso used a generic Jacobian to obtain the Lyapunov exponents3 for the Chaotic

Waterwheel system. While forming a complete system, the Jacobian method represents

a bucket by bucket view of the system and does not appropriately show the distinction

between chaos and order. As Alonso says, “It is the long term permanent transition

between buckets that eventually allows the butterfly effect to take over, and no local

and short term analysis will show this” [1, pg.97]. Using a common numerical method,

not based on the Jacobian, he obtained Lyapunov exponents that properly make the

distinction between chaos and order.

The paper is somewhat unusual in having used analytical mechanics to derive the

Waterwheel’s governing equations. However this is useful as it provided a more rigorous

3A quantity indicating the rate of separation of infinitesimally close trajectories.

16

derivation of the equations of motion. For that reason, it is used as the primary source of

the derivation for the Waterwheel’s equations of motion in both Continuous and Discrete

models in this paper.4

Illing, Fordyce, Saunders and Ormond [8] demonstrated through direct comparison

that the Lorenz equations ‘synchronised’ with the results they obtained from the Water-

wheel. In particular, the angular velocity of their Waterwheel was used as an input signal

for a computer model. High-quality chaos synchronization of the model and the Water-

wheel was achieved. This indicated that the Lorenz equations provide a good description

of the Waterwheel dynamics.

Tongen, Thelwell, and Becerra-Alonso [19] replaced the Waterwheel with a Sandwheel.

They modelled both in parallel where possible noting where methods may be extended

and where lines of enquiry seem exhausted. Numerical simulations were used to compare

and contrast the behavior of the Sandwheel with the Waterwheel. We aim to take a similar

approach with our Refined model. Simulations confirmed that the Sandwheel retained

many of the elements of chaotic Lorenzian dynamics. However, bifurcation diagrams

showed dramatic differences in where the order-chaos-order transitions occurred.

4Dr Alonso was most helpful with advice concerning his work.

17

3 The Mathematics of the Waterwheel

3.1 The Chaotic Waterwheel

The original version developed by Malkus is the simplest and can be described as a toy

Waterwheel with leaky paper cups suspended from its rim (Figure 6). However, Strogatz

[18] was the one to popularise the Waterwheel and in time update it.

Figure 6: Strogatz’s Original Waterwheel [18].

According to Strogatz, water is poured in steadily from the top. If the flow rate is

too slow, the top cups never fill up enough to overcome friction, so the wheel remains

motionless. For faster inflow, the top cup gets heavy enough to start the wheel turning

(Figure 6 (a)). Eventually, the wheel settles into a steady rotation in one direction or the

other (Figure 6 (b)). The system is symmetric so rotation in either direction is equally

possible. Clockwise or counter-clockwise rotations will arise based on initial conditions.

Strogatz goes on to say: “By increasing the flow rate still further, we can destabilize

the steady rotation. Then the motion becomes chaotic: the wheel rotates one way for a

few turns, then some of the cups get too full and the wheel doesn’t have enough inertia

to carry them over the top, so the wheel slows down and may even reverse its direction

[Figure 6 (c)]. Then it spins the other way for a while. The wheel keeps changing direction

18

erratically. Spectators have been known to place bets (small ones, of course) on which

way it will be turning after a minute” [18, pg.302].

In order to understand this behaviour and to be able to produce a simulation, it is

necessary to be able to describe the wheel in terms of a set of equations - specifically

differential equations that map the wheel’s dynamics. We can do this by considering all

the incoming and outgoing flows for mass and using the dissipative Lagrange equations

for the Waterwheel’s motion. First we consider the simple wheel with a discrete number

of buckets and then a Continuous case with infinitesimally small chambers occupying the

entire rim of the wheel.

The aim of the following sections, credit for which goes to Strogatz [18] and Alonso [1],

is to derive governing equations for mass and angular velocity and subsequently show

how they relate to the Lorenz equations. Conservation of mass and torque balance are

used to find these equations. The principal differences between the two cases is that in

the Discrete case we focus on the buckets and in the continuum case we focus on a generic

region of the rim between two points θ1 and θ2.

3.2 Equations of Motion

Variables and parameters pertinent to the Waterwheel are introduced as they appear. A

full list is provided in Appendix A.

Figure 7: Diagram of the Discrete version of the Chaotic Waterwheel withfour buckets.

19

The Discrete case (Figure 7) is the simpler to calculate. It has N buckets of equal size

and shape spaced equally around the rim of a Waterwheel. The buckets are triangular

prisms. This ensures there is no gap under the nozzle for inflowing water to go amiss

and that there is no risk of the buckets colliding. Their mass (empty or with water) is

assumed to be centered on the rim of the Waterwheel. The buckets are labelled from 0

to N − 1. The angular position, θ, is calculated from bucket 0 and the ith bucket has

position (θi = θ + i2πN

). In the Discrete case the frame of reference is anchored to each of

the buckets. Therefore, only incoming and leak flows, from each bucket, are relevant (see

Figure 9 (a)).

Figure 8: Diagram of the Continuous version of the Chaotic Waterwheel asdesigned by Strogatz [18].

In the years since Malkus developed the Chaotic Waterwheel a more sophisticated

20

setup has been established [18, pg.303]. This model has a continuous ring of separate

chambers with no gaps between them. The wheel is illustrated in Figure 8. It rotates in

a plane that is tilted slightly from the horizontal. In this case, the frame of reference is

fixed outside the rotation of the wheel. Water can move into or leave any given section

dθ as the rotation of the Waterwheel carries it (see figure 9 (b)). During the derivation,

we consider sections of the rim between angles θ1 and θ2.

Figure 9: Diagram illustrating the different focuses of Discrete andContinuous models [1].

3.3 Mass Balances

The driving force behind the motion of the Waterwheel is the distribution of water about

the wheel. Thus, the first step in modelling is to conduct a mass balance.

Assumptions:

1. The buckets never overflow.

2. The leakage of water from each bucket is proportional to the mass of water within

said bucket.

21

3. In the Discrete case, there is always a bucket under the nozzle and the buckets never

collide.

3.3.1 Discrete Case

In the Discrete case (Figure 9 (a)) each bucket is considered separately. The change in

mass, for the ith bucket is, therefore, dependent only on:

1. Inflow of water, (qi).

2. Leakage from the buckets, (−kmi).

Where qi is the water inflow rate for the ith bucket, it is equal to either the total water

inflow for the system, q, if the bucket is closest to the top (cos(θi) > cos( πN

)) or 0 for

other positions. Whilst, mi(t) is the mass in the ith bucket and k is the constant of

proportionality for the leakage from each bucket. Taking the above factors into account

the mass balance equation for the Discrete case is:

dmi

dt= qi − kmi . (3.1)

Note that the total mass, M(t), of the system obeys:

dM

dt= q − kM (3.2)

which, therefore, means that (starting from an empty waterwheel):

M(t) =q

k

(1− e−kt

)(3.3)

which tends to a constant as t → ∞. For sufficiently large time, we can therefore treat

mass, and related variables such as the moment of inertia, as constant.

22

3.3.2 Continuous Case

The mass balance equation is somewhat harder to derive in the Continuous case as there

are four factors (see Figure 9 (b)) influencing a change (∆M∗) in mass (M∗(t)) in our

target region (between θ1 and θ2):

1. The inflow of water, ([∫ θ2θ1

(Q)dθ]∆t).

2. Leakage from the buckets, (−[∫ θ2θ1

(km)dθ]∆t). The ‘m’ in the integral implies that

leakage is directly proportional to the mass. The fluid mechanics of leakage are

complicated, but our formula agrees with direct measurements on the Waterwheel

itself, to a good approximation [18].

3. As the wheel rotates, it carries water into the region we are observing, (m(θ1)ω∆t).

4. The corresponding mass carried out of the region, (−m(θ2)ω∆t).

The latter two factors are known as ‘transport terms’. Q(θ) is the water inflow in our

target region. For the Continuous case the mass m(θ, t) is general rather than per bucket

so it is dependant on θ as well as t and ω is the angular velocity of the Waterwheel (ω = θ).

Together the effect is:

∆M∗ = ∆t

[∫ θ2

θ1

(Q)dθ −∫ θ2

θ1

(km)dθ + ω [m(θ1)−m(θ2)]

]. (3.4)

We then combine the transport terms into an integral term

m(θ1)−m(θ2) = −∫ θ2

θ1

∂m

∂θdθ . (3.5)

Dividing by ∆t and letting ∆t→ 0 gives us:

dM∗

dt=

∫ θ2

θ1

(Q− km− ω∂m

∂θ

)dθ . (3.6)

23

By definitiondM∗

dt=∫ θ2θ1

(∂m

∂t)dθ, thus:

∫ θ2

θ1

(∂m

∂t)dθ =

∫ θ2

θ1

(Q− km− ω∂m

∂θ

)dθ . (3.7)

This must be true for any region irrespective of θ1, θ2 giving us the mass balance equation

in the Continuous case:

∂m

∂t= Q− km− ω∂m

∂θ. (3.8)

Note that this is a partial derivative as opposed to a full derivative. This is because

in the Continuous case mass is a function of time and position, m(θ, t), as opposed to the

Discrete case where the mass in a bucket is dependent only on the time, mi(t).

