Attractors

Published 2000, Update 04-January-2014

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Attractors Quotes

Peitgen H. O., Jurgens H. and Saupe D. (1992) "*Chaos
and Fractals, New Frontiers Of Science*", Springer-Verlag. p. 656.

"Strange attractors are the
point where chaos and fractals meet in an unavoidable and most natural fashion:

as geometrical patterns, strange attractors are fractals; as dynamical objects,
strange attractors are chaotic."

●

Bak P. (1997) "*How Nature Works: the science of self-organized
criticality*", Cambridge University Press. p. 31.

"Strange attractors have fractal properties, but they do not represent geometrical fractals in real space like those we see in nature."

My Fragments - From My Personal Point of View, The Complete list see Tamari page.

MF1: Fractals relate to attractors as cobblestones in roads. added 1 January 2000.

Attractors, in this page, are mathematical entities; models, tools, machines, engines, ..., for handling natural phenomena as climate, chemistry, economics, ...

1D - One dimensional attractor:

Logistic Equation - Cobweb model - Populations

x' = ax (1-x)

a = 3.9

2D - Two dimensional attractors:

x_{1} =
l_{1}**sin**θ_{1}

y_{1} = -l_{1}**cos**θ_{1}

x_{2} = l_{1}**sin**θ_{1}
+ l_{2}**sin**θ_{2}

y_{2} = -l_{1}**cos**θ_{1 }
- l_{2cosθ2}

http://en.wikipedia.org/wiki/Double pendulum

x' = y

y' = x -
x^{3} - ay +
b**cos**(ct)

a = 0.25, b = 0.3, c = 1

x_{0} , y_{0}
, z_{0} = 0,
-2 ≤ x , y ≤ 2

x' = 1 - ax^{2}
+ y

y' = bx

a = 1.4 , b= 0.3

x_{0} ,y_{0}
= 1 , -1.5 ≤ x , y ≤ 1.5

x' = x^{2}
- y^{2 + ax
+ by
y' = 2xy - cx
+ dy}

a = 0.9 , b = -0.6 , c = 2 , d = 0.5

KAM Torus Kolmogorov Arnold Moser

x' = x**cos**(ζ)
+ (x^{2} - y)**sin**(ζ)

y' = x**sin**(ζ)
- (x^{2<} - y)**cos**(ζ)

x' = **sin**(ay)
- **cos**(bx)

y' = **sin**(cx)
- **cos**(dy )

3D - Three dimensional attractors:
**Chua**, **Ikeda**, **Lorenz**, **Pickover**, **
Rossler **, and **Tamari** attractors figured with the software
Eco.

Chua attractor - electric circuits

x' = a(y - x - g)

y' = x - y + z

z' = by
- cz

g = ex + (d-e)[abs(x+1)-abs(x-1)]/2 , a = 15.6 , b = 28 , c = 0 , d = -1.143 , e = -0.714

x_{0} = 0.7 ,
y_{0} = 0 ,
z_{0} = 0 , -2 ≤ x,≤ 2, 3 ≤y,z,≤3

Ikeda attractor - Laser

x' = a +
b(x**cos**z
- y**sin**z )

y' = b (x**sin**z
+ y**cos**z)

z' = c -
d / (1+x^{2} +
y^{2})

a = 1 , b = 0.9 , c = 0.4 , d = 6

x_{0}, y_{0},
z_{0} = 0 , - 2 ≤ x, y ≤ 2

Lorenz attractor - meteorology

x' = x - (ax
+ ay), "x"the
convective flow.

y' = y + (bx
- y -

z' = z - (cz
+
xy),
"z"
the vertical temperature distribution.

a = 28 , b = 8/3 . c= 10

x_{0} = 0 ,
y_{0}
= 1 , z_{0} = 0 ,
-1 ≤ x, <y,
z, ≤ 2

x' = **sin**(ax)
- z**cos**(by)

y' = z**sin**(cx)
- **cos**(dy)

z' = e / **sin**(x)

a = 2.24 , b = 0.43 , c = - 0.65 , d = - 2.43 , e = 1

x_{0},
y_{0}, z_{0} = 0 , - 2
≤ x, y ≤ 2

Rossler attractor - chemistry

x' = x - (y
- x)

y' = y + (x
+ ay)

z' = z + (b
+
xz - cz)

a = 0.2 , b = 0.2 , c= 5.7

x_{0} = 1 , y_{0} = 0 ,
z_{0} = 0 , -1 ≤ x , y, z, ≤ 1

Tamari attractor - Economic attractor - economics

variables:

x' = (x -
ay)**cos**(z)
- by**sin**(z)
"x" the output.

y' = (x +
cy)**sin**(z)
+ dy**cos**(z)
"y" the money.

z' = e +
fz + g**atan**{
[(1 - u)y] / [(1 -
i)x] } "z"
the pricing - spiral version.

z' = e +
fz + g**atan**{
(1 - u) / (1 - i)
xy } "z"
the wealth - attractor version.

