Eleganle
Chaotic Flows
J. C. Sprott
Department of Physics
University of Wisconsin
– Madison (USA)
Presented at
American University
in Cairo, Egypt
on May 12, 2011
Modern Beginnings of Chaos
Edward Norton Lorenz, 1917–2008
Photo: MIT
Lorenz System
(1963)
x   ( y  x )
y   xz  rx  y
z  xy  bz
16
1
4
with chaotic solutions for  = 10, r
= 28, and b = 8/3, and Lyapunov
exponents  = (0.9056, 0, –14.5723)
(0.3359, 0, -6.3359)
Lorenz Attractor
x  4( y  x )
y   xz  16x  y
z  xy  z
x   ( y  x ) Elegance
y   xz  rx  y
z  xy  bz
x  x  y
y  xz  2 y
z  xy  z
a = (-1, 1, 1, 0, -2, 1, -1)
Inelegance = 7
a = (4, -4, -1, 16, -1, 1, -1)
Inelegance = 11
x  a1 y  a2 x 0
y  a3 xz  a4 x  a5 y
z  a6 xy  a7 z
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 System
(1976)
x   y  z
y  x  ay
z  b  z ( x  c)
which is chaotic for a = b = 0.2, c = 5.7
More elegant case: a = 0.5, b = 1, c = 3
Inelegance: 10  6
Rössler Attractor
x   y  z
y  x  y / 2
z  1  z ( x  3)
Plus 278 additional such cases in the book…
Some are simplifications of systems already known,
but most are new.
Sprott (1994)
J. C. Sprott,
Phys. Rev. E 50,
R647 (1994)

14 examples with 6
terms and 1 quadratic
nonlinearity

5 examples with 5 terms
and 2 quadratic
nonlinearities
Diffusionless Lorenz System
Van der Schrier &
Maas (2000)
Munmuangsaen &
Srisuchinwong (2009)
x  y  x
y   xz
z  xy  1
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
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  x2 – 1
x  ax  bx  cx2  xx – 1
...
Simplest Chaotic Jerk
Function (Sprott, 1997)
Munmuangsaen,
Srisuchinwong & Sprott
(2011)
f (x )
x  ax  x 2  x
a  2.02
Zhang and Heidel (1997)
3-D quadratic systems with
fewer than 5 terms cannot be
chaotic.
They would have no
adjustable parameters.
Linz (1997)



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
Simplest Piecewise-linear
System (Sprott & Linz, 1999)
x  ax  x  x  1
a  0.6
Halvorsen’s System
x  1.3x  4 y  4 z  y 2
y  1.3 y  4 z  4 x  z 2
z  1.3z  4 x  4 y  x 2
Thomas’ System
x   x  4 y  y 3
y   y  4 z  z 3
z   z  4 x  x 3
Nonautonomous System
x  sgn x  sin t
Nosé-Hoover Oscillator
x  y
y  yz  x
z  1  y
2
Simplest Conservative
Chaotic Flow
x  8x  x  1
Simplest Circulant System
x  y  z
2
y  z  x
2
z  x  y
2
Labyrinth Chaos
x  sin y
y  sin z
z  sin x
Dixon System (2-D !)
x 
xy
x2  y2
y2
y  2
 0.7 y  0.3
2
x y
Simplest Hamiltonian System
(4-D)
x  xy
y   y  x
2
Lorenz-Emanuel System
(101-D)
xi  ( xi 1  xi 2 ) xi 1
Hyperlabyrinth System (101-D)
xi  sin xi 1
Kuramoto-Sivashinsky PDE
ut  uux  uxx  uxxxx
Simple Chaotic DDE
x  sin xt 5
Chua’s Circuit
12 Components
Simplest(?) Inductorless
Circuit
9 Components
Simplest(?) Chaotic Circuit
7 Components
Chaos Circuit
Bifurcation Diagram
References

http://sprott.physics.wisc.edu/
lectures/elegant.ppt (this talk)

http://www.worldscibooks.com/chaos/7
183.html (info about the book)

sprott@physics.wisc.edu (contact me)