As with the Discrete case, the total mass of the system tends to a constant ensuring

that, after a transient period, we can treat the moment of inertia as constant. To show

this we need to use the Fourier Series, we will return to this in Section 4.1.2.

3.4 Torque Balances

The torque, or turning force, (τ) experienced by the wheel is equal to the rate of change of

the product of the wheel’s moment of inertia (I), sometimes called the angular mass, and

the angular velocity (ω). So τ =d

dt(Iω). We make the following simplifying assumptions:

1. Water leaving each bucket exerts no torque on the wheel.

2. The unfilled wheel is symmetrical.

3. Buckets freely rotate around their support and are always oriented with their open

tops vertically upwards.

4. Enough time has elapsed for the mass to be approximately constant and thus the

moment of inertia for the system is constant.

24

Assumption 4 is particularly important as it allows us to simplify the torque equation

to the form τ = Iω.

3.4.1 Discrete Case

Alonso’s approach, which we adopt, uses the dissipative Lagrangian equations of motion

to establish the torque balance for the Waterwheel:

d

dt

(∂L

∂ω

)− ∂L

∂θ= −∂F

∂ω. (3.9)

Where L is the Lagrangian

(= Kinetic energy(

Iω2

2)− Potential Energy(gr

N−1∑i=0

micos(θ + i2π

N

))

)and F is Raleigh’s dissipation function (=

νω2

2). By substituting in these values for L and

F we arrive at the more specific:

Iω − grN−1∑i=0

misin

(θ + i

2π

N

)= −νω , (3.10)

which we can rearrange to get an equation specifically for ω’s rate of change:

ω =

grN−1∑i=0

misin (θi)− νω

I. (3.11)

3.4.2 Continuous Case

In the traditional model presented by Strogatz [18] the Continuous case is treated as

standard and the torque is resolved as a combination of driving gravitational torque and

the damping torque.

The latter is a mix of frictional torque and ‘inertial damping’. The inertial factor

represents bringing water entering the system up to speed. Both factors are proportional

25

to ω so we can represent both with −νω (ν > 0).

In our target section of the rim (dθ) the gravitational torque produced by the mass is:

dτ = mgrsin(θ)dθ . (3.12)

Integrating the gravitational torque over the entire Waterwheel gives us:

gr

∫ π

−πm(θ, t)sin(θ)dθ . (3.13)

So the factors influencing the torque balance are:

1. Damping torque, (−νω).

2. Gravitational Torque, (gr∫ π−πm(θ, t)sin(θ)dθ).

Combining these, we arrive at an ‘integro-differential’ equation (an equation involving

both integrals and derivatives) for the Continuous case torque balance:

ω = −νωI

+gr

I

∫ π

−πm(θ, t)sin(θ)dθ . (3.14)

We simplify the torque equation later using Fourier series, see Section 4.1.2.

26

4 Evaluating the Discrete and Continuous Cases

4.1 Simplifying equations, equivalence to Lorenz

There are several ways to simplify the mass and torque balancing equations from the

previous section and show their equivalence to the Lorenz system. In the case of infinitely

many buckets, our summation turns into an integral:

N−1∑i=0

mi(t)→∫ π

−πm(θ, t)dθ (4.1)

which agrees with the different torque balance equations for Discrete/Continuous cases.

We can then substitute in Fourier series for the mass and inflow terms into our torque

balance equations. When we integrate the orthogonality of the sin and cos terms gets rid

of most of the modes. This method technically works for the Discrete case and is becomes

more accurate as the number of buckets increases, but it is not as rigorous as we would

like. Instead, we adopt a centre of mass derivation as laid out by Alonso [1].

4.1.1 Discrete Case, Centre of Mass based derivation

We approach the centre of mass derivation using the governing equations for the Discrete

model (See Equations 3.1, 3.11). Before proceeding we change the definition of qi from

a discontinuous function to a generic function which is symmetric with respect to the

vertical axis of the Waterwheel (q(θi)). This new definition means we can make the

approximation:N−1∑i=0

q(θi)sin(θi) ∼= 0 . (4.2)

The (normalised) centre of mass can be found separately in vertical (y) and horizontal

(x) axes, as the sum of the products of position and mass, divided by the total mass in

27

the system thus we have:

yCentre of Mass = ycm =k

q

N−1∑i=0

mi [rsin(θi)] (4.3a)

xCentre of Mass = xcm =k

q

N−1∑i=0

mi [rcos(θi)] . (4.3b)

Substituting Equation 4.3(a) into Equation 3.11 gives us a new definition for the rate

of change of ω:

ω =gq

Ikycm −

ν

Iω . (4.4)

To find governing equations for ycm we first take the derivative with respect to time

(remembering that θ is a function of time):

ycm =rk

q

N−1∑i=0

misin(θi) +rkω

q

N−1∑i=0

micos(θi) (4.5)

which we put in terms of ω and xcm by multiplying the terms of each bucket’s mass

balance equation (Equation 3.1) by sin(θi) and considering the sum thereof:

N−1∑i=0

misin(θi) =N−1∑i=0

q(θi)sin(θi)− kN−1∑i=0

misin(θi) . (4.6)

The first term on the RHS is, according to our earlier assumption, 0. We replace the LHS

with a suitably rearranged version of the equation for ycm, thus we have:

q

rkycm − ω

N−1∑i=0

micos(θi) = −kN−1∑i=0

misin(θi) . (4.7)

Multiplying through byrk

qand replacing terms with ycm, xcm as appropriate we find the

28

equation for rate of change of the centre of mass’s vertical position as:

ycm = ωxcm − kycm . (4.8)

Similar methodology (omitted due to repetitiveness) leads to the equivalent equation

for xcm:

xcm =rk

q

N−1∑i=0

q(θi)cos(θi)− ωycm − kxcm . (4.9)

With the xcm equation we were not able to get rid of theN−1∑i=0

q(θi)cos(θi) term. This

necessitates the inclusion of a governing equation for θ to complete the system:

θ = ω . (4.10)

With that inclusion the Equations 4.4, 4.8, 4.9 make the ‘Truncated’ system for the centre

of mass derivation.

Turning this system into the Lorenz system is fairly straightforward and uses the

following change of variables:

ω = kx (4.11a)

ycm =νk2

gqy (4.11b)

xcm =r

q

N−1∑i=0

q(θi)cos(θi)−νk2

gqz (4.11c)

t =τ

k. (4.11d)

In this case σ =ν

Ikand ρ =

gr

νk2

N−1∑i=0

q(θi)cos(θi). β = 1 for all parameter values of

the Waterwheel so it actually corresponds to a subset of the Lorenz equations.

29

4.1.2 Continuous Case, use of Fourier Series

Since the mass distribution m(θ, t) and the inflow Q(θ) are periodic in θ, we can write

them as Fourier series:

m(θ, t) =∞∑j=0

[aj(t)sin(jθ) + bj(t)cos(jθ)] (4.12a)

Q(θ) =∞∑j=0

[q∗j cos(jθ)] . (4.12b)

Q is of cos exclusively because it is symmetric around θ = 0. By substituting these

expressions into the equation for mass (Equation 3.8) and equating coefficients we can

obtain a set of coupled ordinary differential equations for the amplitudes aj, bj:

∂∂t

∞∑j=0

[aj(t)sin(jθ)+bj(t)cos(jθ)]=

∞∑j=0

[q∗j cos(jθ)]

−k∞∑j=0

[aj(t)sin(jθ)+bj(t)cos(jθ)]−ω ∂∂θ

∞∑j=0

[aj(t)sin(jθ)+bj(t)cos(jθ)] .

