Elegant Automation

The following sections are included:

• Dedication
• Preface
• Contents

This chapter will describe the computer program and methods used to write the subsequent chapters. The program identifies systems of coupled non-linear ordinary differential equations whose solutions are chaotic, simplifies them to the extent possible, analyzes them using conventional methods, produces the figures, and writes the accompanying text. Collecting some common information here minimizes repetition.

The system with the odd name JCS-08-13-2022 is the one that motivated this book. The entire research was done, the paper was written, and it was submitted to the journal in one 24-hour period on August 13, 2022. This system is of no particular interest except that it serves as a simple illustrative example of the method to be applied throughout the book.

The Lorenz system is the oldest and most famous example of an autonomous chaotic system. Despite being extensively studied, apparently no one had attempted to simplify the parameters to the extent possible. The resulting simplification gives a strange attractor coexisting with two stable equilibrium points. It is symmetric with respect to a 180° rotation about the z axis.

The Rössler system is nearly as old and well-known as the Lorenz system. It has the same number of terms as the Lorenz system but only a single nonlinearity. Thus it has been widely used as a simple example of chaos.

The Nosé–Hoover system was devised to model a simple harmonic oscillator in thermal equilibrium with a heat bath at constant temperature. Since it is a conservative system, it does not have an attractor, but rather it has a chaotic sea coexisting with regions of quasiperiodicity. With only five terms, it is the simplest example of a thermostatted oscillator, but its dynamics is not ergodic.

The diffusionless Lorenz system is a reduced form of the classical Lorenz system with only five terms and two quadratic nonlinearities. It shares many of the properties of its more familiar relative but with a single parameter.

The Sprott C system is a slight modification of the diffusionless Lorenz system in which the xy term in the $\dot{z}$ equation is replaced with y². Unsurprisingly, the resulting dynamics is very similar.

The Sprott D system is another system with five terms and two quadratic nonlinearities that was discovered in 1994. It has the unusual feature that it is dissipative but time reversible with a symmetric attractor–repellor pair that exchange roles when time is reversed.

The Sprott E system is the final case with five terms and two quadratic nonlinearities that was discovered in 1994. Of the five such cases that were found, it occurred the least frequently. For some choice of the parameters, it produces bursting oscillations.

The Sprott F system is the first and most frequently found of the fourteen chaotic cases with six terms and one quadratic nonlinearity that were discovered in 1994. The remaining thirteen cases are given in the chapters that follow.

The Sprott G system is another case with six terms and one quadratic nonlinearity that was discovered in 1994.

The Sprott H system is another case with six terms and one quadratic nonlinearity that was discovered in 1994.

The Sprott I system is another case with six terms and one quadratic nonlinearity that was discovered in 1994.

The Sprott J system is another case with six terms and one quadratic nonlinearity that was discovered in 1994.

The Sprott K system is another case with six terms and one quadratic nonlinearity that was discovered in 1994.

The Sprott L system is another case with six terms and one quadratic nonlinearity that was discovered in 1994.

The Sprott M system is another case with six terms and one quadratic nonlinearity that was discovered in 1994.

The Sprott N system is another case with six terms and one quadratic nonlinearity that was discovered in 1994.

The Sprott O system is another case with six terms and one quadratic nonlinearity that was discovered in 1994.

The Sprott P system is another case with six terms and one quadratic nonlinearity that was discovered in 1994.

The Sprott Q system is another case with six terms and one quadratic nonlinearity that was discovered in 1994.

The Sprott R system is another case with six terms and one quadratic nonlinearity that was discovered in 1994.

The Sprott S system is another case with six terms and one quadratic nonlinearity. Despite being the case from 1994 that occurred the least frequently, this system is highly robust.

This is an old but relatively unknown case with six terms and one quadratic nonlinearity that was somehow missed in the extensive 1994 search for such systems.

With only five terms and a single quadratic nonlinearity, this is arguably the simplest example of a dissipative autonomous chaotic flow, although it is relatively fragile. It is in the form of a jerk equation for which several additional examples are given in the chapters that follow.

A variation of the system in the previous chapter is probably the simplest dissipative autonomous chaotic flow with a cubic rather than a quadratic nonlinearity. It is also a jerk system with five terms and a single nonlinearity but with inversion symmetry, and it is relatively fragile.

Only slightly more complicated and considerably predating the Malasoma system is another jerk system with a cubic nonlinearity but with six terms. This system was developed to model the irregular variability in the luminosity of stars. The authors noted the aperiodicity and sensitive dependence on initial conditions but were apparently unaware of the similar concurrent work of Lorenz. The system is inversion symmetric with a symmetric pair of strange attractors for the given parameters.

Perhaps the simplest chaotic jerk system with a piecewise-linear nonlinearity is the Linz–Sprott system. It has an absolute value function that makes it especially suitable for electronic implementation using diodes. It also provides an example of a memory oscillator of which there are many other cases.

Perhaps the simplest chaotic jerk system with a signum nonlinearity is the Elwakil–Kennedy system. It has inversion symmetry, produces double scrolls, and is especially suitable for electronic implementation using saturating amplifiers. It is another example of a memory oscillator of which there are many other cases.

Chua’s circuit is one of the oldest and most widely studied chaotic electronic circuits. It can be modeled in a number of ways, one of the simplest of which uses a piecewise-linear nonlinearity to model Chua’s diode. It can produce a wide variety of strange attractors including some that are hidden. The system is inversion symmetric with a symmetric double-scroll strange attractor for some values of the parameters and a symmetric pair of single-scroll strange attractors coexisting with a symmetric pair of limit cycles for the case shown here.

