Chapter
1

**Complex Behavior of Simple Systems**

**Julien Clinton Sprott**

Department of Physics

University of Wisconsin - Madison

sprott@juno.physics.wisc.edu

**1.1. Introduction**

Since the seminal work of Lorenz [1963] and Rössler [1976], it has been known that complex behavior (chaos) can occur in systems of autonomous ordinary differential equations (ODEs) with as few as three variables and one or two quadratic nonlinearities. Many other simple chaotic systems have been discovered and studied over the years, but it is not known whether the algebraically simplest chaotic flow has been identified. For continuous flows, the Poincaré-Bendixson theorem [Hirsch 1974] implies the necessity of three variables, and chaos requires at least one nonlinearity. With the growing availability of powerful computers, many other examples of chaos have been subsequently discovered in algebraically simple ODEs. Yet the sufficient conditions for chaos in a system of ODEs remain unknown.

This paper will review the history of recent attempts to identify the simplest such system and will describe two candidate systems that are simpler than any previously known. They were discovered by a brute-force numerical search for the algebraically simplest chaotic flows. There are reasons to believe that these cases are the simplest examples with quadratic and piecewise linear nonlinearities. The properties of these systems will be described.

**1.2. Lorenz and Rössler Systems**

The celebrated Lorenz equations are

(1)Note that there are seven terms on the right-hand side of these equations, two of which are nonlinear. Also note that there are three parameters. The other four coefficients can be set to unity without loss of generality since

Although the Lorenz system is often taken as the prototypical autonomous dissipative chaotic flow, it is less simple than the Rössler system given by

(2)which also has seven terms and three parameters, but only a single quadratic nonlinearity.

Other autonomous chaotic flows that are algebraically simpler than Eq. (2) have also been discovered. For example, Rössler [1979] found chaos in the system

(3)which has a single quadratic nonlinearity but only six terms and two parameters.

**1.3. Quadratic Jerk Systems**

More recently, we embarked on an extensive computer search for chaotic systems with five terms and two quadratic nonlinearities or six terms and one quadratic nonlinearity [Sprott 1994]. We found five cases of the former type and fourteen of the latter type. One of these cases was conservative and previously known [Posch 1986], and the others were dissipative and apparently previously unknown.

In response to this work, Gottlieb [1996] pointed out that one of our examples can be recast into the explicit third-order scalar form

(4)which he called a "jerk function" since it involves a third derivative of

In response to this question, Linz [1997] showed that the Lorenz and Rössler models have relatively complicated jerk representations, but that one of our examples can be written as

(5)In a subsequent paper [Eichhorn 1998], Linz and coworkers showed that all of our cases with a single nonlinearity and some others could be organized into a hierarchy of quadratic jerk equations with increasingly many terms. They also derived criteria for functional forms of the jerk function that cannot exhibit chaos.

We also took up Gottlieb’s challenge and discovered a particularly simple case

(6)which has only a single quadratic nonlinearity and a single parameter [Sprott 1997]. With and , this three-dimensional dynamical system has only five terms. It exhibits chaos for

**Figure 1.** Attractor for the
simplest chaotic flow with a quadratic nonlinearity from Eq. (6) with *a*
= 2.017.

This system and most
of the other cases that we found share a common route to chaos. The control
parameter *a* can be considered a damping rate for the nonlinear oscillator.
For large values of *a*, there is one or more stable equilibrium points.
As *a* decreases, a Hopf bifurcation occurs in which the equilibrium
becomes unstable, and a stable limit cycle is born. The limit cycle grows
in size until it bifurcates into a more complicated limit cycle with two
loops, which then bifurcates into four loops, and so forth, in a sequence
of period doublings, until chaos finally onsets. A further decrease in
*a* causes the chaotic attractor to grow in size, passing through
infinitely many periodic windows, and finally becoming unbounded when the
attractor grows to touch the boundary of its basin of attraction (a crisis).
A bifurcation diagram for Eq. (6) is shown in Fig. 2. In this figure, the
local maxima of *x* are plotted as the damping *a* is gradually
decreased. Note that the scales are plotted backwards to emphasize the
similarity to the logistic map. Indeed, a plot of the maximum *x*
versus the previous maximum in Fig. 3 shows an approximate parabolic dependence,
albeit with a very small-scale fractal structure.

**Figure 2.** Bifurcation diagram
for Eq. (6) as the damping is reduced.

**Figure 3.** Return map showing
each value of *x*_{max} versus the previous value of *x*_{max}
for Eq. (6) with *a* = 2.017. The insert shows fractal structure at
a magnification of 10^{4}.

**1.4. Piecewise Linear Jerk Systems**

Having found what appears to be the simplest jerk function with a quadratic nonlinearity that leads to chaos, it is natural to ask whether the nonlinearity can be weakened. In particular, the term in Eq. (6) might be replaced with . A numerical search did not reveal any such chaotic solutions. However, the system

(7)which is equivalent to Eq. (6) for

**Figure 4.** Regions of *a-b*
space for which chaos occurs in Eq. (7).

In an extensive numerical search for the algebraically simplest dissipative chaotic flow with an absolute-value nonlinearity, Linz and Sprott [Linz, 1999] discovered the case

(8)which exhibits chaos for

**Figure 5.** Attractor for the
simplest chaotic system with an absolute-value nonlinearity from Eq. (8)
with *a* = 0.6 and *b* = 1.