(4.13)

Making the appropriate differentiations and then equating coefficients for sin(jθ) and

cos(jθ) terms gives us:

aj = jωbj − kaj (4.14a)

bj = −jωaj − kbj + q∗j . (4.14b)

Likewise with the equation for torque (Equation 3.14):

ω = −νωI

+gr

I

∫ π

−π

∞∑j=0

[aj(t)sin(jθ) + bj(t)cos(jθ)]sin(θ)dθ . (4.15)

30

Orthogonality means that only a1 survives when we perform the integration. Rearranging

for ω gives us:

ω =πgra1 − νω

I. (4.16)

Our formulas for aj and bj imply a1, b1 and ω form a closed system independent of aj, bj

where j ≥ 2. Therefore, our overall equations of motion are:

a1 = −ωb1 − ka1 (4.17a)

b1 = −ωa1 − kb1 + q∗1 (4.17b)

ω =πgra1 − νω

I. (4.17c)

The advantage of these equations is that they are considerably simpler to deal with

than the original integro-differential equation. Before moving on we return to the To-

tal mass in the Continuous case (see Section 3.3.2). Using the Fourier series we have

introduced for m(θ, t) and our equations for bj(t) we obtain:

Total Mass =

∫ π

−πm(θ, t)dθ = 2πb0(t) . (4.18)

But b0(t) is given by b0 = q0 − kb0 which is effectively the same as the Discrete case. We

can therefore reuse our result. Thus, (for b0(0) = 0) the total mass in the Continuous

case is:

Total Mass = 2πq0k

(1− e−kt) . (4.19)

This, like the Discrete case, tends to a constant exponentially fast.

We can convert our equations of motion (Equation 4.17) into a subset of the Lorenz

31

equations using the change of variables and rescaling of time detailed in Equation 4.20:

a1 =kν

πgry (4.20a)

b1 =−kνπgr

z +q1k

(4.20b)

ω = kx (4.20c)

t =τ

k. (4.20d)

The algebra is straightforward. The final result is:

x =ν

Ik(y − x) (4.21a)

y = x

(πgqr

k2ν− z)− y (4.21b)

z = xy − z . (4.21c)

Which, writing σ =ν

Ik, ρ =

πgqr

k2νand setting β = 1, are the Lorenz equations:

dx

dt= σ(y − x) (4.22a)

dy

dt= x(ρ− z)− y (4.22b)

dz

dt= xy − βz . (4.22c)

4.2 The Stability of the Waterwheel

Previously, we have discussed the chaos that originates in a cup of coffee (Section 2.1).

We described how the qualities of the system change (bifurcate) in response to different

temperature differences. We have not, however, touched upon the mathematics used to

find the bifurcation points and the properties around them. In this section we calculate

the values for which these bifurcations occur. Identifying some general trends for the

32

default Lorenz system and the Truncated system we derived from the Discrete bucket

model. Once again credit goes to Alonso [1].

Stability analysis is based around fixed points so these are identified first. Next the

Jacobian is found and by evaluating the Jacobian at the fixed points we are able to classify

their stability.

4.2.1 Lorenz System

The fixed points (x∗, y∗, z∗) of the Lorenz system (Equation 2.1) are well known:

(0, 0, 0) (4.23a)

(√β(ρ− 1),

√β(ρ− 1), (ρ− 1)) (4.23b)

(−√β(ρ− 1),−

√β(ρ− 1), (ρ− 1)) . (4.23c)

The fixed point at the origin exists for all parameter values. Physically it represents

no convection or a stationary Waterwheel. The latter two exist only for ρ > 1 with a

pitchfork bifurcation occurring at ρ = 1. They correspond to steady convection/rotation

of the system. For the Lorenz system the Jacobian (J) is:

J =

−σ σ 0

ρ− z −1 −x

y x −β

. (4.24)

33

The Jacobian at the origin fixed point is:

J(0,0,0) =

−σ σ 0

ρ −1 0

0 0 −β

. (4.25)

The characteristic equation for this matrix is:

(λ+ β)[λ2 + (σ + 1)λ− σ(ρ− 1)

]= 0 , (4.26)

which clearly has one eigenvalue of λ1 = −β the others being given by applying the

quadratic formula, so the other eigenvalues are:

λ± =−(σ + 1)±

√(σ + 1)2 + 4σ(ρ− 1)

2(4.27)

(for future reference we set λ+ = λ2, λ− = λ3). Working from the standard assumption

that σ, ρ and β are positive we find that λ1,3 are always the same sign, negative. The

only eigenvalue that could change sign (and thus the stability of the fixed point) is λ2.

The sign change occurs when ρ = 1. For ρ < 1 we have λ2 < 0 and the origin is stable.

If ρ > 1 we have λ2 > 0 so the origin becomes a saddle point.

Assessing the Jacobian at the (±√β(ρ− 1),±

√β(ρ− 1), (ρ− 1)) fixed points we get

the matrices:

J(±√β(ρ−1),±

√β(ρ−1),(ρ−1)) =

−σ σ 0

1 −1 ∓√β(ρ− 1)

±√β(ρ− 1) ±

√β(ρ− 1) −β

. (4.28)

34

The characteristic equations for these matrices are the same and take the form:

λ3 + (1 + β + σ)λ2 + β(σ + ρ)λ+ 2βσ(ρ− 1) = 0 . (4.29)

If we let ρ = 1 then the inhomogeneous term is equal to 0 and the characteristic

equation is of the form:

λ[λ2 + (β + 1 + σ)λ+ β(1 + σ)

]= 0 (4.30)

which has eigenvalues of λ = 0, λ = −β and λ = −(1 + σ). In general, the varying

values of ρ, σ and β allow two of the eigenvalues to be complex (complex conjugate to one

another) and change the sign of their real part. The Hopf boundary is the point where

the real part of our complex conjugate eigenvalues change sign and thus the stability of

the system. This is a form of bifurcation and when it occurs a limit cycle is created. For

the Waterwheel/Lorenz system the Hopf bifurcation is subcritical meaning the limit cycle

is unstable. This instability is why we don’t find the limit cycle as a periodic solution.

The complex eigenvalues of the system are linearly stable (real part < 0) in the region

outside the Hopf boundary [15] and are non-stable in the region within. The real root is

negative for all values of ρ [17].

For the Lorenz system the Hopf bounday is given by:

ρ = ρH = σσ + β + 3

σ − (β + 1). (4.31)

When 1 < ρ < ρH both of the fixed points are stable and so the Lorenz system will

spiral towards the nearest one. If ρ > ρH however, then the complex eigenvalues will both

have positive real parts and states in the nearby area will spiral outwards. This spiral

forms the Lorenz attractor.

The reason that states do not spiral off to infinity is because all trajectories of the

Lorenz equations are confined to paths of zero volume [15].

35

To show that this property holds we write (x, y, z) as a vector Y with the Lorenz

system being a function f(Y) = Y. Multi variable calculus tells us that the volume has

the property:

V =

∫V

(∇ · f)dV (4.32)

where:

∇ · f =∂

∂x[σ(y − x)] +

∂

∂y[x(ρ− z)− y)] +

∂

∂z[xy − βz] = −(σ + 1 + β) < 0 . (4.33)

By inserting this value into our equation for V we obtain:

V = −∫V

(σ + 1 + β)dV (4.34)

which means that our solution is:

V (t) = e−(σ+1+β)t . (4.35)

So, the trajectories tend to paths of zero volume (such as the Lorenz attractor) ex-

ponentially quickly. This result ensures that our fixed points cannot spiral off to infinity

but instead orbit the strange attractor. It also rules out the possibility of repelling fixed

points, quasiperiodic and repelling orbits as solutions to the Lorenz system. Figure 10 is

a plot of the Hopf Boundary in (σ, ρ) space for β = 1 (as in the Waterwheel system).

Whilst the non-stable fixed points found within the Hopf bounday are necessary for

chaotic systems they are not sufficient on their own, therefore, the blue region in Figure

10 is not entirely chaotic.

36

Figure 10: ρ against σ plot of the regimes of the Lorenz system [1].

4.2.2 Truncated System

As shown in section 4.1.1, representing the Waterwheel in terms of its angular velocity

and vertical/horizontal centres of mass, allowed us to show the equivalence to the Lorenz

system in the set of cases where β = 1. This means, with the appropriate rescaling of

variables laid out in Equation 4.11, that the systems are the same.

By repeating our stability analysis we ensure that the systems are indeed equal and

an algebra mistake has not crept in somewhere. It also serves to check our earlier analysis

for any errors. The stability analysis is not affected by the way we define qi, and thus we

can adopt the approximationN−1∑i=0

q(θi)cos(θi) ' q [1, pg.48].