Exogens:

a ≡ Inertia = 1.013

b ≡ Productivity = - 0.011

c ≡ Printing = 0.02

d ≡ Adaptation= 0.96

e ≡ Exchange = 0

f ≡ Indexation = 0.01

g ≡ Elasticity/Expectations= 1

u ≡ Unemployment = 0.05

i ≡ Interest= 0.05

x_{0}, y_{0},
z_{0}, = 1 , 1 ≤ x, y, ≤ 4

source: Tamari (1997) Conservation and Symmetry Laws and Stabilization Programs in Economics.

**N** "Tamari attractor",
Enrique Zeleny, From the
Wolfram Demonstrations Project.

Note 1, 12-Augoust-2006 A note on Tamari Space - the Nest

Sprott J. C. (1993) "*Strange Attractors: Creating Patterns in Chaos*",
M&T books, NY.

"To assess how common chaos is in nature, we must address the more complicated and subjective issue of whether the equations we have examined are a representative sample of the equations that describe natural processes. ... Chaos is not the most common behavior, but it is neither particularly rare."

The Origin of the Nest is Tamari Space on which the the economic system, that is Tamari Attractor, sail.

O^{'} = (O -
aM)**cos**(P)
- bM**sin**(P),
"O" Output.

M^{'} = (O
+ cM)**sin**(P)
+ dM**cos**(P),
"M"
Money.

P^{'} = e +
fP + g**arctan**{[(1-u)M] / [(1-i)O]}, "P" Pricing.

(1-u) ≡ **Gresham** coefficient, (1-u)M represent demand.

(1-i) ≡ **Gresham** coefficient, (1-i)O represent supply.

The **Nest** is the economic surface on which economic
systems sail (in the deterministic sense) and wander (in the random sense). In a
particular range of parameters, the Nest constitutes an attractor.

Some scientists have scientific problems and are looking for tools (mathematical, statistical or graphic) to solve them and some scientists have the tools and are looking for scientific problems on which to use them. This is identical to a person who has a nail and is looking for a hammer or alternatively, has a hammer and is looking for a nail. Anyone dealing with a science, e.g., Biology, Chemistry, Economy, Physics, Psychology (and today also 'Science-Fiction' and 'Speculations' are considered legitimate sciences) finds himself in one of these two positions.

When I was working as an economist in the Economic Planning
Authority, in the Seventies, we were asked to perform annual forecasts and
prepare 5-year Economic Plans. Usually, the planning and the forecasting were
conducted by the naive (*inertia*) method, according to which one took the
last growth rates of the various parameters and projected (*extrapolated*)
them to the future. This worked as long as no inflation existed and the country
was growing economically. That is, the *inertia* was enough for the
forecasting and planning.

With the change of the ruling party in Israel (1977), the economical navigation policy changed as well. From a social-democratic society (characterized by solidarity and planning), we have turned, slowly but surely, into a capitalist-liberal state (Darwinian and competitive). Inflation raised its head and forecasting and planning were made more difficult and problematic, and were left out and disappeared from public-economic thinking and navigation processes. What happened was that the economic think tanks of government departments were closed one by one and in the end even the Economic Planning Authority, which was located within the Prime Minister's office, was dismantled.

The planning procedure at that time was using econometric models for forecasting and planning. These together with computerized spreadsheets, gave us an extensive computing ability, but little mathematical capacity or knowledge (in the Sprott sense, quoted above).

As one who believes that **'Economics is Physics with Pricing**'
or '**Physics is Economics without Pricing'** (how **Nature** prices
itself, and what kind of *money* **He** uses, is not relevant to this
remark), it was clear to me that what was lacking in a proper planning and
forecasting processes were 'Behavior Equations' of the Economy. My first attempt
to add an appropriate Mathematical tool to the system (shelf solution) was in
the article "*Prices And Quantities In The Israeli Housing Market*" (1981),
the main idea of which was using the mechanism of cobwebs to describe the
behavior of the housing market in Israel.

With the rise of inflation it seemed that the important variable involved would be the quantity of Money (M) and that it had a crucial influence on the rest of the variables in the system. However, unexpectedly, the expected correlation between money (M) and the Pricing (P) could not be found and that needed an explanation in itself.

My explanation was (to show) that the mechanism of money
transfer into the market is not direct but involves the impact of "*Gresham
Law*" (1983). During those days the accepted way to deal with the influence
of money on the economy was with what is called The "Real Balances Effect"
(M/P), and it was common to assume that as long as the economic variables were
linked to the Consumer Price Index - the inflation would be neutral.