This system also exhibits
a period-doubling route to chaos as shown in Fig. 6 and otherwise resembles
the simple quadratic case previously described. This example relates to
the quadratic flows as the tent map does to the logistic map. We claim
it is the most elementary piecewise linear chaotic flow. Linz [2000] has
recently proved that chaos cannot exist in Eq. (8) if any of the terms
are set to zero. Furthermore, the piecewise linear nature of the nonlinearity
allows for an analytic solution to Eq. (8) by solving two linear equations
and matching the boundary conditions at *x* = 0 [Linz 1999].

**Figure 6.** Bifurcation diagram
for Eq. (8) with *b* = 1 as the damping is reduced.

Equation (8) is a special case of the more general system

(9)in which

**1.5. Electrical Circuit Implementations**

Piecewise linear forms
of *G*(*x*) lend themselves to electronic implementation using
diodes and operational amplifiers [Sprott 2000]. One example of such a
circuit that solves Eq. (8) electronically is shown in Fig. 7. In this
circuit, all the capacitors are 0.1 microfarads, and the resistors are
1 kilo-ohms except for the variable resistor, whose value is the inverse
of the damping constant *a* in Eq. (8) in units of kilo-ohms. The
non-inverting inputs to the amplifiers are grounded and not shown. The
fundamental frequency at the onset of oscillation is 1592 Hz (5000/pi).
The period doublings, periodic windows, and chaos are easily made audible
by connecting the output *x* to an amplifier and speaker. Such circuits
are similar in spirit to Chua’s circuit [Matsumoto 1985] but are easier
to implement and analyze.

**Figure 7.** A chaotic circuit using inverting operational
amplifiers and diodes that solves Eq. (8).

**1.6. Conclusions**

Two new dissipative chaotic systems, given by Eq. (6) and Eq. (8), have been described that are algebraically simpler than the Lorenz and Rössler attractors. One has a quadratic nonlinearity, and the other has an absolute value nonlinearity. Each system is apparently the algebraically simplest dissipative chaotic system of its type. The latter case is especially suited for electronic implementation using diodes and operational amplifiers.

**References**

Eichhorn, R., Linz,
S.J., & Hänggi, P., 1998, Transformations of Nonlinear Dynamical
Systems to Jerky Motion and its Application to Minimal Chaotic Flows. *Phys.
Rev. E*, **58**, 7151.

Fu, Z., & Heidel,
J., 1997, Non-Chaotic Behavior in Three-Dimensional Quadratic Systems.
*Nonlinearity*, **10**, 1289.

Gottlieb, H.P.W., 1996,
What is the Simplest Jerk Function that gives Chaos? *Am. J. Phys*.,
**64**, 525.

Heidel, J., & Fu,
Z., 1999, Nonchaotic Behavior in Three-Dimensional Quadratic Systems II.
The Conservative Case. *Nonlinearity*, **12**, 617.

Hirsch, H.W., &
Smale, S., 1974, *Differential Equations, Dynamical Syustems and Linear
Algebra*, Academic Press (New York), **11**, 239.

Linz, S.J., 1997, Nonlinear
Dynamical Models and Jerky Motion. *Am. J. Phys*., **65**, 523.

Linz, S.J., & Sprott,
J.C., 1999, Elementary Chaotic Flow. *Phys. Lett. A*, **259**,
240.

Linz, S.J., 2000, No-chaos Criteria for Certain Jerky Dynamics. submitted for publication.

Lorenz, E.N., 1963,
Deterministic Nonperiodic Flow. *J. Atmos. Sci.*, **20**, 130.

Matsumoto, T., Chua,
L.O., & Komoro, M., The Double Scroll. *IEEE Trans. Circuits Syst.*,
**CAS-32**, 797.

Posh, H.A., Hoover,
W.G., & Vesely, F.J., 1986, Canonical Dynamics of the Nosé Oscillator:
Stability, Order, and Chaos. *Phys. Rev. A*, **33**, 4253.

Rössler, O.E.,
1976, An Equation for Continuous Chaos. *Phys. Lett. A*, **57**,
397.

Rössler, O.E.,
1979, Continuous Chaos – Four Prototype Equations. *Am. (N.Y.) Acad.
Sci*., **316**, 376.

Schot, S.H., 1978,
The Time Rate of Change of Acceleration. *Am. J. Phys.*, **46**,
1090.

Sprott, J.C., 1994,
Some Simple Chaotic Flows. *Phys. Rev. E*, **50**, R647.

Sprott, J.C., 1997,
Simplest Dissipative Chaotic Flow. *Phys. Lett. A*, **228**, 271.

Sprott, J.C., 2000,
A New Class of Chaotic Circuit. *Phys. Lett A*, **266**, 19.