The fixed points (ω∗, y∗cm, x∗cm) of the Truncated system are:

(0, 0, r) (4.36a)

(

√gqr

ν− k2,

√νk2

gq(r − νk2

gq),νk2

gq) (4.36b)

(−√gqr

ν− k2,−

√νk2

gq(r − νk2

gq),νk2

gq) . (4.36c)

37

The (0, 0, r) fixed point corresponds to the origin in the Lorenz system. It matches

up as the change of variables would suggest. Fixed points 4.36 (b), (c) match up to the

Lorenz system’s accordingly. The Jacobian of the Truncated system is:

J =

−νI

gq

Ik0

xcm −k ω

−ycm −ω −k

. (4.37)

At fixed point 4.36 (a) the Jacobian takes the form:

J(0,0,r) =

−νI

gq

Ik0

r −k 0

0 0 −k

(4.38)

which has the characteristic equation:

(λ+ k)

[(λ+

ν

I)(λ+ k)− gqr

Ik

]= 0 . (4.39)

This has one eigenvalue of λ1 = −k and the others are given by applying the quadratic

formula to:

λ2 + (ν

I+ k)λ+

(νk

I− gqr

Ik

)= 0 . (4.40)

Thus the other eigenvalues are given by:

λ =−(νk + Ik2)±

√(νk + Ik2)2 + 4Ik(gqr − νk2)

2Ik. (4.41)

Like the Lorenz system we set λ+ = λ2, λ− = λ3. As with the default Lorenz system

λ2 is the only eigenvalue which can change sign. The (0, 0, r) fixed point is stable when

gqr

νk2< 1 and a saddle point for

gqr

νk2> 1 which corresponds to the ρ < 1, ρ > 1 cases in

38

the Lorenz system’s origin fixed point. The similarity is consistent with our definition of

ρ in terms of k, ν, g and r.

In our Waterwheel we intend to keep most parameters the same and vary q as our

bifurcation parameter. Consequently, the bifurcation point should be presented as a value

of this parameter (qc). Rearranging the equations above we arrive at:

qc =νk2

gr. (4.42)

Before moving on we should interpret the physical situation represented by the fixed

point. (0, 0, r) represents the position at the top centre of the Waterwheel with no move-

ment. Large values of q relate to an unstable case as we would expect for a bucket perched

at the top of the Waterwheel. The model also allows for the fixed point to be stable which

is clearly not possible. The reason for this oddity is the definition of dampening in the

model (see Equation 3.11). As long as |grN−1∑i=0

misin (θi) | > |νω| it damps the motion

as it should, if the condition is not met then the νω term counteracts the motion in an

unrealistic fashion. This irregularity is very localised and numerical analysis of the system

usually takes place at values of q where the systems acts in line with physical properties

[1].

Assessing the Jacobian at the (±√

gqrν− k2,±

√νk2

gq(r − νk2

gq), νk

2

gq) fixed points we get

the matrices:

J(±√

gqrν−k2,±

√νk2

gq(r− νk2

gq), νk

2

gq)

=

−νI

gqIk

0

νk2

gq−k ±

√gqrν− k2

∓√

νk2

gq(r − νk2

gq) ∓

√gqrν− k2 −k

(4.43)

39

which have the same characteristic equation. It takes the form:

λ3 + (2k +ν

I)λ2 + (

νk

I+gqr

ν)λ+

1

I

√gqr

ν− k2

[ν

√gqr

ν− k2 +

√ν(gqr − νk2

]= 0 .

(4.44)

The case where the inhomogeneous term is 0 arises when q = qc which, accounting

for rescaling, matches our ρ = 1 condition in the Lorenz system. We have omitted the

algebra but the Hopf boundary where eigenvalues change sign is also analogous to the

Lorenz system. The similarities are as expected with the systems being a rescaling of

one another. This supports our findings and has served to double-check our algebra

throughout.

40

5 Refining the Model

We now come to the refinements of the Waterwheel model. Generally water leaking out

of buckets is assumed to leave the system without further interaction. This work tests

the impact of water being caught in other buckets.

We approached this problem from the Discrete bucket case believing that it would be

simpler than the Continuous case.

5.1 Bucket positioning and assumptions

Simplifying Assumptions:

1. Our assumptions regarding the mass and torque balance from previous sections still

hold.

2. Water falls instantly between buckets and matches velocity with them at the moment

of impact.

These assumptions are problematic in regards to outflowing water. Leakage is treated

as directly proportional to the mass in the bucket. The fluid dynamics of the leaking

buckets are complicated but match well with Strogatz’s practical models. In those models

however, tubes are affixed to the bottom of the buckets. This means the fluid dynamics are

reduced to much simpler Poiseuille flow in a pipe. Our Refined model cannot incorporate

such an addition as these tubes carry water away from the system.

The second assumption is a simplifying one to ensure that water being caught does

not affect the total mass through splashing and that the dampening torque factor ν is

unchanged compared to the Basic model.

The scope of this paper makes this adaptation to the model exclusively in the Discrete

case and focuses on the effect on the mass of the system, giving us this revised set of

equations of motion:

41

dmi

dt= qi − kmi + cijkmj (5.1a)

ω =

grN−1∑i=0

misin (θi)− νω

I0 + r2N−1∑i=0

mi

(5.1b)

where cij is the capture coefficient indicating whether bucket i is catching water from

bucket j.

cij =

1 bucket i is catching water from bucket j

0 otherwise

. (5.2)

The conditions for bucket’s catching water from each other are derived in the next

section.

5.1.1 Calculating the capture coefficient

In the model θ is measured from vertically upwards. We can, therefore, calculate the

vertical position of the ith bucket as rcos(θi) and horizontal position as rsin(θi) where

the center of the Waterwheel is the origin, see Figure ??.

Figure 11: Diagram illustrating the horizontal and vertical coordinates ofbuckets.

42

In the N bucket case, buckets are spaced at 2πN

radian intervals around the rim of

the Waterwheel. Two buckets equidistant from the inflow pipe must have their edges

in contact in order to ensure that at least one bucket is being filled at all times. The

distance between the buckets’ centres is 2 halves of their width and is 2rsin( πN

). Meaning

the buckets have a width of 2rsin( πN

) at their rim. The holes through which water leaks

out of the buckets is treated as a single hole directly below the centre of each bucket at

position rsin(θi). The capture coefficient is 1 when the hole in the top bucket (bucket i)

through which water leaks (rsin(θi)) is less than half the bucket width (rsin( πN

)) from the

centre of the catching bucket (bucket j) (rsin(θj)). It is 0 otherwise. We can, therefore,

define the capture coefficient thus:

ci,j =

1 |sin(θi)− sin(θj)| < sin( π

N), cos(θi) < cos(θj)

0 otherwise

. (5.3)

The first condition represents the horizontal displacement between the buckets being

less then their width. The second conditions ensures that the catching bucket is lower

vertically than the leaking bucket so that water is not flowing upwards. In Section 5.2 we

will perform stability analyses on the 2 and 3 bucket models, first on the Basic systems

and then on the Refined versions.

5.2 Examples of the Basic and Refined systems

Before proceeding we explicitly take note that the N bucket waterwheel exhibits N -fold

rotational symmetry. The upshot of this is that any fixed point will be repeated, stability

and all, after a2π

Nrotation. Different buckets assume different positions but as they are

identical this has no effect. This symmetry means that when finding fixed points and

assessing their stability it is sufficient to find one. The other fixed points will appear at

predictable intervals. In addition, analysing one fixed point will give results that can be

applied to all the other fixed points.

43

5.2.1 Example I: 2 Bucket Basic model

We shall now examine the simplest possible Waterwheel, the two bucket case without

refinement. The equations of motion are:

m0 = q0 − km0 (5.4a)

m1 = q1 − km1 (5.4b)

ω =gr(m0 −m1)sin(θ)− νω

I0 + r2(m0 +m1)(5.4c)

θ = ω . (5.4d)

The fixed points (m∗0,m∗1, ω

∗, θ∗) of the system are:

m∗0 =q0k

(5.5a)

m∗1 =q1k

(5.5b)

ω∗ = 0 (5.5c)

θ∗ = 0 or π (mod 2π) . (5.5d)

These correspond to bucket 0 and bucket 1 being situated directly at the top of the

Waterwheel respectively. We define I∗ = I0 + r2(m∗0 + m∗1) as the moment of inertia at

the fixed points. The Jacobian of the system is given by:

44

J =

−k 0 0 0

0 −k 0 0

gr

Isin(θ) −gr

Isin(θ) −ν

I

gr

I(m0 −m1)cos(θ)

0 0 1 0

. (5.6)

In this particular system the Jacobians for both the fixed points are the same. This

is due to (m∗0 −m∗1) and cos(θ∗) both change sign in the different cases. Thus the sign

changes cancel each other out. The Jacobian for both is therefore:

J( qk,0,0,0) = J(0, q

k,0,π) =

−k 0 0 0

0 −k 0 0

0 0 − ν

I∗gqr

I∗k

0 0 1 0

. (5.7)

which has the characteristic equation:

(λ+ k)2[I∗kλ2 + νkλ− gqr

I∗k

]= 0 . (5.8)

Two of the eigenvalues are immediately apparent (λ1,2 = −k), with the other two

being given by:

λ3,4 =−νk ±

√(νk)2 + 4gqI∗rk

2I∗k. (5.9)

We set λ3 = + case and λ4 = - case. λ4 is clearly always negative. For λ3 the sign of

4gqI∗rk determines the overall sign. 4gqI∗rk is always positive meaning λ3 > 0 always.

