convert

Some simple chaotic jerk functions

J. C. Sprott

Department of Physics, University of Wisconsin, Madison, Wisconsin 53706

Received 9 September 1996; accepted 3 January 1997

A numerical examination of third-order, one-dimensional, autonomous, ordinary differential
equations with quadratic and cubic nonlinearities has uncovered a number of algebraically simple
equations involving time-dependent accelerations

jerks

that have chaotic solutions. Properties of

some of these systems are described, and suggestions are given for further study. ©

1997 American

Association of Physics Teachers.

I. INTRODUCTION

One of the most remarkable recent developments in clas-

sical physics has been the realization that simple nonlinear
deterministic equations can have unpredictable

chaotic

long-term solutions. Chaos is now thought to be rather com-
mon in nature, and the study of nonlinear dynamics has
brought new excitement to one of the oldest fields of science.
The widespread availability of inexpensive personal comput-
ers has brought many new investigators to the subject, and
important research problems are now readily accessible to
undergraduates. An interesting and yet unsolved problem is
to determine the minimum conditions necessary for chaos.
This paper will describe several examples of chaotic flows
that are algebraically simpler than any previously reported
and will suggest further lines of promising investigation.

The chaotic system to which one is usually first introduced

is the logistic equation,

which is remarkably simple and yet exhibits many of the
common features of chaos. For most values of A in the range
3.5699... to 4, it produces a sequence of x values that exhibit
sensitive dependence on initial conditions and long-term un-
predictability. Its behavior can be studied with a simple com-
puter program or even a pocket calculator.

Equation

is a one-dimensional iterated map in which

the variable x advances in discrete time steps or jumps. Most
of the equations of physics, and science in general, are more
naturally expressed in the form of differential equations in
which the variables evolve continuously in time. Newton’s
second law is the prototypical example of such a continuous
dynamical process.

Whereas chaos can arise in discrete-time systems with

only a single variable, at least three variables are required for
chaos in continuous-time systems.

The reason is that the

trajectory has to be nonperiodic and bounded to some finite
region, and yet it cannot intersect itself because every point
has a unique direction of flow. Newton’s second law in one
dimension

inherently contains two variables because it

involves a second derivative. It is really two equations, a
kinematic one defining the velocity, dx/dt

, and a dy-

namic one describing the rate of change of this velocity,
d

/dt

F/m. Thus Newton’s second law in 1D with a force

that depends only on position and velocity cannot produce
chaos since there are only two phase-space variables

x and

In two spatial dimensions, there are four phase-space vari-

ables, and thus chaos is possible. For example, a planet or-
biting a single massive star is described by two spatial com-

ponents and two velocity components since the orbit lies in a
plane. The equation of motion is nonlinear because the force
is proportional to the inverse square of the separation. How-
ever, this system does not exhibit chaos because there are
two constants of the motion—mechanical energy and angular
momentum—which reduce the phase-space dimension from
four to two. With a third object, such as a second star, the
planet’s motion can be chaotic, even when the motions are
coplanar, because the force on the planet is no longer central,
and its angular momentum is thus not conserved. The three-
body problem was known 100 years ago to have chaotic
solutions and has never been solved in the sense of deriving
an analytic expression for the position of the bodies as a
function of time.

A one-dimensional system can exhibit chaos if the force

has an explicit time dependence. For example, a sinusoidally
driven mass on a nonlinear spring with a cubic restoring
force

and linear damping

bx˙

obeys an equation

x¨

bx˙

A sin

where x˙

[

dx/dt. This is a special case of Duffing’s

equation

whose chaotic behavior has been studied by Ueda.

It is a useful model for any symmetric oscillator such as a
mass on a spring driven to a sufficiently large amplitude that
the restoring force is no longer linear. Note that x and t can
be rescaled so as to eliminate two of the four parameters

k, A, and

. For example, we can take k

1 without loss

of generality. Thus the behavior of the system is determined
entirely by two parameters

b and A in this case

, and by the

initial conditions, x

and x˙

. Equation

is known to

have chaotic solutions

for b

0.05 and A

7.5, among other

values.