Actually a feedback mechanic process was working which feeds itself and creates a "Perpetuum Mobile". The printing of money accelerates inflation and that, through the linked mechanisms, accelerates the printing. Very quickly it was discovered that the Real Balances are in fact a two-edged sword. When they reach over a certain critical amount their tendency reverses. The public doesnt accumulate them anymore but actually withdraws from them and this accelerates the inflation.

I demonstrated this idea in the article "*Anatomy Of Tragedy*"
(1985). In it I tried to show that the scale itself doesn't really matter, be it
a large country or a small one, the same rule applies (i.e., what we call today,
a Fractal Structure or Universality). The conclusion was that the important
variable is not the Real Balances, but the amount of money relative to output
(M/O). This idea was developed in my article "*Theoretical And Empirical
Inflation*" (1986).

The main question in my work is: what happens when money (M) is
introduced as a variable in the *utility and production functions,* and an
especially important question is, what happens in an economic system when the
amount of money is raised indefinitely. It quickly turns out that the real
issues of debate are ones of *measurer* and *measured*, whereas the
Measurer (**M**) is an imminent part in a measured system [O_{t}, M_{t};
Output, Money, time].

Being an economist by training, and not a mathematician, I was
looking for a mathematical shelf-solution in order to progress. At first, I
worked only in an Output-Money Space [O_{t}, M_{t}]. I described
this in the article "*Why There Is No Growth In Israel?*" (1990). In the
article I showed that a surplus of money curved the economic space and caused
the economic units to deviate from their optimal conditions. Since I found that
an economic system in the output and money space is a *conservative* one,
I chose the * Cremona equations* (which have a conservative quality
in a two dimensional space) as suitable for the description of the output-money
[O

It is likely that the real equations, in the sense of the quote
given above, are modifications of the * Cremona
equations*. It is also likely that the econometric findings that the real
equations produce are not supposed to change on principle, especially not the
sign of the money parameter (- b) in the first equation. These ideas I wrote in
the book

The basic hypothesis in my work was that the correct
mathematical surfaces for the description of economic systems are *Minimal
Surfaces* because of their minimization property (see for example **
Osserman** "*A survey of Minimal surfaces*" Dover 1969). However, all my
efforts to prove this econometrically failed.

In the next working phase, I tried to check the dynamics of the
system by adding the third Pricing-equation (**P**) to the two dimensional *
Cremona equations*, this is supposed to function as a

The Economic equations were implemented in
Mathematica software in order to better learn the graphic structure of the
system. The result was published in the
Conservation and Symmetry Laws and Stabilization Programs in Economics
(1997, English). Now it is the time to study these "behavior-equations (The
Nest)" through a dynamic, more advanced shelf-software for the purpose of
mapping the **Nest**.

Most of the time, the economy of a country is within the green boundary of the Nest, the surface of which, is almost linear (in rates) and the situation not chaotic, and no high sensitivity to initial conditions exist. Forecasting and planning are possible with a reasonable level of accuracy and likelihood, in the short term of up to 3 years.

Sometimes, during politically chaotic times (especially in times of armed conflicts), some countries find themselves, unwillingly, in a rush towards the red and dangerous zone. In this zone, a country approaches the critical chaotic areas together with a continuous rise of sensitivity to the initial conditions of the economic system and it looses control.

The mapping of the Nest, in its variables and parameters ranges, will serve as an efficient tool for navigation and management of the economy of a country, see Ecometry .

Note 2, 13-April-2008 A note on Feedback in Attractors

Pickover attractor ● Ikeda attractor ● Tamari attractor

Equations | ||

x^{'} = sin(ax)
- zcos(by), |
x^{'} = a
+ b(xcosz
- ysinz), |
x^{'}
= (x -
ay)cos(z)
- bysin(z), "x"
the output, Cremona/Conservative |

y^{'} = zsin(cx)
- cos(dy), |
y^{'} = b(xsinz
+ ycosz), |
y^{'}
= (x +
cy)sin(z)
+ dycos(z), "y" the
money, -------equations------------ |

- - - - - - - - - - - - - - - | - - - - - - - - - - - - - - - - | - - - - - - - - - - - - - - - - - -- - - -- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - |

z^{'} = e /
sin(x), |
z = c -
d / (1+x^{2}+y^{2}) |
z^{'}
= e +
fz +
garctan{ [(1-u)y]
/ [(1-i)x]
}, "z"
pricing version, Feedback
equation.(probably belongs to the spiral
attractor family).z ^{'} =
e + fz
+ garctan{ (1-u)
/ (1-i)
xy
}, "z"
wealth version, Feedback equation. |