Thus all fixed points of the 2 Bucket Basic model are saddle points for all parameter

values.

45

5.2.2 Example II 2 Bucket Refined model

We shall now examine the Refined version of the two bucket Waterwheel. The capture

coefficient is:

cij =

1 |sin(θ)| < 1

2, cos(θi) < cos(θj)

0 otherwise

(5.10)

and the equations of motion are:

m0 = q0 − km0 + c01km1 (5.11a)

m1 = q1 − km1 + c10km0 (5.11b)

ω =gr(m0 −m1)sin(θ)− νω

I0 + r2(m0 +m1)(5.11c)

θ = ω . (5.11d)

The fixed points (m∗0,m∗1, ω


m∗0 =q0k

+ c01m1 (5.12a)

m∗1 =q1k

+ c10m0 (5.12b)

ω∗ = 0 (5.12c)

−π6

< θ∗ <π

6or

5π

6< θ∗ <

7π

6(mod 2π) . (5.12d)

In this system (m0 − m1) = 0 whenever either bucket is in a position to capture

leaked water and the appropriate capture coefficient is non-zero (see Figure 12). This

means instead of individual fixed points (as in the Basic case) we have a continuum.

46

These correspond to bucket 0 and bucket 1 being situated at the top of the Waterwheel

respectively. The issues in applying a derivative on a discontinuity (of cij) are the reason

the continuum of fixed points is defined with a strict inequality. We again define I∗ =

I0 + r2(m∗0 +m∗1) as the moment of inertia at the fixed points.

Figure 12: Diagram showing the two edge positions in the continuum of fixedpoints for the 2 Bucket Refined case.

The Jacobian of the system is given by:

J =

−k c01k 0 0

c10k −k 0 0

gr

Isin(θ) −gr

Isin(θ) −ν

I

gr

I(m0 −m1)cos(θ)

0 0 1 0

. (5.13)

At the fixed points with bucket 0 centred on top the Jacobian is:

J( qk, qk,0,0) =

−k 0 0 0

k −k 0 0

0 0 − ν

I∗0

0 0 1 0

. (5.14)

The characteristic equation of which is:

(λ+ k)2λ(λ+ν

I∗) = 0 . (5.15)

47

This means all the eigenvalues are non-positive. The λ = 0 eigenvalue reflects the fact

that a small perturbation will result in being in a different fixed point a small perturbation

away, not moving towards or away from this fixed point. At the edges of the stable ranges

we have the Jacobians:

J( qk, qk,0,±sin−1( 1

2)) =

−k 0 0 0

k −k 0 0

gr

2I∗− gr

2I∗− ν

I∗0

0 0 1 0

. (5.16)

This equates to bucket 0 on top with bucket 1 catching the water leaking from bucket

0.

The characteristic equations corresponding to these Jacobians are the same as for the

Jacobian with θ = 0 and indeed for any Jacobian in the range. The stability analysis is

therefore the same. As previously, discussed the Waterwheel system exhibits rotational

symmetry. The stability analysis for fixed points with bucket 1 on top would yield the

same results. We therefore have a complete picture of the behaviour of the Waterwheel.

5.2.3 Example III: 3 Bucket Basic model

We shall now examine the second simplest Waterwheel, the three bucket case without

refinement. The equations of motion are:

48

m0 = q0 − km0 (5.17a)

m1 = q1 − km1 (5.17b)

m2 = q2 − km2 (5.17c)

ω =gr[(m0 − m1

2− m2

2)sin(θ) +

√32

(m1 −m2)cos(θ)]− νω

I0 + r2(m0 +m1 +m2)(5.17d)

θ = ω . (5.17e)

The fixed points (m∗0,m∗1,m

∗2, ω


m∗0 =q0k

(5.18a)

m∗1 =q1k

(5.18b)

m∗2 =q2k

(5.18c)

ω∗ = 0 (5.18d)

θ∗ = 0 or2π

3or

4π

3(mod 2π) (5.18e)

which correspond to bucket 0, bucket 1 or bucket 2 being situated directly at the top

of the Waterwheel respectively. We define I∗ = I0 + r2(m∗0 +m∗1 +m∗2) as the moment of

inertia at the fixed points. The Jacobian of the system (using S to represent sin and C

to represent cos) is:

49

J =

−k 0 0 0 0

0 −k 0 0 0

0 0 −k 0 0

gr

IS(θ)

gr

IS(θ +

2π

3)

gr

IS(θ +

4π

3) −ν

I

gr

I

[(m0C(θ) +m1C(θ +

2π

3) +m2C(θ +

4π

3)

]0 0 0 1 0

.

(5.19)

The Jacobian for the fixed point with bucket 0 centred on top (θ∗ = 0 mod 2π) is:

J(qk,0,0,0,0)

=

−k 0 0 0 0

0 −k 0 0 0

0 0 −k 0 0

0

√3gr

2I∗−√

3gr

2I∗− ν

I∗gqr

I∗k

0 0 0 1 0

. (5.20)

Which has the characteristic equation:

(λ+ k)3[I∗kλ2 + νkλ− gqr

I∗k

]= 0 . (5.21)

This is the same as the 2 Bucket system but with an extra eigenvalue of λ = −k. The

stability analysis is therefore the same as for that system and we omit the calculation.

5.2.4 Example IV: 3 Bucket Refined model

For the Refined three Bucket Waterwheel the equations of motion are:

50

m0 = q0 − km0 + c01km1 + c02km2 (5.22a)

m1 = q1 − km1 + c10km0 + c12km2 (5.22b)

m2 = q2 − km2 + c20km0 + c21km1 (5.22c)

ω =gr[(m0 − m1

2− m2

2)sin(θ) +

√32

(m1 −m2)cos(θ)]− νω

I0 + r2(m0 +m1 +m2)(5.22d)

θ = ω . (5.22e)

The fixed points (m∗0,m∗1,m

∗2, ω

∗, θ∗) of the system are non-existent. To show this first

consider the possible positions of the Waterwheel:

Figure 13: Diagram illustrating the possible positions a 3 Bucket Waterwheelcan take.

As can be identified from the image there is always a bucket in top position and another

in position to capture leaked water. This system has no fixed points. A Hypothetical fixed

point with bucket 0 on top would either have c01 = 1 and 0 < θ < π3

(case I) or c02 = 1

andπ

3< θ < 0 (case II). We now consider our values:

51

m∗0 =q

k(5.23a)

m∗1 =

q

kcase I

0 case II

(5.23b)

m∗2 =

0 case I

q

kcase II

(5.23c)

ω∗ = 0 (5.23d)

0 =

12sin(θ∗) +

√32cos(θ∗) case I

12sin(θ∗)−

√32cos(θ∗) case II

. (5.23e)

Rearranging Equation 5.23 (e) we find values for θ∗:

tan(θ∗) =

−√

3 case I

√3 case II

⇒ θ∗ =

−π

3case I

π

3case II

. (5.24a)

In both cases the only value of θ∗ that allows for a fixed point lies outside of our

possible range of values showing that there is no fixed point with bucket 0 on top. The

rotational symmetry of the system means that this holds true for bucket 1 and bucket 2

as well. Thus the Refined 3 Bucket system has no fixed points.

52

6 Effect of Refinement on the System

In this section we discuss the results Matlab has produced using the Refined equations

we derived previously. See Appendix B for the code used to model the Waterwheel.

6.1 2 Bucket systems

Figure 14 shows example plots of the 2 Bucket Basic/Refined systems with identical initial

conditions and parameter values. The system parameters are R = 2, I0 = 0.04, Q = 0.025,

g = 9.81, ν = 1.5 and k = 0.013. The initial conditions are within the continuum of fixed

points for the Refined wheel but not one of the two fixed points for the Basic system. The

resulting plots clearly show aperiodic, bounded values in the Basic system supporting the

idea that it’s chaotic whereas the Refined system is in one of it’s neutrally stable fixed

points and remains there in perpetuity.

(a) 2 Bucket Basic system

(b) 2 Bucket Refined system

Figure 14: Example plots of the 2 Bucket Basic/Refined systems.

53

The red and blue lines in each case are the masses of water in bucket 0 and bucket 1

respectively (in the Refined system they overlap leading to only one being visible). The

purple line represents the total mass and the light blue line is the angular velocity. Total

mass tending to a constant, chaos etc all arise in line with the stability analyses and

derivations from previous sections.

