^{1,a)}

### Abstract

The Malkus–Lorenz water wheel is analyzed in a more direct way. Two new dimensionless parameters associated with properties of the water wheel are used in place of the traditional Lorenz parameters, which relate only to Lorenz’s fluid model. The primary result is a performance map in the new dimensionless parameter space, which shows where the major behavior types occur and associated bifurcations. A slice across this map is examined by a bifurcation plot that shows details of the transitions between various types of behavior. An example of a 12-cup water wheel model is provided, and its behavior is compared over a wide range of parameters on the performance map to that of the Malkus-Lorenz water wheel. It is shown how the map with its water wheel parameters can be simply converted to the map with Lorenz’s fluid dynamics parameters.

The work was initiated following suggestions by Professor Robert L. Zimmerman, University of Oregon, who provided essential guidance. Consultations with Professor Godfrey Gumbs, Hunter College, are greatly appreciated, as is assistance in the required file format conversions by Rakesh Venkatesh, Tufts University. Many excellent suggestions by the unknown reviewers were gratefully incorporated.

I. INTRODUCTION

II. MALKUS WATER WHEEL EQUATIONS

III. DIMENSIONLESS PARAMETERS

IV. MAJOR BIFURCATIONS

A. The pitchfork bifurcation

B. The onset of preturbulence, the B bifurcation

C. The onset of stable chaos: The C bifurcation

D. The Hopf bifurcation

V. THE WATER WHEEL BEHAVIOR MAP

A. Periodic orbit examples

B. Bifurcation diagrams

VI. RELATION TO THE LORENZ EQUATIONS AND PARAMETERS

VII. SUGGESTIONS FOR EXPERIMENTS

A. Finding values of the parameters for a nonideal system

B. Computer model of a nonideal water wheel

C. Performance examples

VIII. SUMMARY

### Key Topics

- Bifurcations
- 31.0
- Friction
- 11.0
- Water masses
- 9.0
- Differential equations
- 8.0
- Overflows
- 8.0

## Figures

Twelve-cup water wheel. The wheel’s axis is horizontal; water is added at the top, and the hanging cups leak.

Twelve-cup water wheel. The wheel’s axis is horizontal; water is added at the top, and the hanging cups leak.

Continuous-cups water wheel. The axis is tilted so that the fixed cups can form a continuous ring; water enters at several locations, symmetric about the high point.

Continuous-cups water wheel. The axis is tilted so that the fixed cups can form a continuous ring; water enters at several locations, symmetric about the high point.

Water wheel coordinates. The coordinate system is in the plane of the wheel, with the cups located on the circle of radius . A single input stream of water enters the highest cup at the point . The center of mass of the water in all the cups is located at a point within the circle.

Water wheel coordinates. The coordinate system is in the plane of the wheel, with the cups located on the circle of radius . A single input stream of water enters the highest cup at the point . The center of mass of the water in all the cups is located at a point within the circle.

Example of preturbulence. (a) Plot of the angular velocity versus time. (b) The plot of versus shows the path of the center of mass motion in the plane of the wheel. A possibly very long initial period of apparent chaos is followed by convergence to a steady rotation.

Example of preturbulence. (a) Plot of the angular velocity versus time. (b) The plot of versus shows the path of the center of mass motion in the plane of the wheel. A possibly very long initial period of apparent chaos is followed by convergence to a steady rotation.

Bifurcation parameter . Plots of (a) angular velocity versus time and (b) the center of mass motion are shown for a pair of values of that straddle the value of . The initial conditions are very close to the stationary point at , and .

Bifurcation parameter . Plots of (a) angular velocity versus time and (b) the center of mass motion are shown for a pair of values of that straddle the value of . The initial conditions are very close to the stationary point at , and .

Bifurcation parameter . Same comments as for Fig. 5, but straddling the point.

Bifurcation parameter . Same comments as for Fig. 5, but straddling the point.

Example of a chaotic orbit.

Example of a chaotic orbit.

Water wheel map for the water wheel parameters . Above the A curve the wheel always comes to a stop. Between the A and C bifurcation curves the only stable behavior is steady unidirectional motion. Between the C and D curves, steady turning, periodic motion, or chaos results depending on the initial conditions. Below the D curve there are only periodic orbits or chaos. Periodic orbit examples are symmetric pendulum motion within the E contour, a “fill and fall” motion between I and J, and others in an infinite number of narrow strips such as the partial curves F, G, and H, which lie in the chaotic region. The horizontal dashed line shows where the bifurcation plot of Fig. 10 is located, with only changing. The curved dashed line is an example with only the input water rate changing.

