September 22, 1998

Last revised March 11, 2004

Previous work has identified numerous examples of algebraically simple chaotic flows with quadratic nonlinearities. Here we report the result of a search for simple chaotic flows in which the single nonlinearity has the form of the absolute value of one of the variables. Such an example is to quadratic flows as the tent map is to the logistic map.

Following a suggestion by Stefan
Linz, equations of the form d^{3}*x*/d*t*^{3}
= *A*d^{2}*x*/d*t*^{2} + *B*d*x*/d*t*
+ *C*|*x*| + *D* were examined. Many chaotic solutions
were found for various choices of the parameters *A*, *B*, *C*,
and *D*, and different initial conditions. One simple example
is d^{3}*x*/d*t*^{3} = *A*d^{2}*x*/d*t*^{2}
- d*x*/d*t* + |*x*| - 1 with initial conditions at *x*
= d*x*/d*t* = d^{2}*x*/d*t*^{2} = 0.

A bifurcation diagram for the above equation was produced using the
BASIC source and (DOS) object
code provided. The parameter *A* was varied from -0.8 to
-0.5, and the local maximum in *x* was plotted. The resulting
period-doubling route to chaos is shown below:

A Hopf bifurcation occurs at *A* = -1 (off the left edge of the
figure), where the stable fixed point at *x* = -1 bifurcates into
a limit cycle with unit angular frequency. For *A* = -0.6, variation
of *B* and *C* reveal a similar period-doubling route to chaos.
Variation of *D* controls the size of the attractor and appears to
produce chaotic solutions for all *D* < 0. For *A* =
-0.6 (just to the left of the period-5 window), the calculated
Lyapunov exponents are (0.035, 0, -0.635), and the Kaplan-Yorke dimension
is 2.055. An animated view of the attractor with *A* = -0.6
is shown below:

In the image above, the d*x*/d*t* axis is toward the top of
the screen and the object rotates in the *x*-d^{2}*x*/d*t*^{2}
plane. Lucas Finco
has written a Java applet that allows you
to vary *A* and rotate the attractor arbitrarily. This system
is ripe for further exploration. The discontinuity in the flow is
not evident in the image of the attractor because it occurs only in the
fourth derivative (d^{4}*x*/d*t*^{4}).
The attractor is strikingly similar to other simple
flows with quadratic nonlinearities. Several other
simple systems with the absolute value nonlinearity are also known.

An operational amplifier circuit that simulates the dynamics of the above system is given below:

In this circuit, if all components except the variable resistor have
unit magnitude (ohms, farads, and volts), the output should exactly solve
the equation in real time. Alternately, the resistors can be 1 kilo-ohm
and the capacitors can be 1000 microfarads. If the capacitors are
decreased to 0.1 microfarad, the frequency should increase by a factor
of 10,000 making the output in the audio range (1592 Hz at the first Hopf
bifurcation). This permits the bifurcations and chaos to be heard
with a loudspeaker as the variable resistor (value 1/*A*) is adjusted
from 1000 to 2000 ohms. Germanium signal diodes should be used to
reduce the forward voltage drop, although the circuit will work with silicon
diodes at a slightly smaller value of *A*.

The circuit has been successfully simulated with SPICE
using real operational amplifiers and diodes, and the only possible difficulty
occurs if the right-most operational amplifier has a dc offset that produces
a positive output in the absence of any input. A real circuit using
parts available for about $10 from Radio Shack has also been constructed
and tested. Its output is available in WAV
format and in RealAudio format as 1/*A*
is increased. The digitized signal is actually the integral of *x*
so as to accentuate the low frequencies during the period doublings.
You should be able to hear the period-1, period-2, period-4, and period-8
bifurcations, chaos, the period-6 window, chaos again, the period-5 window
with doubling to period-10, and chaos again.

Many other chaotic electrical circuits have been devised. The
virtue of this one is that all components are linear except for the diodes,
and their nonlinearity is entirely in their *I-V* characteristic and
not in some subtle and hard-to-model property such as junction capacitance.
For this reason, the circuit can be scaled to operate over a nearly arbitrary
range of frequencies, and quantitative comparisons can be made with theory
to the precision of the electrical components (a percent or better).
This raises the question of what is the simplest chaotic circuit that contains
only linear operational amplifiers, resistors, capacitors, diodes, and
batteries. The circuit above has 18 components. At least
one
other such circuit with 18 components has been devised. Other
circuits with as few as 11 components have recently been found and
are under study. A longer paper
on chaotic circuits with more detail is also available.

The circuit is similar in spirit to Chua's circuit, which is also piecewise linear but uses an inductor and diodes or saturated operational amplifiers to provide a negative resistance. The circuit here has a number of advantages over Chua's circuit:

- It doesn't require inductors which are hard to model precisely because of their frequency dependent resistive loss.
- It is less sensitive to the values and quality of the components.
- It can be scaled to operate at essentially any frequency.
- When the forward voltage drop of the diodes is taken into account, the circuit permits automated bifurcation plots using a swept voltage source in place of the battery.
- It is a piecewise linear approximation to a quadratic rather than cubic nonlinearity.
- Its representation in terms of a third order ODE is much simpler.

The circuits are published in Phys. Lett. A and Am. J. Phys.

See also K. Kiers, D. Schmidt, and J. C. Sprott,