6.1.1 Fixed points

In the Basic model we have two fixed points corresponding to either bucket being located

at the top of the Waterwheel. The Refined model has a continuum whenever the buckets

are in a position to catch water from one another (see Figure 12). This is a property we

would expect to see in systems with different (but even) numbers of buckets albeit for

smaller ranges of θ.

6.1.2 Bifurcation Values

Whilst neither system undergoes a bifurcation, as system parameters vary, we have shown

that in at least some ranges of θ the Total mass of the Refined system will adopt a constant

value. This value is twice as high as in the Basic system reflecting the elongated route

that water must take to leave the system.

6.2 3 Bucket Systems

Figure 15 shows example plots of the 3 Bucket Basic/Refined systems with identical initial

conditions and parameter values. The system parameters are R = 2, I0 = 0.04, Q = 0.02,

g = 9.81, ν = 1.5 and k = 0.013. The starting point represents the fixed point with bucket

0 on top in the Basic case but as our analysis indicated no such fixed point exists for the

Refined case. The resulting plots clearly show aperiodic, bounded values in the Refined

system which supports the idea that it is chaotic whereas the Basic system remains in its’

fixed points in perpetuity.

54

(a) 3 Bucket Basic system

(b) 3 Bucket Refined system

Figure 15: Example plots of the 3 Bucket Basic/Refined systems.

Again the light blue line represents angular velocity and the purple line Total mass.

We can see that total mass in the Refined case tends to a constant. Red, blue and green

lines represent the mass in each bucket.

6.2.1 Fixed points

In the Basic model we have three fixed points corresponding to each of the three buckets

occupying the θi = 0 position. The Refined model by contrast has no fixed points at

all. As Figure 13 is meant to illustrate the top bucket will always be leaking in to one of

the others, this ensures that even when the top bucket is dead center there is a source of

gravitational torque from the catching bucket that serves to make the system unstable.

Like the even buckets cases this lack of fixed points is a property we would expect to find

in all Refined systems with an odd number of buckets

55

6.2.2 Bifurcation Values

Like the two Bucket systems neither of the 3 bucket examples undergo a bifurcation as

system parameters vary. In this Refined system we know that the total mass tends to a

constant value (2q

k) (more clearly than the two Bucket Refined case) since there is always

one, and only one, bucket in a position to catch leaked water. This value is the same as

in the two Bucket Refined case. We do not have enough examples or a theoretical proof

but it does appear that the total mass is the same for all Refined Systems.

56

7 Conclusions and Scope for Future Work

The aim of this paper has been to refine the existing model of the chaotic Waterwheel

in order to produce more accurate governing equations. To this end, we investigated the

equations and dynamics of the basic system. Then we refined the model by considering

the water flow between buckets. So deriving new governing equations and repeating our

investigation of refined systems dynamics, in comparison to the basic model.

Section 2 summarises the literature review undertaken for this project. Of note is

the fact that although Lorenz established his equations in 1963, it was not until Malkus

built his waterwheel at MIT in the early 1970s that the concept of Chaos theory took

off. Gleick provides some very useful insights regarding Lorenz’s work and Malkus’s initial

scepticism of chaos theory when the concept was first proposed. A significant detail in light

of Malkus’s subsequent developments in the field. Of the more recent research reviewed

Alonso’s 2010 paper [1] on ‘Deterministic Chaos in Malkus’ Waterwheel’ is particularly

instructive for our study. This paper differs from previous works in using analytical

mechanics for derivations of the Discrete and Continuous versions of the Waterwheel,

reducing the scope for error. An examination of this approach would benefit future

researchers in this field.

In Section 3 we outline the governing equations of the basic Waterwheel by balancing

the effects of mass and torque. This approach is useful for finding fixed points within,

and conducting short-term analysis upon, the Waterwheel system. Its limitations revolve

around the difficulty of analysing the transitions between which bucket occupies the top

position, as this masks the butterfly effect. In addition, the amount of algebra needed for

analysis skyrockets as the number of buckets in the model increases.

Given these limitations, in Section 4, we decided to analyse the basic Waterwheel

using a Centre of Mass approach. This has the advantage of limiting our variables to

three, ensuring a consistent amount of algebra regardless of the number of buckets and

giving a clear indication of when the system bifurcates into chaos. Disadvantages of this

57

approach include an inability to distinguish between buckets. This makes it difficult to get

a physical idea of the fixed points. When introducing new physical factors to the governing

equations, the lack of a clear link between physics and equations also complicates matters.

As ‘Deterministic Chaos in Malkus’ Waterwheel’ is unclear about the relation between

the Discrete case and the Lorenz system, we contacted Dr Alonso to clarify this point.

He established that for a direct relationship, one needs to assume that there are infinitely

many buckets. In this and the following section, we also established that the Discrete case

is best used for physical systems and the Continuous case for comparison to the Lorenz

system.

Section 5 distinguishes itself from previous literature with the introduction of the

capture coefficient (cij). Here we evaluate the two and three bucket versions of the Wa-

terwheel with and without this factor. This helps us identify the difference in dynamics

caused by the refinement. Matlab is invaluable for testing our models and creating exam-

ples of the Lorenz system and chaotic properties. However, this project has emphasised

how weak our coding skills are and the need for further study if we continue to use the

programme.

In Section 6 we examine the differences between the basic and refined models, such as

their altered range of fixed points and differing physical limits. Our conclusion is that the

seemingly simple refinement of flow between buckets has a drastic impact on the dynamics

of the Waterwheel.

If we were to undertake this study again, we would attempt to adopt the Centre of

Mass approach for the Refined Waterwheel to get a better idea of chaos-order transitions

rather than fixed points. We believe that the fluid dynamics within the Waterwheel is a

somewhat neglected area of study. Work on the effect of inertia on leak rate, the inertia

of water flowing between buckets and the effect on ν, specifically, of bringing water of

variable speed to the same velocity as the Waterwheel, would be particularly useful.

58

A List of Nomenclature, subdivided by type of model

General NomenclatureVariable Description Units

θ The angle of rotation relative to the labora-tory, increases counter clockwise

Radians

ω(t) Angular velocity of Waterwheel, (= θ) Radians per Secondq Rate at which water is pumped into the sys-

temKilograms per Second

M(t) Total mass of water in the Waterwheel

M(t) =

∑i

mi(t) Discrete∫ π−πm(θ, t)dθ Continuous

Kilograms

r Radius of the wheel Metresg Gravity Newtonsν Rotational dampening rate, controllable by

merit of brakeNewton Metre Seconds

I0 The moment of inertia if the Waterwheel isempty

Kilogram Metres squared

I The moment of inertia of thewheel, (constant after sufficient time)

I =

(I0 + r2

N−1∑i=0

mi

) Kilogram Metres squared

τ The torque of the wheel, (=d

dtIω) Newton Metres

k Leak rate from buckets per Second

Discrete NomenclatureVariable Description Unitsmi(t) The mass of water in the ith bucket Kilogramsθi The position of the ith bucket Radiansqi Water flowing into the ith bucket(

qi =

{q if cos( π

N) < cos(θ + i2π

N)

0 otherwise

) Kilograms per Second

N The number of buckets None/DimensionlessT The Kinetic energy:

12

(I0 + r2

N−1∑i=0

mi

)θ2 → 1

2(I) θ2

Newton Metres

V The Potential energy: grN−1∑i=0

micos(θ + i2π

N

)Newton Metres

L The Lagrangian (= T − V ) Newton Metres

F Raleigh’s dissipation function: (=1

2νθ2) Newton Metres per Second

59

Continuous NomenclatureVariable Description Unitsm(θ, t) Mass distribution of water Kilograms per Radianα The tilt of the Waterwheel axis Radiansg0 Effective gravity, (= gsinα) Newtons

M∗(t) Total mass of water between θ1 and θ2 KilogramsQ(θ) Rate at which water is pumped into buckets,

(q =∫ 2π

0Qdθ)

Kilograms per Radian Second

60

B Matlab Code

This Appendix provides the Matlab code used to create the diagrams in the paper not

taken (and cited) as from an outside source

B.1 Basic Lorenz system

1 f unc t i on dy = simple ( t , y )% The Lorenz system with sigma = 10 , rho = 28 ,

beta = 8/3

dy = ze ro s (3 , 1 ) %a coloumn vecto r . y (1 ) r e p r e s e n t s x , y (2 ) r e p r e s e n t s y

and y (3) r e p r e s e n t s z

3 dy (1) = 10 ∗ ( y (2 ) − y (1 ) ) ;

dy (2 ) = y (1) ∗ (28 − y (3 ) ) − y (2 ) ;

5 dy (3) = y (1) ∗ y (2 ) − (8/3) ∗ y (3 ) ;