Systems such as Eq.

with an explicit time dependence

can be rewritten in autonomous form

t does not appear ex-

plicitly

by defining a new variable

t, leading to a sys-

tem of three, first-order, ordinary differential equations

ODEs

such as

x˙

A sin

The new variable

is a periodic phase, and thus the global

topology of the system is a torus. Other standard examples of
chaotic autonomous ODEs with three variables include the
Lorenz

and Ro¨ssler

attractors, which have only quadratic

nonlinearities, but each of which has a total of seven terms
on its right-hand side.

An earlier paper

described a computer search that re-

vealed 19 examples of chaotic flows that are algebraically
simpler than the Lorenz and Ro¨ssler systems. These autono-

537

Am. J. Phys. 65

, June 1997

mous equations all have three variables

x, y , and z

and

either six terms and one quadratic nonlinearity or five terms
and two quadratic nonlinearities.

II. JERK FUNCTIONS

Recently Gottlieb

pointed out that the simplest ODE in a

single variable that can exhibit chaos is third order, and he
suggested searching for chaotic systems of the form

j(x,x˙,x¨), where j is a jerk function

time derivative of

acceleration

10,11

He showed that case A in Ref. 8,

x˙

y ,

y˙

y z,

z˙

can be recast into the form

x˙

x¨

/x˙,

and he wondered whether yet simpler forms of the jerk func-
tion exist that lead to chaos. Equation

is a special case of

the Nose´–Hoover thermostated dynamic system,

12,13

which

exhibits time-reversible Hamiltonian chaos.

Careful examination of the other 18 equations in Ref. 8

shows that most if not all of them can be reformulated into a
third-order ODE in a single variable. This exercise is well
suited for a student with an elementary knowledge of differ-
ential calculus. For example, case I in Ref. 8 can be written
as

x¨

0.2x˙

5x˙

0.4x

which in some ways is more appealing than Eq.

since it

has only polynomial terms and a single quadratic nonlinear-
ity.

It is interesting to ask under what conditions a system of m

first-order ODEs in m variables can be written as an mth
order ODE in a single variable. A theorem by Takens

as-

sures us that almost any variable from an m-dimensional
system can be used to reconstruct the dynamics provided a
sufficient number of additional variables are constructed
from the original variable by successive time delays, x(t),x(t

),x(t

),..., but it may require as many as

1 such time lags

called the ‘‘embedding dimension’’

This condition was subsequently relaxed to 2m.

It is rea-

sonable to assume that a differential equation of order 2m
would also suffice. However, if we are free to choose the
variable optimally, there is reason to hope that an embedding
of m might suffice in most cases.

Systems for which this is apparently not the case include

ones with periodic forcing such as Eq.

. The reason is that

one of the variables,

, lies on a circle that requires a Eu-

clidean dimension of 2 to embed it. If we replace the sin

in Eq.

with a new variable y that obeys the harmonic

oscillator equation, y¨

y , Eq.

with k

1 can be

written as a fourth-order ODE,

¨x¨

x¨

bx˙

6xx˙

It is interesting to note that the trigonometric nonlinearity

sin

in Eq.

has been replaced exactly by a small num-

ber of polynomial terms, suggesting that equations such as
Eq.

are rather more general than would at first appear.

The term ¨x¨

[

x/dt

is the time derivative of the jerk,

which might be called a ‘‘spasm.’’ It has also been called a
‘‘jounce,’’ a ‘‘sprite,’’ a ‘‘surge,’’ or a ‘‘snap,’’ with its suc-
cessive derivatives, ‘‘crackle’’ and ‘‘pop.’’

Note that the

parameter A does not appear in Eq.

, but it does appear in

the initial conditions, whose number is one greater than in

Eq.

. The initial conditions must satisfy the constraint

(0)

bx¨(0)

(0)x˙(0) for Eq.

to be equivalent

to Eq.