Water wheel map for the water wheel parameters . Above the A curve the wheel always comes to a stop. Between the A and C bifurcation curves the only stable behavior is steady unidirectional motion. Between the C and D curves, steady turning, periodic motion, or chaos results depending on the initial conditions. Below the D curve there are only periodic orbits or chaos. Periodic orbit examples are symmetric pendulum motion within the E contour, a “fill and fall” motion between I and J, and others in an infinite number of narrow strips such as the partial curves F, G, and H, which lie in the chaotic region. The horizontal dashed line shows where the bifurcation plot of Fig. 10 is located, with only changing. The curved dashed line is an example with only the input water rate changing.

Examples of stable periodic orbits. The motion of the water’s center of mass in the plane of the wheel is shown for five stable orbits, all with , located at five dots on Fig. 8. (a) Plot of the initial transient behavior leading to the periodic pendulum stable orbit. The final pendulum orbit and the other final orbits are shown enlarged in (b)–(f). The starting points are all close to .

Examples of stable periodic orbits. The motion of the water’s center of mass in the plane of the wheel is shown for five stable orbits, all with , located at five dots on Fig. 8. (a) Plot of the initial transient behavior leading to the periodic pendulum stable orbit. The final pendulum orbit and the other final orbits are shown enlarged in (b)–(f). The starting points are all close to .

Bifurcation diagram for (see dotted line in Fig. 8). For each value of from 0.005 through 3.0 a run was started close to as was done for Fig. 8 and run for for the initial transient to be removed. For the next a dot is placed in the figure at each extremum (change in direction) of the coordinate of the center of mass. Abrupt changes as is varied indicate bifurcations.

Bifurcation diagram for (see dotted line in Fig. 8). For each value of from 0.005 through 3.0 a run was started close to as was done for Fig. 8 and run for for the initial transient to be removed. For the next a dot is placed in the figure at each extremum (change in direction) of the coordinate of the center of mass. Abrupt changes as is varied indicate bifurcations.

(a) The top and right borders indicate the standard Lorenz parameters (, , with ) for this plot and for Fig. 8. (b) As in Fig. 8, and are proportional to the inverse of the cup leakage and wheel slowdown time constants, respectively. The A, C, D, and E contours are for the ideal Malkus–Lorenz wheel, for which steady unidirectional motion occurs only between curves A and D, chaos is found only below curve C, and symmetrical pendulum motion occurs only within curve E. The labels Un, Pe, Ch, ChSt, and Aff indicate the type of motions (Unidirectional, Pendulum, Chaos, Chaos-then-Stationary, and Asymmetric fill-and-fall) found for a computer model of a 12-cup wheel, at the corresponding locations on this simplified version of Fig. 8. The values were found using the procedure outlined in Sec. VII A. The behavior of the 12-cup wheel is generally similar to that of the ideal water wheel, except for large .

(a) The top and right borders indicate the standard Lorenz parameters (, , with ) for this plot and for Fig. 8. (b) As in Fig. 8, and are proportional to the inverse of the cup leakage and wheel slowdown time constants, respectively. The A, C, D, and E contours are for the ideal Malkus–Lorenz wheel, for which steady unidirectional motion occurs only between curves A and D, chaos is found only below curve C, and symmetrical pendulum motion occurs only within curve E. The labels Un, Pe, Ch, ChSt, and Aff indicate the type of motions (Unidirectional, Pendulum, Chaos, Chaos-then-Stationary, and Asymmetric fill-and-fall) found for a computer model of a 12-cup wheel, at the corresponding locations on this simplified version of Fig. 8. The values were found using the procedure outlined in Sec. VII A. The behavior of the 12-cup wheel is generally similar to that of the ideal water wheel, except for large .

## Tables

Water wheel parameters.

Water wheel parameters.

Scaling of distance and time units.

Scaling of distance and time units.

Variables and parameters for the 12-cup water wheel. SI units are used.

Variables and parameters for the 12-cup water wheel. SI units are used.

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