B.2 Comparison plot

1 [ T1 , Y1 ] = ode45 ( @simple , [ 0 5 0 ] , [ 1 1 1 ] ) ; %Lorenz system with i n t i a l

c o n d i t i o n s (1 , 1 , 1 ) from time 0 to 50

[ T2 , Y2 ] = ode45 ( @simple , [ 0 5 0 ] , [ 1 . 0 0 1 1 1 ] ) ; %Lorenz system with i n t i a l

c o n d i t i o n s ( 1 . 0 0 1 , 1 , 1 ) from time 0 to 50

3 p lo t (T1 , Y1 ( : , 1 ) ,T2 , Y2 ( : , 1 ) ) ; %p lo t o f the Lorenz systems above

t i t l e ( ’ Comparison o f x aga in s t t f o r l o r e n z equat ions with d i f f e r i n g

i n i t i a l c o n d i t i o n s ’ , ’ f o n t s i z e ’ , 24)

5 x l a b e l ( ’Time ( t ) ’ , ’ f o n t s i z e ’ , 20)

y l a b e l ( ’ Po s i t i on x ( t ) ’ , ’ f o n t s i z e ’ , 20)

61

B.3 Lorenz attractor

[T,Y] = ode45 ( @simple , [ 0 1 0 0 ] , [ 1 1 1 ] ) ; %Lorenz system with i n t i a l

c o n d i t i o n s (1 , 1 , 1 ) from time 0 to 100

2 p lo t (Y( : , 1 ) ,Y( : , 3 ) ) ; %p lo t o f x aga in s t z

t i t l e ( ’ Strange Att ractor ’ , ’ f o n t s i z e ’ , 24)

4 x l a b e l ( ’ x ( t ) ’ , ’ f o n t s i z e ’ , 20)

y l a b e l ( ’ z ( t ) ’ , ’ f o n t s i z e ’ , 20)

B.4 Basic 2 Bucket system

1 %Function to c a l c u l a t e the motion o f a 2 bucket Chaotic Waterwheel without

%re f inement

3 f unc t i on myODE(R, I ,Q, g , nu , k )

IC = [ 1 . 9 2 3 1 1 .9231 0 0 . 0 0 1 ] ; %I n i t i a l c o n d i t i o n s o f the Waterwheel

5 opt ions = odeset ( ’ AbsTol ’ ,1 e−10, ’ RelTol ’ ,1 e−10) ;

R = 2 %Radius o f the Wheel

7 I = 0 .04 %Moment o f I n e r t i a o f empty wheel . . .

Q = 0.025 %Water in f l ow ra t e

9 g = 9.81 %E f f e c t i v e Gravity

nu = 1 .5 %I n e r t i a l Dampening term

11 k = 0.013 %Leakage Rate

[ t ,X] = ode45 (@TwoBB, [ 0 500 ] , IC , opt ions ,R, I ,Q, g , nu , k ) ;

13 f i g u r e

p l o t ( t ,X( : , 1 ) , ’ r ’ , t ,X( : , 2 ) , ’ b ’ , t ,X( : , 1 ) + X( : , 2 ) , ’m’ , t ,X( : , 3 ) , ’ c ’ )

15 x l a b e l ( ’ t ’ )

y l a b e l ( ’m 0 , m 1 , M, \omega ’ )

17 g r id

end

19 f unc t i on dx = TwoBB( t , x ,R, I ,Q, g , nu , k )%x (1) , x (2 ) are masses o f water in

buckets 0 and 1 , x (3 ) i s angular v e l o c t i y and x (4) i s the angular

p o s i t i o n o f bucket 0

62

i f cos ( p i /2) < cos ( x (4 ) )%Bucket 0 on top

21 dx (1) = Q − k∗x (1 ) ;

dx (2 ) = −k∗x (2 ) ;

23 dx (3) = ( g∗R∗ ( x (1 ) ∗ s i n ( x (4 ) ) + x (2) ∗ s i n ( x (4 ) + pi ) ) − nu∗x (3 ) ) /( I + Rˆ2

∗ ( x (1 ) + x (2) ) ) ;

dx (4 ) = x (3) ;

25

e l s e i f cos ( p i /2) < cos ( x (4 ) + pi ) %Bucket 1 on top

27 dx (1) = −k∗x (1 ) ;

dx (2 ) = Q − k∗x (2 ) ;


∗ ( x (1 ) + x (2) ) ) ;

dx (4 ) = x (3) ;

31 end

dx=dx ’ ;

33 end

B.5 Refined 2 Bucket system

1 %Function to c a l c u l a t e the motion o f a 2 bucket Chaotic Waterwheel with

%re f inement


IC = [ 1 . 9 2 3 1 1 .9231 0 0 . 0 0 1 ] ; %I n i t i a l c o n d i t i o n s o f the Waterwheel








[ t ,X] = ode45 (@TwoBR, [ 0 500 ] , IC , opt ions ,R, I ,Q, g , nu , k ) ;

63

13 f i g u r e

p l o t ( t ,X( : , 1 ) , ’ r ’ , t ,X( : , 2 ) , ’ b ’ , t ,X( : , 1 ) + X( : , 2 ) , ’m’ , t ,X( : , 3 ) , ’ c ’ )

15 x l a b e l ( ’ t ’ )

y l a b e l ( ’m 0 , m 1 , M, \omega ’ )

17 g r id

end

19 f unc t i on dx = TwoBR( t , x ,R, I ,Q, g , nu , k )%x (1) , x (2 ) are masses o f water in

buckets 0 and 1 , x (3 ) i s angular v e l o c t i y and x (4) i s the angular

p o s i t i o n o f bucket 0

i f cos ( x (4 ) ) > 0 & abs ( s i n ( x (4 ) ) ) > 1/2 %Bucket 0 on top , too f a r apart to

catch water

21 dx (1) = Q − k∗x (1 ) ;

dx (2 ) = −k∗x (2 ) ;


∗ ( x (1 ) + x (2) ) ) ;

dx (4 ) = x (3) ;

25

e l s e i f cos ( x (4 ) ) > 0 & abs ( s i n ( x (4 ) ) ) < 1/2 %Bucket 0 on top , c l o s e enough

to catch water

27 dx (1) = Q − k∗x (1 ) ;

dx (2 ) = k∗x (1 ) − k∗x (2 ) ;


∗ ( x (1 ) + x (2) ) ) ;

dx (4 ) = x (3) ;

31

e l s e i f cos ( x (4 ) ) < 0 & abs ( s i n ( x (4 ) ) ) > 1/2 %Bucket 1 on top , too f a r

apart to catch water

33 dx (1) = − k∗x (1 ) ;

dx (2 ) = Q − k∗x (2 ) ;


∗ ( x (1 ) + x (2) ) ) ;

dx (4 ) = x (3) ;

37

64

e l s e i f cos ( x (4 ) ) < 0 & abs ( s i n ( x (4 ) ) ) < 1/2 %Bucket 1 on top , c l o s e enough

to catch water

39 dx (1) = k∗x (2 ) − k∗x (1 ) ;

dx (2 ) = Q − k∗x (2 ) ;


∗ ( x (1 ) + x (2) ) ) ;

dx (4 ) = x (3) ;

43 end

dx=dx ’ ;

45 end

B.6 Basic 3 Bucket system

1 %Function to c a l c u l a t e the motion o f a 3 bucket Chaotic Waterwheel without

%re f inement


IC = [ 0 0 0 0 0 ] ; %I n i t i a l c o n d i t i o n s o f the Waterwheel








[ t ,X] = ode45 (@ThreeBB , [ 0 500 ] , IC , opt ions ,R, I ,Q, g , nu , k ) ;

13 f i g u r e

p l o t ( t ,X( : , 1 ) , ’ r ’ , t ,X( : , 2 ) , ’ b ’ , t ,X( : , 3 ) , t ,X( : , 1 ) + X( : , 2 ) + X( : , 2 ) , ’m’ , t ,X

( : , 3 ) , ’ g ’ , t ,X( : , 4 ) , ’ c ’ )

15 t i t l e ( ’ 3 Bucket Bas ic ’ )

x l a b e l ( ’ t ’ )

17 y l a b e l ( ’m 0 , m 1 , m 3 , M, \omega ’ )

65

g r id

19 end

func t i on dx = ThreeBB( t , x ,R, I ,Q, g , nu , k )

21 %x (1) , x (2 ) , x (3 ) are masses o f water in buckets 0 , 1 and 2 r e s p e c t i v e l y .

x (4 ) i s angular v e l o c t i y and x (5) i s the angular p o s i t i o n o f bucket 0

i f cos ( x (5 ) ) > cos ( p i /3) %Bucket 0 on top

23 dx (1) = Q − k∗x (1 ) ;

dx (2 ) = − k∗x (2 ) ;

