of a Chaotic Lorenz System

Ned J. Corron

The Lorenz system is one of a few standard oscillators commonly used to explore chaos [1]. An accessible description of its mathematical features and chaotic dynamics is presented by Thompson and Stewart [2]. This important system was originally developed as a simplified mathematical model of atmospheric instabilities and does not derive from a basic circuit topology. As such, realizing an electronic Lorenz oscillator requires an explicit analog computer implementation of the oscillator equations. A few designs for realizing a Lorenz attractor have been published in the literature [3‑14]. Here, we present an original circuit design that is derived directly from the Lorenz system equations.

The Lorenz system is defined by the system of ordinary differential equations

_{} (1)

where *X*, *Y*, and *Z* are the states of the
system and *s*, *R*, and *b*
are parameters. Typical parameter values that yields chaotic dynamics are *s* = 10, *R* = 30,
and *b* = 8/3. For these values, the attractor generated by
numerically integrating (1) is shown in Figure 1. Typical waveforms for the
three dynamical states are shown in Figure 2.

Figure 1.
Lorenz attractor generated by numerical integration of
the system equations with *s* = 10, *R* = 30,
and *b* = 8/3.

Figure 2.
Typical Lorenz waveforms generated by numerical
integration of the system equations with *s* = 10, *R* = 30,
and *b* = 8/3.

To obtain a practical circuit design for the Lorenz system, it is necessary to scale the system equations. To this end, we define the new system states

_{} (2)

where *a* is a scale parameter to be chosen. With the
scaling (2), the system dynamics are governed by

_{} (3)

For a practical circuit, it is desirable to use ±10‑V
signals to represent the system states. Based on the magnitude of the
oscillations shown in Figure 1, choosing *a* = 1/3
yields states that map directly to the desired voltage range. As such, the
scaled system is

_{} (4)

where *x*, *y*, and *z* are measured in
volts. The Lorenz attractor for the scaled system (4) is shown in Figure 3.

Figure 3.
Numerical Lorenz attractor for the scaled system with *a* = 1/3.

A circuit
schematic that closely realizes the scaled Lorenz system is shown in Figure 4. The voltages at the nodes
labeled *x*, *y*, and *z* correspond to the states of the scaled
system (4). The multipliers are each AD633, for which the additional summing
node is grounded. The operational amplifiers are all TL082 or equivalent. The
circuit is powered with ±15 V. Since the input to the AD633 is very
high impedance, the three-volt reference for *V _{a}*
can be implemented using resistors as a voltage divider; however, it is
preferable to use a Zener diode or other fixed
voltage reference to make the circuit independent to changes in the supply
voltages. As drawn, the circuit oscillates near 2 kHz; however, to vary
the frequency without affecting the chaotic nature, all three capacitors may be
scaled equally.

Figure 4. Lorenz circuit schematic.

The
oscillator parameters *R*, *s*,
and *b* for the Lorenz circuit are controlled with three circuit
resistors. The *R* parameter is set with the variable resistor R* _{R}* by the relation

_{} (5)

Similarly, the *s*
and *b* parameters are set by the resistors R* _{s}* and R

_{} (6)

and

_{}. (7)

Often for bifurcation studies of the Lorenz system dynamics,
the parameter *R* is varied while *s*
and *b* are held fixed; as such, R* _{R}*
is implemented here using a variable resistor. Clearly, variable resistors can
also be used for R

_{} (8)

where V* _{a}* is measured in
volts.

As drawn in
Figure 4, the circuit is configured
for *s* = 10, *b* = 8/3,
*a* = 1/3, and a variable *R*. Using a 10 kW potentiometer for R* _{R}* yields the range
3 ≤

Figure 5. Lorenz attractor measured from circuit.

Figure 6. Typical waveforms measured from Lorenz circuit.

It is
important to note that the Lorenz system equations are symmetric under the
transformation _{}. Technically, this symmetry makes the system structurally
unstable. That is, arbitrarily small deviations in the mathematical description
of the oscillator will destroy this symmetry and result in a qualitatively
different system. In terms of a physical circuit, this is bad news, since an
exact realization of the equations is impossible due to limited precision of
the circuit components. As a result, a Lorenz circuit will exhibit asymmetric
oscillations, although the degree of asymmetry can be minimized using
high-precision components and trimmers. Unfortunately, the effects of the
asymmetry can be significant in circuits with analog multipliers, which are
notable for output voltage offset errors. Such errors, if uncompensated, can
lead to a obvious imbalance in the attractor exhibited by the circuit.

To
illustrate the asymmetry in a practical implementation, we consider the peak
return map on the strictly positive state *z*. In Figure 7, a return map for successive
maxima in the *z* state obtained from a
numerical integration of the Lorenz system equations is plotted. As evident in
this figure, the dynamics of the Lorenz system is well modeled by a unimodal, one-dimensional return map [15]. Such a simplification is
important, especially in the application of symbolic dynamics to this system [16]. In particular, it is known
that a generating partition exists at the maximum in such a return map, and
that an infinite sequence of returns above and below this partition (*i.e.*, symbols 1 and 0, respectively) uniquely
defines a trajectory for the iterated system. However, for the Lorenz system,
there is an ambiguity in the sign of the states *x* and *y* due to
the symmetry inherent in the system equations. With each return above the
partition (*i.e.*, with each symbol 1),
the state of the system switches across the symmetry, flipping the sign of the *x*
and *y* states; however, determining the absolute state of the system with
respect to this symmetry requires knowing the sign of the initial state of the
system.

Figure 7.
Peak return map on *z* from numerical integration
of the Lorenz system equations.

In Figure 8, we show the peak return map
measured for an actual Lorenz circuit. Here, it is apparent that the return map
is more complex and the dynamics of the circuit are not well defined by a
simple one-dimensional return map. This complication is due to the inevitable
loss of symmetry due to a physical implementation. To illustrate the asymmetry,
peak returns with positive *x* and *y* are shown as open circles in
the figure, while those with negative *x* and *y* are shown filled.
It is evident that the loss of symmetry has resulted in twin maps, offset by
roughly 0.05 V, corresponding to returns on each side of the attractor.
Trimming the offsets of the two analog multipliers in the circuit could be used
to reduce this asymmetry, but true symmetry is impossible to achieve in any
physical system.

In conclusion, we have shown a simple circuit implementation of the chaotic Lorenz system. This oscillator circuit is capable of producing the interesting nonlinear dynamics of this important system, including chaos. Besides obvious pedagogical applications, this circuit may be also well suited for technological applications, including communications via controlled symbolic dynamics [16]. However, it is important to realize that the idealized symmetry of Lorenz’s original mathematical model cannot be perfectly achieved in any physical realization, and any experiments designed to use this circuit must not hinge critically upon realizing perfect symmetry.

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