The Chen system is another relatively old and widely studied chaotic system. It is a time-reversed version of the Lorenz system with a particular constraint on the parameters. It is symmetric with respect to a 180° rotation about the z axis.

This is an unpublished chaotic system proposed by Arne Dehli Halvorsen with the property that the equations are cyclically symmetric so that the strange attractor is symmetric with respect to a 90° rotation about any of the three axes. The resulting attractor has a special elegance and beauty.

René Thomas proposed an even simpler cyclically symmetric system, variants of which have been widely studied. Its strange attractor is symmetric with respect to a 90° rotation about any of the three axes as well as inversion symmetric, and it has a special elegance and beauty. It has an infinite line of equilibrium points, only a sample of which is shown.

This is an old system originally devised to model the onset of turbulence in a fluid. With ten terms and five nonlinearities, it is more complicated than most of the previous systems, but it is included here for historical interest. The equations are symmetric under a 180° rotation about the z axis with a symmetric pair of strange attractors for the chosen parameters.

This is another old mathematical chaotic system describing Euler’s rigid body rotation with linear feedback control. It is one of the first systems studied in which there is a pair of strange attractors, although the simplified case shown here has a single attractor.

This is another old chaotic system that was proposed to investigate the onset of turbulence. Like the Lorenz system, this system is invariant under a 180° rotation about the z axis, but with a cubic nonlinearity.

This system was proposed by Lorenz as a simplification of the equations used by Hadley to describe the global circulation of winds around the Earth. With eleven terms, five quadratic nonlinearites, and two constant forcing terms, it is one of the most complicated systems in this book, but it is included here for historical reasons.

This system is a modification of the Sprott D system with an additional constant term that removes the equilibrium. It is one of the earliest and simplest cases with a hidden strange attractor. Many other such examples are now known. It is also unusual because it is dissipative but time reversible with a symmetric attractor–repellor pair.

This system is a modification of the Sprott E system with an additional constant term that gives a single stable equilibrium in addition to the strange attractor. Thus the strange attractor is hidden, and with only six terms and two quadratic nonlinearities, this might be the simplest such example.

Previous chapters have included systems with inversion symmetry (three variables) and rotational symmetry (two variables). This system has reflection symmetry (one variable) taken as x → −x. Since $\dot{x}=0$ at x = 0, the orbit cannot cross the x = 0 plane, and any attractor on one side of the plane must have a mirror image on the other side.

This system has four cross product terms and admits as many as five coexisting attractors including some with a multi-wing butterfly shape for certain values of the parameters. For the chosen parameters, a symmetric strange attractor coexists with two neutrally stable equilibrium points. Like the Lorenz system, it is symmetric under a 180° rotation about the z axis.

This is a simple example of a system with an infinite line of equilibria coexisting with a strange attractor that is hidden by virtue of the fact that nearly all initial conditions chosen in the vicinity of the line do not lead to the attractor. The system is a dissipative extension of the Nose–Hoover system.

Most of the previous examples have more linear than nonlinear terms. This system has all quadratic terms except for a single constant term. It is apparently the simplest such system. It has inversion symmetry, time-reversal invariance, and an attractor–repellor pair.

This is an unusual system with a dissipative region for some initial conditions and a conservative region for other initial conditions for the same value of the parameters. There are no equilibrium points, and thus the strange attractor is hidden. The system is time-reversal invariant with a symmetric attractor–repellor pair.

Previous examples have included dissipative systems that are time-reversible with inversion and rotational symmetry. The case here completes the picture with a time-reversible system that is reflection invariant with a symmetric attractor–repellor pair. It is also unusual in that it has three lines of equilibria at (0, 0, z) and (±1, y, 0) and conservative periodic orbits coexisting with the strange attractor.

Chaotic systems can be constructed with surfaces of equilibria, one simple example of which is shown here. If the surface is closed or extends to infinity in a three-dimensional state space, it divides the space into two regions that the orbit cannot cross since the flow vector is zero on the surface. More complicated or more numerous surfaces can divide the space into even more segments, some of which may contain coexisting attractors. This system is invariant under a 180° rotation about the z axis with a symmetric pair of strange attractors.

There are many examples of two-dimensional periodically forced nonlinear oscillators that are chaotic. They can all be written in a three-dimensional autonomous form. The case shown here is one of the oldest and simplest such example. Periodically forced systems do not have equilibrium points, and so the attractor is hidden despite being globally attracting.

The previous chapter showed a periodically forced system with the nonlinearity in the restoring force. The system here is a linear oscillator with a spatially periodic nonlinearity in the damping, resulting in a countable infinity of nested attractors, including limit cycles, tori, and strange attractors. Such systems are said to be ‘megastable’.

Tori are common in three-dimensional conservative systems and in two-dimensional periodically driven systems, but attracting tori are rare in three-dimensional autonomous systems. The case here, which is a variant of the Nosé–Hoover system, is perhaps the simplest example where such a torus exists along with a strange attractor or by itself.

Buncha Munmuangsaen studied a variant of the Nosé –Hoover system in which the thermostat controls the average speed rather than the average energy of the linear oscillator. The resulting system has strange attractors, limit cycles and invariant tori. The system is time-reversible with an attractor–repellor pair and is apparently fully ergodic for some choices of the parameters despite lacking equilibrium points and thus being hidden.

This chapter concludes the book with an even fifty systems by considering a simple conservative linear oscillator with a signum thermostat that makes it fully ergodic with a chaotic sea that has a precisely Gaussian distribution of the x and y variables. It is a variant of the Nosé –Hoover system in which the ‘bang-bang’ thermostat turns on and off abruptly rather than linearly. The system is time-reversible and rotationally invariant with no equilibrium points.

The following sections are included:

• Bibliography
• Index
• About the Author