In one sense, equations involving the jerk (x

) and its de-

rivatives are unremarkable. Newton’s second law (x¨

F/m)

leads necessarily to a jerk whenever the force F in the frame
of reference of the mass m has a time dependence, either
explicitly [F(t)] or implicitly [F(x,x˙)]. Except in a few
special cases such as a projectile moving without drag in a
uniform gravitational field, nonzero jerks exist. For example,
a mass on a linear spring has a sinusoidally varying jerk, as
well as all higher derivatives. However, in such a case, only
two phase-space variables

x and x˙

are required to describe

the motion. In the cases considered here, not only is the jerk
nonzero, but the acceleration (x¨) is an independent phase-
space variable necessary to describe the motion. Such an
equation might arise naturally in a case like a planet orbiting
a pair of fixed massive stars where there are two spatial
dimensions

and thus four phase-space variables

but with

the trajectory constrained to a three-dimensional subset of
the phase space by the conservation of energy.

III. NUMERICAL SEARCH PROCEDURE

Since Eq.

demonstrates the existence of chaotic jerk

functions with only quadratic nonlinearities, it is interesting
to identify the simplest such function. The most general
second-degree polynomial jerk function is

x˙

x¨

x˙

for which the goal is to find chaotic solutions with the fewest
nonzero coefficients and with the fewest nonlinearities.
Equation

ensures us that chaotic jerk functions with four

terms and one nonlinearity exist. Thus we seek cases with
four or fewer terms and one nonlinearity or fewer than four
terms and two nonlinearities. Such cases would be at least as
simple as the 19 cases in Ref. 8.

The numerical procedure was to choose randomly three or

four of the coefficients

through a

, set them to uni-

formly random values in the range

5 to 5, and then calcu-

late the trajectory for randomly chosen initial conditions

x˙, and x¨

in the range

5 to 5. The range

5 to 5 is arbitrary

and poses no significant restriction because x and t can be
rescaled. A fourth-order Runge–Kutta integrator with a step
size of

0.05 was used. The process was repeated the

order of 10

times. The most common dynamic was for the

trajectory to escape to infinity, and this was detected by stop-
ping the calculation whenever

x˙

x¨

exceeded 10

Because of the quadratic nonlinearity, unbounded cases are
usually identified within a few dozen iterations. The remain-
ing solutions most often settled to a fixed point or limit
cycle. Rare cases

the order of one in 10

exhibited chaotic

solutions. Thus, in some sense, it is reasonable to conclude
that chaos is relatively rare in algebraically simple systems
of ODEs.

The simplest way to detect chaos is to use its characteristic

sensitive dependence on initial conditions. The calculation
could be done twice in parallel with initial conditions that
differ by a small

. The quantity

can be chosen in

al-

most

any direction and assigned a value

1 but several

orders of magnitude greater than the computational preci-
sion. This is best done after a few thousand iterations to let
the orbit converge to the attractor and to avoid unnecessary

538

Am. J. Phys., Vol. 65, No. 6, June 1997

J. C. Sprott

calculations for unbounded cases. The signature of chaos is
that the separation of the orbits usually reaches a value of
order unity

0.1

quickly.

A more careful procedure and the one used here is to

calculate the Lyapunov exponent.

This was done in a man-

ner similar to that described above, except that after each
iteration, the new orbit separation

was determined, and the

separation was readjusted to

along the direction of

. The

largest Lyapunov exponent was then determined by averag-
ing the natural logarithm of

along the orbit. A decidedly

positive Lyapunov exponent is a signature of chaos. Since
there is always a zero Lyapunov exponent for a periodic
flow, corresponding to a direction parallel to the flow, the
search condition was for a Lyapunov exponent that remains
in excess of 0.01 for 10

iterations. Typically about one cha-

otic solution emerged per hour of computing on a 66-MHz
486 personal computer. All the candidate chaotic cases found
in this way were then tested with a smaller iteration step size

0.01

for at least 10

iterations.

IV. SEARCH RESULTS

Chaotic flows in three dimensions

can be character-

ized as either dissipative or conservative, according to
whether the trajectory is attracted to a region of space with
fractal dimension less than 3, a so-called strange attractor.

Dissipative systems have this property, and the attractor is
independent of the initial conditions provided they lie in the
basin of attraction. By contrast, a conservative system has a
trajectory whose dimension depends on initial conditions,
and is three for a chaotic trajectory, two for a quasiperiodic
trajectory

two incommensurate frequencies

, one for a peri-

odic trajectory, and zero for an equilibrium point.

