Elegant Chaos: Algebraically Simple Chaotic Flows
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Transcript Elegant Chaos: Algebraically Simple Chaotic Flows
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 ax 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 ax 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)
xi ( xi 1 xi 2 ) xi 1
Hyperlabyrinth System (101-D)
xi 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)
[email protected] (contact me)