Published 2000, Update 10-December-2012
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."
Fractals relate to attractors as cobblestones in roads. (added: 1.1.2000)
Attractors, in this page, are mathematical entities (tools, machines, engines, ...) for handling natural phenomena as climate, chemistry, economics, ...
One dimensional attractor - 1D :
Logistic Equation - Cobweb model - Populations
x' = ax (1-x)
a = 3.9
Two dimensional attractors - 2D :
x1 = l1sinθ1 y1 = -l1cosθ1 x2 = l1sinθ1 + l2sinθ2 y2 = -l1cosθ1 - l2cosθ2
x' = y y' = x - x3 - ay + bcos(ct)a = 0.25
x0 , y0 , z0 = 0 -2 ≤ x , y ≤ 2
x' = 1 - ax2 + y
a = 1.4
x0 ,y0 = 1
-1.5 ≤ x , y ≤ 1.5
x' = x2 -
y2 + ax + by
y' = 2xy - cx + dy
a = 0.9
b = -0.6
c = 2
d = 0.5
KAM Torus Kolmogorov Arnold Moser
x' = xcos(ζ) + (x2 - y)sin(ζ)
y' = xsin(ζ) - (x2 - y)cos(ζ)
de Jong Attractor
x' = sin(ay) - cos(bx)
Three dimensional attractors - 3D
Chua, Ikeda, Lorenz, Pickover, Rossler, and Tamari attractors figured with the software Eco.
N 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
x0 = 0.7
y0 = 0
z0 = 0
Ikeda attractor - Laser
x' = a + b(xcosz - ysinz )
Lorenz attractor - climate
x' = x - (ax + ay), "x"the convective flow.
y' = y + (bx - y - zx), "y" the horizontal temperature distribution.
z' = z - (cz + xy), "z" the vertical temperature distribution.
a = 28
b = 8/3
x' = sin(ax) - zcos(by)
y' = zsin(cx) - cos(dy)
z' = e / sin(x)
a = 2.24
b = 0.43
c = - 0.65
d = - 2.43
e = 1
x0, y0, z0 = 0
- 2 ≤ x, y ≤ 2
Rossler attractor - chemistry
x' = x - (y
y' = y + (x + ay)
z' = z + (b + xz - cz)
a = 0.2
b = 0.2
x0 = 1
y0 = 0
z0 = 0
Tamari attractor , Economic attractor - economics
x' = (x - ay)cos(z) - bysin(z) "x"
y' = (x + cy)sin(z) + dycos(z) "y" the money
z' = e + fz + gatan[(1 - u)y / (1 - i)x] "z" the pricing
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
x0, y0, z0, = 1
1 ≤ x, y, ≤ 4
source: Tamari (1997) Conservation and Symmetry Laws and Stabilization Programs in Economics.
Note 1, 12-Augoust-2006 A note on Tamari Attractor - 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 - Tamari Attractor
O' = (O - aM)cos(P) - bMsin(P), "O" Output.
M' = (O + cM)sin(P) + dMcos(P), "M" Money.
P' = e + fP + garctan[(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 [Ot, Mt; 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 [Ot, Mt]. 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 [Ot, Mt] space.
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 Ecometry - Foundations of Economics (1991, Hebrew). The name Ecometry is derived from the combinations of the words Eco-nomy and Geo-metry.
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 feedback equation. The equation was formulated during a process of trial and error in order for it to combine well with the statistical data. It expresses the idea that the amount of money which is brought to the market for exchange is sensitive to: Interest rates (i), Unemployment (u), Exchange rates (e) and Linking (f). Thus, a three-dimensional space is in fact created: Output, Money and Pricing, and its rates of velocity, Growth, Printing and Inflation respectively [O,M,P,; o,m,p,]. The econometric results show that the economic systems tend to be a symmetrical systems. These ideas and findings were brought forward in my book Dynamic Economy (1995, Hebrew).
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 attractorz Ikeda attractor Tamari attractor - probably belongs to the spiral attractor family.
|x' = sin(ax) - zcos/b>(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+x2+y2)||z' = e + fz + garctan[(1-u)y / (1-i)x], "z" the pricing, Feedback equation|
|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.|
|-2 ≤ x , y ≤ 2||-2 ≤ x , y ≤ 2||1 ≤ x , y ≤ 4|
|x0, y0, z0, = 0||x0, y0, z0, = 0||x0, y0, z0, = 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.
N Arneodo, Coullet, and Tresser C. (1981) "Possible New Strange Attractors With Spiral Structure", Com. Math. Phys. 79, 573-579. (added 18-11-2012)
Crilly A. J., Earnshaw R. A., JonesH. (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..
N 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.