Three-dimensional chaotic flows must have one positive

Lyapunov exponent, one zero exponent, and one negative
exponent. The sum of the exponents is the rate of volume
expansion for a cluster of initial conditions. This sum cannot
be positive for bounded trajectories. If it is negative

dissi-

pative

, the initial conditions are drawn to an attractor whose

volume is zero because its dimension is less than three

just

as a 2-D surface has zero volume

. If the sum of the expo-

nents is zero

conservative

, there is no contraction, and the

chaotic trajectory fills some 3-D region, perhaps with a frac-
tal boundary.

It is relatively easy to calculate numerically the rate of

volume expansion, and from that, the negative Lyapunov
exponent for a chaotic jerk function. It is given in terms of
the sum of the Lyapunov exponents by V

dV/dt

(

]

j /

]

x¨

x˙

x¨. Since this expression de-

pends on x and its derivatives, in general it must be averaged
along the trajectory. The dimension can then be estimated
using the Kaplan–Yorke conjecture,

where L

is the positive exponent and L

is the negative

exponent. The exponent L

is zero. Dissipative systems are

usually easier to identify because they are less sensitive to
initial conditions, have lower dimension, and are more robust
to errors in the numerical method. They are also more rep-
resentative of real physical systems since dissipation is
nearly always present in some degree. These systems will be
discussed first.

A. Dissipative systems

The simplest chaotic dissipative system that was found has

all its coefficients equal to zero except a

, a

, and a

. Two

of these coefficients can be set to unity without loss of gen-
erality, and the remaining coefficient was arbitrarily taken as
a

[

A, leading to the equation,

Ax¨

x˙

It is unlikely that any algebraically simpler form of an au-
tonomous chaotic flow exists because the above equation has
the minimum number of terms that allows an adjustable pa-
rameter and it has only a single quadratic nonlinearity. It can
be equivalently written as three, first-order, ordinary differ-
ential equations with a total of five terms,

x˙

a˙

This is one fewer term or nonlinearity than in any of the 19
cases previously found and two fewer than in the Ro¨ssler
equations. This case has been described in detail elsewhere.

It is a special case of Eq.

with the x˙ term absent. It was

presumably not discovered previously because the range of
A for which chaos occurs is very narrow.

Equation

has bounded solutions for 2.017...

2.082..., a period-doubling route to chaos, and a nearly

parabolic return map that strongly resembles the logistic
equation. The positive Lyapunov exponent is largest for
A

2.017, and the exponents

base-e

at that value are

0.0550, 0,

2.0720, corresponding to a Kaplan–Yorke

dimension of 2.0265. The attractor is approximately a Mo¨-
bius strip, and the basin of attraction is shaped like a tadpole
with a tail that apparently extends to infinity along the

axis.

Figure 1 shows a projection of the attractor onto the

x –

plane including a portion of the trajectory as it spirals

outward to the attractor from an initial condition near

but

not at!

the unstable saddle-focus at the origin. The eigenval-

ues, given by the characteristic equation,

0, are

within about 1% of

2.24, 0.1

0.66i over the range of A

for which bounded solutions exist.

A similar example of a chaotic flow was found in which

the x˙

term is replaced with xx˙,

Ax¨

xx˙

but this case is equivalent to Eq.

to within a constant as

can be seen by differentiating Eq.

with respect to time

and defining a new variable

[

x˙. It is chaotic over the same

range of A as is Eq.

. Figure 2 shows its attractor for

2.017 projected onto the x –

plane including a portion

Fig. 1. Strange attractor for Eq.

, with A

2.017.

539

Am. J. Phys., Vol. 65, No. 6, June 1997

J. C. Sprott

of the trajectory as it spirals outward to the attractor from an
initial condition near

but not at!

the unstable saddle focus

at the origin. These appear to be the only two examples of
dissipative chaotic jerk equations with three terms and one
quadratic nonlinearity.

Two cases were found with three terms and two quadratic

nonlinearities, a form analogous to Eq.

Axx¨

x˙

and one analogous to Eq.

Axx¨

xx˙

Equation

has an attractor that strongly resembles Fig. 1

for A

0.645, and Eq.

has an attractor that resembles

Fig. 2 for A

0.113, although the initial conditions must be

chosen carefully since their basins of attraction are relatively
small.