25 dx (3) = − k∗x (3 ) ;

dx (4 ) = ( g∗R∗ ( x (1 ) ∗ s i n ( x (5 ) ) + x (2) ∗ s i n ( x (5 ) +(2/3)∗ pi ) + x (3) ∗ s i n ( x (5 )

+(4/3)∗ pi ) ) − nu∗x (4 ) ) /( I + Rˆ2∗ ( x (1 ) + x (2) + x (3) ) ) ;

27 dx (5) = x (4) ;

29 e l s e i f cos ( x (5 ) +(2/3)∗ pi ) > cos ( p i /3) %Bucket 1 on top

dx (1) = − k∗x (1 ) ;

31 dx (2) = Q − k∗x (2 ) ;

dx (3 ) = − k∗x (3 ) ;

33 dx (4) = ( g∗R∗ ( x (1 ) ∗ s i n ( x (5 ) ) + x (2) ∗ s i n ( x (5 ) +(2/3)∗ pi ) + x (3) ∗ s i n ( x (5 )

+(4/3)∗ pi ) ) − nu∗x (4 ) ) /( I + Rˆ2∗ ( x (1 ) + x (2) + x (3) ) ) ;

dx (5 ) = x (4) ;

35

e l s e i f cos ( x (5 ) +(4/3)∗ pi ) > cos ( p i /3) %Bucket 2 on top

37 dx (1) = − k∗x (1 ) ;

dx (2 ) = − k∗x (2 ) ;

39 dx (3) = Q − k∗x (3 ) ;


+(4/3)∗ pi ) ) − nu∗x (4 ) ) /( I + Rˆ2∗ ( x (1 ) + x (2) + x (3) ) ) ;

41 dx (5) = x (4) ;

end

43 dx=dx ’ ;

end

66

B.7 Refined 3 Bucket system

%Function to c a l c u l a t e the motion o f a 3 bucket Chaotic Waterwheel with

2 %ref inement

func t i on myODE(R, I ,Q, g , nu , k )

4 IC = [ 0 0 0 0 0 ] ; %I n i t i a l c o n d i t i o n s o f the Waterwheel

opt ions = odeset ( ’ AbsTol ’ ,1 e−10, ’ RelTol ’ ,1 e−10) ;

6 R = 2 %Radius o f the Wheel

I = 0 .04 %Moment o f I n e r t i a o f empty wheel . . .

8 Q = 0.02 %Water in f l ow ra t e

g = 9.81 %E f f e c t i v e Gravity

10 nu = 1 .5 %I n e r t i a l Dampening term

k = 0.013 %Leakage Rate

12 [ t ,X] = ode45 (@ThreeBR , [ 0 500 ] , IC , opt ions ,R, I ,Q, g , nu , k ) ;

f i g u r e

14 p lo t ( t ,X( : , 1 ) , ’ r ’ , t ,X( : , 2 ) , ’ b ’ , t ,X( : , 3 ) , t ,X( : , 1 ) + X( : , 2 ) + X( : , 2 ) , ’m’ , t ,X

( : , 3 ) , ’ g ’ , t ,X( : , 4 ) , ’ c ’ )

t i t l e ( ’ 3 Bucket Ref ined ’ )

16 x l a b e l ( ’ t ’ )

y l a b e l ( ’m 0 , m 1 , m 3 , M, \omega ’ )

18 g r id

end

20 f unc t i on dx = ThreeBR( t , x ,R, I ,Q, g , nu , k )

%x (1) , x (2 ) , x (3 ) are masses o f water in buckets 0 , 1 and 2 r e s p e c t i v e l y .

x (4 ) i s angular v e l o c t i y and x (5) i s the angular p o s i t i o n o f bucket 0

22 i f cos ( x (5 ) ) > cos ( p i /3) & abs ( s i n ( x (5 ) )−s i n ( x (5 ) +(2/3)∗ pi ) ) > s i n ( p i /3)

& abs ( s i n ( x (5 ) )−s i n ( x (5 ) +(4/3)∗ pi ) ) < s i n ( p i /3) %Bucket 0 on top , Bucket

2 catch ing water

dx (1 ) = Q − k∗x (1 ) ;

24 dx (2) = − k∗x (2 ) ;

dx (3 ) = k∗x (1 ) − k∗x (3 ) ;


+(4/3)∗ pi ) ) − nu∗x (4 ) ) /( I + Rˆ2∗ ( x (1 ) + x (2) + x (3) ) ) ;

dx (5 ) = x (4) ;

67

28

e l s e i f cos ( x (5 ) ) > cos ( p i /3) & abs ( s i n ( x (5 ) )−s i n ( x (5 ) +(2/3)∗ pi ) ) < s i n ( p i

/3) & abs ( s i n ( x (5 ) )−s i n ( x (5 ) +(4/3)∗ pi ) ) > s i n ( p i /3) %Bucket 0 on top ,

Bucket 1 catch ing water

30 dx (1) = Q − k∗x (1 ) ;

dx (2 ) = k∗x (1 ) − k∗x (2 ) ;

32 dx (3) = − k∗x (3 ) ;


+(4/3)∗ pi ) ) − nu∗x (4 ) ) /( I + Rˆ2∗ ( x (1 ) + x (2) + x (3) ) ) ;

34 dx (5) = x (4) ;

36 e l s e i f cos ( x (5 ) +(2/3)∗ pi ) > cos ( p i /3) & abs ( s i n ( x (5 ) +(2/3)∗ pi )−s i n ( x (5 ) ) )

> s i n ( p i /3) & abs ( s i n ( x (5 ) +(2/3)∗ pi )−s i n ( x (5 ) +(4/3)∗ pi ) ) < s i n ( p i /3) %

Bucket 1 on top , Bucket 2 catch ing water

dx (1 ) = − k∗x (1 ) ;

38 dx (2) = Q − k∗x (2 ) ;

dx (3 ) = k∗x (2 ) − k∗x (3 ) ;


+(4/3)∗ pi ) ) − nu∗x (4 ) ) /( I + Rˆ2∗ ( x (1 ) + x (2) + x (3) ) ) ;

dx (5 ) = x (4) ;

42

e l s e i f cos ( x (5 ) +(2/3)∗ pi ) > cos ( p i /3) & abs ( s i n ( x (5 ) +(2/3)∗ pi )−s i n ( x (5 ) ) )

< s i n ( p i /3) & abs ( s i n ( x (5 ) +(2/3)∗ pi )−s i n ( x (5 ) +(4/3)∗ pi ) ) > s i n ( p i /3) %


44 dx (1) = k∗x (2 )− k∗x (1 ) ;

dx (2 ) = Q − k∗x (2 ) ;

46 dx (3) = − k∗x (3 ) ;


+(4/3)∗ pi ) ) − nu∗x (4 ) ) /( I + Rˆ2∗ ( x (1 ) + x (2) + x (3) ) ) ;

48 dx (5) = x (4) ;

50 e l s e i f cos ( x (5 ) +(4/3)∗ pi ) > cos ( p i /3) & abs ( s i n ( x (5 ) +(4/3)∗ pi )−s i n ( x (5 ) ) )

> s i n ( p i /3) & abs ( s i n ( x (5 ) +(4/3)∗ pi )−s i n ( x (5 ) +(2/3)∗ pi ) ) < s i n ( p i /3) %

68


dx (1 ) = − k∗x (1 ) ;

52 dx (2) = k∗x (3 ) − k∗x (2 ) ;

dx (3 ) = Q − k∗x (3 ) ;


+(4/3)∗ pi ) ) − nu∗x (4 ) ) /( I + Rˆ2∗ ( x (1 ) + x (2) + x (3) ) ) ;

dx (5 ) = x (4) ;

56

e l s e i f cos ( x (5 ) +(4/3)∗ pi ) > cos ( p i /3) & abs ( s i n ( x (5 ) +(4/3)∗ pi )−s i n ( x (5 ) ) )

< s i n ( p i /3) & abs ( s i n ( x (5 ) +(4/3)∗ pi )−s i n ( x (5 ) +(2/3)∗ pi ) ) > s i n ( p i /3) %


58 dx (1) = k∗x (3 )− k∗x (1 ) ;

dx (2 ) = − k∗x (2 ) ;

60 dx (3) = Q − k∗x (3 ) ;


+(4/3)∗ pi ) ) − nu∗x (4 ) ) /( I + Rˆ2∗ ( x (1 ) + x (2) + x (3) ) ) ;

62 dx (5) = x (4) ;

64 end

dx=dx ’ ;

66 end

69

C Website

70

D References

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87

modelling the chaotic waterwheel

Education

bucket waterwheel

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bucket systems

basic system

continuous system

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