Simple Chaotic
Systems and Circuits
J. C. Sprott
Department of Physics
University of Wisconsin Madison
Presented at
University of Catania
In Catania, Italy
On July 15, 2014
Outline

Abbreviated History

Chaotic Equations
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Chaotic Electrical Circuits
Abbreviated History

Poincaré (1892)
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Van der Pol (1927)
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Ueda (1961)
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Lorenz (1963)
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Knuth (1968)
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Rössler (1976)
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May (1976)
Lorenz Equations (1963)
dx/dt = Ay – Ax
dy/dt = –xz + Bx – y
dz/dt = xy – Cz
7 terms, 2 quadratic
nonlinearities, 3 parameters
Rössler Equations (1976)
dx/dt = –y – z
dy/dt = x + Ay
dz/dt = B + xz – Cz
7 terms, 1 quadratic
nonlinearity, 3 parameters
Lorenz Quote (1993)
“One other study left me with mixed
feelings. Otto Roessler of the
University of Tübingen had formulated
a system of three differential equations
as a model of a chemical reaction. By
this time a number of systems of
differential equations with chaotic
solutions had been discovered, but I
felt I still had the distinction of having
found the simplest. Roessler changed
things by coming along with an even
simpler one. His record still stands.”
Rössler Toroidal Model (1979)
“Probably the simplest strange attractor of a 3-D ODE”
(1998)
dx/dt = –y – z
dy/dt = x
dz/dt = Ay – Ay2 – Bz
6 terms, 1 quadratic
nonlinearity, 2 parameters
Sprott (1994)


J. C. Sprott,
Phys. Rev. E 50,
R647 (1994)
14 additional examples
with 6 terms and 1
quadratic nonlinearity
5 examples with 5
terms and 2 quadratic
nonlinearities
Gottlieb (1996)
What is the simplest jerk
function that gives chaos?
x  J ( x, x , x )
Displacement: x
Velocity: x = dx/dt
Acceleration: x = d2x/dt2
Jerk: x = d3x/dt3
Linz (1997)
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

Lorenz and Rössler systems
can be written in jerk form
Jerk equations for these
systems are not very “simple”
Some of the systems found by
Sprott have “simple” jerk
forms:
x  x
  xx  ax – b
Sprott (1997)
“Simplest Dissipative Chaotic Flow”
dx/dt = y
dy/dt = z
dz/dt = –az + y2 – x
x  ax  x 2  x
5 terms, 1 quadratic
nonlinearity, 1 parameter
Zhang and Heidel (1997)
3-D quadratic systems with
fewer than 5 terms cannot
be chaotic.
They would have no
adjustable parameters.
Eichhorn, Linz and Hänggi
(1998)

Developed hierarchy of
quadratic jerk equations with
increasingly many terms:
x  ax  x 2  x
x  ax
  bx  xx – 1
x  ax  bx  x 2 – 1
x  ax  bx  cx 2  xx – 1
...
Weaker Nonlinearity
dx/dt = y
dy/dt = z
dz/dt = –az + |y|b – x
b
x  ax  x  x
Seek path in a-b space that gives
chaos as b  1.
Regions of Chaos
Linz and Sprott (1999)
dx/dt = y
dy/dt = z
dz/dt = –az – y + |x| – 1
x  ax  x  x  1
6 terms, 1 abs nonlinearity, 2
parameters (but one =1)
General Form
dx/dt = y
dy/dt = z
dz/dt = – az – y + G(x)
x  ax  x  G(x )
G(x) = ±(b|x| – c)
G(x) = ±b(x2/c – c)
G(x) = –b max(x,0) + c
G(x) = ±(bx – c sgn(x))
etc….
Universal Chaos Approximator?
Operational Amplifiers
First Jerk Circuit
x  ax  x  x  1
18 components
Bifurcation Diagram for
First Circuit
Strange Attractor
for First Circuit
Calculated
Measured
Second Jerk Circuit
15 components
x   Ax  x  B x  C
Chaos Circuit
Third Jerk Circuit
11 components
x   Ax  x  x  sgn( x )
Simpler Jerk Circuit
9 components
x   Ax  Bx  C (sgn x - x )
Inductor Jerk Circuit
7 components
x   Ax  Bx  C (sgn x - x )
Delay Lline Oscillator
6 components
x  sgn x - x
References

http://sprott.physics.wisc.edu/
lectures/cktchaos/ (this talk)

http://sprott.physics.wisc.edu/c
haos/abschaos.htm

sprott@physics.wisc.edu