Eight functionally distinct cases were found of jerk func-

tions with four terms and one quadratic nonlinearity that
have strange attractors. These are mostly generalizations of
the simpler chaotic cases previously described with an addi-
tional linear term. Most if not all of them are functionally
equivalent to cases in Ref. 8. For example, one can add a
term proportional to x˙ or a constant term in either Eq.

and find chaotic solutions. These cases are characterized

by a pair of parameters, and hence it is more tedious to
explore their properties.

Other examples of dissipative chaotic flows were found of

the form,

Ax¨

x˙

where f (x) is a second-degree

or higher

polynomial given

in the notation of Eq.

by f (x)

. One

such case is

Ax¨

x˙

which has chaotic solutions for parameters in the neighbor-
hood of A

0.5 and B

0.25 and a period-doubling route to

chaos as A is decreased or B is increased. Its attractor is
shown in Fig. 3 including a portion of the trajectory as it
spirals outward from an initial condition near

but not at!

one of the unstable saddle-foci at x

1/2

, x˙

0, x¨

with

eigenvalues

0.8, 0.152

1.105i. Equation

is func-

tionally equivalent to case S in Ref. 8.

B. Conservative systems

In contrast to the cases above, conservative systems are

ones in which the phase-space volume is conserved. A suf-
ficient but not necessary condition is a

The only such cases found were of this type and had the
form of Eq.

with A

0. Since chaos requires a nonlin-

earity, the function f (x), must contain a quadratic or other
nonlinearity.

chaotic

solutions

were

found

for

f (x)

, but with an added linear or constant term

both

, many such solutions were found. However, in each

case, the trajectories eventually escaped, although the chaotic
transient can persist for hundreds of cycles. Unbounded tra-
jectories in a conservative

volume-conserving

system may

seem paradoxical, but an example is a spacecraft launched
from the earth with an initial velocity just sufficient for it to
escape from the solar system.

An example of a conservative system that exhibits chaos is

Eq.

with A

x˙

Positive values of B less than about 0.05 produce chaotic
solutions for selected initial conditions. Large values of B

0.05

are chaotic for most initial conditions, but the tra-

jectory quickly escapes. As B approaches zero, the range of
initial conditions that produce chaos shrinks to zero, and the
escape time approaches infinity. An appropriate intermediate
value is B

0.01. As with Eq.

this system has two equi-

librium points at x

1/2

, x˙

0, x¨

0. They are both un-

stable saddle foci, with the trajectory spiraling out from the
one at

1/2

and into the one at

1/2

, producing a toroidal

structure. The eigenvalues are given by the characteristic
equation,

1/2

0. Equation

may represent the

algebraically simplest example of a conservative chaotic
flow, analogous to Eq.

for dissipative chaotic flows. It

may also be the simplest formulation of a torus for suitable
initial conditions.

The behavior of such a system is best exhibited in a Poin-

care´ section, where, for example, the location of the trajec-
tory as it punctures the x¨

0 plane is plotted for various

initial conditions. Figure 4 shows such a plot for Eq.

with B

0.01. Twenty-one initial conditions are shown, uni-

form over the interval x

0.011

x˙

0, x¨

The global topology is a set of nested tori produced by in-
commensurate periodic oscillations in x

vertical in Fig. 4

Fig. 2. Strange attractor for Eq.

, with A

2.017.

Fig. 3. Strange attractor for Eq.

, with A

0.5 and B

0.25.

540

Am. J. Phys., Vol. 65, No. 6, June 1997

J. C. Sprott

and x˙

horizontal in Fig. 4

. However, there are an infinite

number of surfaces where the frequency ratio is rational,
producing chains of islands in the Poincare´ section. The
period-8, -9, -10, and -11 islands are evident in Fig. 4. Each
of these islands is surrounded by a separatrix in the vicinity
of which chaos occurs. Most of these regions are invisibly
small in Fig. 4. However, the period-10 and higher islands
overlap, producing a large connected stochastic region that
extends to infinity in the

x direction. The islands with pe-

riod less than 10 are apparently enclosed by KAM tori

Kolomorogov–Arnold–Moser

and are thus bounded.