Parameters | ||

a = 2.24 | a = 1 | a ≡ Inertia = 1.013 |

b = 0.43 | b = 0.9 | b ≡ Productivity = -0.011 |

c = -0.65 | c = 0.4 | c ≡ Printing = 0.02 |

d = -2.43 |
d = 6 |
d ≡
Adaptation = 0.96 |

e = 1 | e ≡ Exchange = 0, | |

f ≡ Indexation = 0.01 | ||

g ≡ Elasticity/Expectations = 1 | ||

u ≡
Unemployment rate = 0.05 (1-u) ≡ Gresham coefficient(1-u)y represent demand. |
||

i ≡
Interest rate = 0.05 (1-i) ≡ Gresham coefficient(1-i)x represent supply. |
||

Domain | ||

-2 ≤ x , y ≤ 2 | -2 ≤ x , y ≤ 2 | 1 ≤ x , y ≤ 4 |

Initial Conditions |
||

x_{0}, y_{0},
z_{0}, = 0 |
x_{0}, y_{0},
z_{0}, = 0 |
x_{0},
y_{0}, z_{0},
= 1 |

In dynamic systems the output of yesterday is the input (feedback) of today, and the feedback (height, potential, pricing, ..., ) equation makes the difference among the phenomena in conservative systems.

Books and Articles:

Anishchenko, Astakhow, Neiman, Vadivasova, Schimansky-Geier (2002) "*NonLinear Dynamics of Chaotic and Stochastic Systems*", Springer.

Arneodo, Coullet, and Tresser C. (1981) "*Possible New
Strange Attractors With Spiral Structure*", Com. Math. Phys. 79, 573-579.

Crilly A. J., Earnshaw R. A., Jones H. (Editors) (1991) "*Fractals and Chaos*",
Springer-Verlag.

Cvitanovic' P. (editor) (1989) "*Universality in Chaos*", 2ed,
Institute of Physics Publishing, Bristol and Philadelphia.

Feder J. (1988) *"Fractals"*, Plenum Press.

Holden A.V. (editor) (1986) *"Chaos",* Princeton UP.

Ivancevic V. G. and Ivancevic T. T. (2006) "*Geometrical Dynamics of Complex
Systems*", Springer.

Ivancevic V. G. and Ivancevic T. T. (2007) "*High-Dimensional Chaotic and
Attractor Systems: A Comprehensive Introduction*", Springer.

Lorenz N. Edward (1993) "*The Essence of Chaos*", University of Washington
Press.

Medio A. with Gallo G. (1992) "*Chaotic Dynamics: Theory and Applications to
Economics*", Cambridge UP.

Peitgen H. O., Jurgens H. and Saupe D. (1992) "*Chaos and Fractals, New
Frontiers Of Science*", Springer-Verlag.

Pickover C. A. (1990) "*Computers, Pattern, Chaos and Beauty*", St.
Martin's Press, N.Y..

Puu T. (1991) *"Nonlinear Economic Dynamics"*, 2ed.,Springer-Verlag.

Puu T. (2003) *"Attractors, Bifurcations, & Chaos; Nonlinear Phenomena in
Economics"*, 2ed., Springer.

Reichl L. E. (1992) *"The Transition to Chaos", *Springer-Verlag.

Rosser J. B. (1991) "*From Catastrophe to Chaos: A General Theory of
Economics Discontinuities*", Kluwer Academic Publishers.

Ruelle David (1995) "*Turbulence, Strange Attractors, and Chaos*", World
Scientific.

Schroeder M. (1991) "*Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise*", W. H. Freeman and Com. N.Y..

Skiadas C. H. Skiadas C. (2009) "*Chaotic Modelling and Simulation: Analysis of Chaotic Models, Attractors and Forms" *, CRC Press.

Sprott J. C. (1993) "*Strange Attractors: Creating Patterns in Chaos*",
M&T Books, NY.

Sprott J. C. (2003) "*Chaos and Time-Series Analysis*", Oxford UP.

Surhone L. M., Tennoe M. T. , Henssonow S. F. (Ed.) (2010) "*Rossler attractor*", βetascript Pub.

Surhone L. M., Tennoe M. T. , Henssonow S. F. (Ed.) (2011) " *Tamari attractor*", βetascript Pub.

Wegner T. and Tyler B. (1993) "*Fractal Creations*" 2ed, Waite Group Press.

Zaslavsky G. M. (1985) *"Chaos in Dynamic Systems",* translated from Russian by Kisin V.I., Harwood AP.

Links

http://demonstrations.wolfram.com/TamariAttractor/

http://en.wikipedia.org/wiki/Attractor

http://www.scholarpedia.org/article/Attractor

http://mathworld.wolfram.com/Attractor.html

http://hypertextbook.com/chaos/21.shtml

http://sprott.physics.wisc.edu/sa.htm (Sprott).

http://pchen.ccer.edu.cn/homepage/Homepage%20Chinese/AED2003/readingpapers/DetermModel/SDR88p.PDF

Software:

http://www.phaser.com/index.html