Equation

can be transformed into a form that re-

sembles the logistic equation in Eq.

by defining a new

variable y

[

x/A

1/2, where A

[

1/2

. The resulting equa-

tion,

y˙

A y

has properties identical to Eq.

except for a scaling fac-

tor. A value of A

0.2 gives results analogous to B

0.01 in

Fig. 4.

One usual characteristic of conservative flows, not shared

by dissipative flows is time-reversal invariance. Equation

has this property as can be verified by replacing t with

t and defining a new variable y

[

x. The resulting equa-

tion for y is identical to Eq.

. Similarly, Eq.

is time-

reversal invariant as can be verified by replacing t with

and defining a new variable x

[

y .

V. NEWTONIAN JERKS

The jerk function in Eq.

is the most general quadratic

polynomial form, but it is not in general derivable by differ-
entiating Newton’s second law with a force that depends
explicitly on the instantaneous position, velocity, and time.
For such a case, we require that F(x˙,x,t) satisfy

dF/dt

x¨

]

x˙

]

m j ,

which in turn implies that F

and hence j

must be of the

form F

x¨U

x˙V

c, where U

[

]

x˙ and V

[

]

x. If j

is of the form j

j (x,x˙,x¨), then c is a constant. If x and x˙ are

independent, then

]

x˙, and Eq.

reduces to

x˙

x¨

x˙

Equation

is the most general quadratic form of what

might be called a Newtonian jerk function.

An extensive search for chaotic solutions of quadratic

Newtonian jerk equations did not produce any such ex-
amples, even with all six coefficients not zero. This result is
reasonable since the force must have an explicit time depen-
dence

unless a

, and this dependence consists of an

additive term of the form ct, which is unbounded as t ap-
proaches infinity. However, the total force must be bounded
since it is proportional to x¨, which is bounded.

VI. CUBIC NONLINEARITIES

The procedure outlined above can be extended to other

types of nonlinearities. For example, consider jerk functions
with only cubic nonlinearities. To impose symmetry, set the
constant and quadratic

even

terms to zero. The most gen-

eral such jerk function is

x˙

x¨

xx˙

xx¨

x˙x¨

x¨

x˙

xx˙

x˙

Note that Eq.

is not a Newtonian jerk because it con-

tains terms higher than linear in x¨.

A search was carried out for chaotic solutions using this

jerk form. The search was less extensive than the one with
quadratic jerks

about 10

cases vs 10

. However, chaotic

solutions were found about ten times more often

about 0.1%

of the cases examined vs 0.01%

, and so many examples

were found. Eight functionally distinct forms were found
with three terms and two cubic nonlinearities, and four were
found with four terms and one cubic nonlinearity. Interest-
ingly, no cases as simple as Eq.

were found,

although it’s difficult to rule out their existence.

An example of a cubic dissipative chaotic flow that oc-

curred often and that has a different structure than the cases
previously described is

x¨

x˙

It is governed by a single parameter A, which over the range
0

1 produces limit cycles of many periodicities inter-

spersed within broad regions of chaos. Figure 5 shows its
attractor for A

0.25, projected onto the x –

plane including

a portion of the trajectory as it spirals outward from an initial
condition near

but not at!

the unstable saddle focus at the

origin, with eigenvalues

1/3

. Similar appearing cha-

Fig. 4. Poincare´ section at x¨

0 for Eq.

, with B

0.01.

Fig. 5. Strange attractor for Eq.

, with A

0.25.

541

Am. J. Phys., Vol. 65, No. 6, June 1997

J. C. Sprott

otic solutions can be found by replacing the x¨

term in Eq.

by x

x¨ or by xx¨

Another dissipative cubic jerk function that has chaotic

solutions is

Ax¨

xx˙

Its strange attractor for A

3.6 as shown in Fig. 6 projected

onto the x –

plane resembles two back-to-back elongated

Ro¨ssler attractors. The origin is a saddle focus with eigen-
values

0, 0,

A. It is linearly neutrally stable, but weakly

unstable because of higher-order nonlinearities.

A cubic system was found that is conservative and cha-

otic. It has three terms and two cubic nonlinearities,

x˙

It consists of two sets of nested tori, one at positive x and the
other at negative x, coupled in such a way that trajectories
near their intersection are chaotic and encircle both tori. The
trajectories are bounded, in contrast to the case in Fig. 4. Its
Poincare´ section in the x¨

0 plane for A

0.01 is shown in

Fig. 7. Twenty-one initial conditions are shown, uniform
over the interval

0.769

x(0)

0.65x˙(0)

0.769, x¨

Island structure is just barely discernible near the last closed
toroidal surface.

VII. SIMPLE NUMERICAL METHOD

It is worth noting that all the results in this paper can be

replicated using an extremely simple numerical algorithm.
The difficulty in obtaining reliable solutions of coupled dif-
ferential equations has inhibited teachers and students from
devoting the same attention to chaotic flows as has been
given to chaotic maps such as the logistic map. Inappropriate
numerical methods can produce spurious results, including
false indications of chaos, and the temptation is to rely on
pedagogically undesirable canned algorithms.

Consider a linear harmonic oscillator, x¨

x, whose

phase-space trajectory is a circle with a radius determined by
the initial conditions and proportional to the square root of
the energy. The most straightforward way to solve such an
equation is the Euler method,

where h is a small increment of time. However, it is easy to
show that each iteration causes the radius to increase by a
factor of 1

/2 and the energy by a factor of 1

. Since

/h iterations are required to complete one cycle, the cu-

mulative error is linear in h, and hence the method is called
first order. The trajectory spirals outward to infinity for any
choice of h.

This problem is not as serious as it appears for dissipative

systems since it merely reduces the dissipation by an amount
that can be made negligible by choosing h sufficiently small.
However, for a conservative system, the Euler method is
essentially useless if the trajectory is followed for many
cycles. A small change in Eq.

suggested by Cromer

leads to a system that conserves energy exactly when aver-
aged over half a cycle:

This form also is very easy to program because it allows the
variables to be advanced sequentially rather than simulta-
neously. It generalizes to higher dimensions and performs
well with jerk systems since two of the derivatives involve
only a single variable. A simple

DOS

BASIC

program that

solves Eq.

by this method is

SCREEN 12
x=.02
v=0
a=0
h=.01
WHILE INKEY$=‘‘ ’’

x=x+h

v=v+h

j=−2.017

a+v

v−x

a=a+h

PSET (320+40

x, 240−40

WEND

This program produces the attractor in Fig. 1. Although all
the chaotic systems described in this paper were verified with
a fourth-order Runge–Kutta integrator, the figures were pro-
duced with minor variations of the code above

sometimes

with a much smaller value of h

to emphasize the usefulness

of this simple algorithm and to encourage experimentation.
For long calculations, double-precision

or higher

is recom-

mended to control round-off errors.

Fig. 6. Strange attractor for Eq.

, with A

3.6.

Fig. 7. Poincare´ section at x¨

0 for Eq.

, with A

0.01.

542

Am. J. Phys., Vol. 65, No. 6, June 1997

J. C. Sprott

VIII. SUMMARY

This paper has shown many examples of previously un-

known chaotic systems that involve a third-order ODE in a
single variable with simple polynomial

quadratic and cubic

nonlinearities and either one or two control parameters. None
of the cases have been examined in great detail, offering the
opportunity for additional exploration. One could search for
other similar and perhaps even simpler examples of chaotic
flows. One could look at other nonlinearities, such as trigo-
nometric, logarithmic, or exponential. The bifurcations and
routes to chaos could be examined. The basins of attraction
could be mapped. The Lyapunov exponents and dimensions
could be calculated. The structure of various Poincare´ sec-
tions and return maps could be studied. One could try to
construct physical models to which these equations apply
and attempt to observe their chaotic behavior. These sugges-
tions represent a wealth of possibilities for student research
projects. The simple computer code described in the previous
section provides an appealing starting point for such studies.

ACKNOWLEDGMENTS

I am grateful to Paul Terry, Dee Dechert, and David

Elliott for useful discussions.

Electronic mail: sprott@juno.physics.wisc.edu

R. May, ‘‘Simple mathematical models with very complicated dynamics,’’
Nature 261, 45–67

1976

M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems
and Linear Algebra

Academic, New York, 1974

G. Duffing, Erzwungene Schwingungen bei Vera¨nderlicher Eigenfrequenz

Vieweg, Braunschweig, 1918

Y. Ueda, in New Approaches to Nonlinear Problems in Dynamics, edited
by P. J. Holmes

SIAM, Philadelphia, 1980

, p. 311.

J. M. T. Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos

Wiley, New York, 1986

E. N. Lorenz, ‘‘Deterministic nonperiodic flow,’’ J. Atmos. Sci. 20, 130–
141

1963

O. E. Ro¨ssler, ‘‘An equation for continuous chaos,’’ Phys. Lett. A 57,
397–398

1976

J. C. Sprott, ‘‘Some simple chaotic flows,’’ Phys. Rev. B 50, R647–R650

1994

H. P. W. Gottlieb, ‘‘Question #38. What is the simplest jerk function that
gives chaos?’’ Am. J. Phys. 64, 525

1996

S. H. Schot, ‘‘Jerk: The time rate of change of acceleration,’’ Am. J. Phys.
46, 1090–1094

1978

T. R. Sandin, ‘‘The Jerk,’’ Phys. Teach. 20, 36–40

1990

S. Nose´, ‘‘Constant temperature molecular dynamics methods,’’ Prog.
Theor. Phys. Suppl. 103, 1–46

1991

W. G. Hoover, Computational Statistical Mechanics

Elsevier, Amsterdam

1992

W. G. Hoover, ‘‘Remark on ‘Some simple chaotic flows’’’ Phys. Rev. E
51, 759–760

1995

F. Takens, in Dynamical Systems and Turbulence, Lecture Notes in Math-
ematics Vol. 898, edited by D. A. Rand and L. S. Young

Springer, Berlin,

1981

T. Sauer, J. A. Yorke, and M. Casdagli, ‘‘Embedology,’’ J. Stat. Phys. 65,
579–616

1991

These terms were suggested by J. Codner, E. Francis, T. Bartels, J. Glass,
and W. Jefferys, respectively, in response to a question posed on the
USENET sci.physics newsgroup.

J. C. Sprott, ‘‘How common is chaos?’’ Phys. Lett. A 173, 21–24

1993

A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, ‘‘Determining
Lyapunov exponents from a time series,’’ Physica 16D, 285–317

1985

J. C. Sprott, Strange Attractors: Creating Patterns in Chaos

M&T Books,

New York, 1993

J. L. Kaplan and J. A. Yorke, in Functional Differential Equations and
Approximations of Fixed Points, edited by H.-O. Peitgen and H.-O.
Walther, Lecture Notes in Mathematics, Vol. 730

Springer-Verlag, Ber-

lin, 1979

, p. 204.

J. C. Sprott, ‘‘Simplest dissipative chaotic flow,’’ Phys. Lett. A

in press

Animated rotational views of this attractor and its basin are available at
http://sprott.physics.wisc.edu/simpjerk.htm

V. I. Arnold, Mathematical Methods in Classical Mechanics

Springer,

New York, 1978

A. Cromer, ‘‘Stable solutions using the Euler approximation,’’ Am. J.
Phys. 49, 455–459

1981

FACULTY SALARIES

There is also too small a compensation allowed to the Professors to make it an object for men

of talents to settle down in this business. The professor in a college is obliged to see much
company—the parents and guardians of the students expect some attention when they visit the
place. Also the price of all articles of living in the vicinity of a college is greater than that in the
country around while the salary is generally so small that with the strictest economy the ends
cannot be made to meet at the close of the year. We have too many colleges. The endowments are
too much scattered to produce the best effect or to allow of salaries which shall secure competent
instructors and the necessary implements of education. The salaries at Yale are but 12 hundred
dollars and those at Schenectady were the same until lately they have been cut down. In our
college none of the Professors are able to live on their salaries. Such is the expense of living in this
place that since I have been in Princeton I have been obliged to expend from 250 to 300 dollars
per year more than I receive from the college. The trustees however are desposed to be as liberal
as the state of the funds will allow but they cannot exceed their means.

Joseph Henry, letter to Peter Bullions

1846

, in The Papers of Joseph Henry, edited by Marc Rothenberg

Smithsonian

Institution Press, Washington, 1992

, Vol. 6, pp. 461–462.

543

Am. J. Phys., Vol. 65, No. 6, June 1997

J. C. Sprott