THE PARADIGM OF COMPLEX SYSTEMS

• M.G.Mahjani

• K.N.Toosi University of Technology

• Mahjani@kntu.ac.ir

THE CENTURY OF COMPLEXITY ?

"I think the next century will be the century of complexity."

Stephen Hawking

From Certainty to Uncertainty

Deterministic

• Newton demonstrated that his three laws of motion, combined through the process of logic, could accurately predict the orbits in time of the planets around the sun, the shapes of the paths of projectiles on earth, and the schedule of the ocean tides throughout the month and year, among other things.

The End of Certainty From mechanical organization to biological

metaphor- evolution , self organization , from simplicity to complexity

From atom to quasi biological entity From being to becoming From “ TIME IS ILLUSON” to “ TIME IS

OPERATOR” From equilibrium thermodynamic to far from

equilibrium thermodynamic

WHY WHAT “TRADITIONAL SCIENCE” DID TO THE

QUESTION MADE THE PRESENT SITUATION INEVITABLE:

• THE MACHINE METAPHOR TELLS US TO ASK “HOW?”

• REAL WORLD COMPLEXITY TELLS US TO ASK “WHY?”

WHY WHAT “TRADITIONAL SCIENCE” DID TO THE

MODELING RELATION MADE THE PRESENT SITUATION

INEVITABLE:

• THE “REAL WORLD” REQUIRES MORE THAN ONE “FORMAL SYSTEM” TO MODEL IT (THERE IS NO “UNIVERSAL MODEL”)

WHY WHAT “TRADITIONAL SCIENCE” DID TO THE MODELING

RELATION MADE THE PRESENT SITUATION INEVITABLE:

• WE MORE OR LESS FORGOT THAT THERE WAS AN

• ENCODING AND DECODING

CODE• A set of rules, a mapping or a transformation

establishing correspondences between the elements in its domain and the elements in its range or between the characters of two different alphabets. information maintaining codes establish one-to-one correspondences. Information loosing codes establish many-to-one and/or one-to-many correspondences. When a code relates a set of signs to a set of meanings by convention, a code can be seen to constitute symbols. When it maps a set of behaviors into a set of legal categories, a code can be seen to be one of law. When it accounts for the transformation of one kind o r signal into another kind of signal it can be seen to describe an input-output device. When applied to linguistic expressions it is a translation. According to Webster's, "to codify" is "to reduce to a code,“ to systematize, to classify. Indeed, any many-to-one code defines an equivalence relation or classification of the elements in its domain. It is incorrect to call a set of signs (to which a code may apply) a code. (Krippendorff)

WHY IS THE WHOLE MORE THAN THE SOME OF THE PARTS?

• BECAUSE REDUCING A REAL SYSTEM TO ATOMS AND MOLECULES LOOSES IMPORTANT THINGS THAT MAKE THE SYSTEM WHAT IT IS

• BECAUSE THERE IS MORE TO REALITY THAN JUST ATOMS AND MOLECULES (ORGANIZATION, PROCESS, QUALITIES, ETC.)

CAN WE DEFINE COMPLEXITY?

• Complexity is the property of a real world system that is manifest in the inability of any one formalism being adequate to capture all its properties. It requires that we find distinctly different ways of interacting with systems. Distinctly different in

• the sense that when we make successful models, the formal systems needed to describe each distinct aspect are NOT

• derivable from each other

Definition of Complexity

Complexity philosophy is an holistic mode of thought and relates to the following properties of systems. Not all these features need be present in all systems, but the most complex cases should include them.

WHY IS ORGANIZATION SPECIAL ? BEYOND MERE ATOMS AND MOLECULES

• IS THE WHOLE MORE THAN THE SUM OF ITS PARTS?

• IF IT IS THERE IS SOMETHING THAT IS LOST WHEN WE BREAK IT DOWN TO ATOMS AND MOLECULES

• THAT “SOMETHING” MUST EXIST

vast new horizons have been opened up for our imaginations requiring new

conceptualizations and innovative research

Unpredictability

Uncontrolled

Fuzzy sets

genetic a logorithm G enetic Program ing

Neural netw ork

Soft Com puting

non-standard instabilaty Non-Uniform

Non-Equilibrium

Nonlinear

C o m p lex S ys tem

complexity

self-M odification self-Reproduction self-Organization Coevolution

Em ergence attractor

Strange Attractor Fractal

Chaos

complex system

Type of Complexity

S ta tic C o m p lex ityF ixed s truc tures , fro zen in tim e

D yn am ic C o m p lex ityS ys tem s w ith tim e reg ula rities

E vo lvin g C o m p lex ity O p en en d ed m uta tio n , in n o va tio n

S e lf-O rg an is in g C o m p lex ityS e lf-m a in ta in in g sys tem s, aw are

Com plex System

Complex System

• Complexity Theory states that critically interacting components self-organize to form potentially evolving structures exhibiting a hierarchy of emergent system properties.

ASPECTS RELATED TO DYNAMIC CONCEPTIONS

Mode Inorganic Organic

Constraints Static Dynamic

Change Deterministic Stochastic

Language Procedural Production

Operation Taught Learning

Interaction Defined Co-evolutionary

Function Specified Fuzzy

Update Synchronous Asynchronous

Future Predictable Unpredictable

State Space Ergodic Partitioned

Causality Linear Circular

Mode Inorganic OrganicConstruction Designed Evolved

Control Central Distributed

Interconnection Hierarchical Heterarchical

Representation Symbolic Relational

Memory Localised Distributed

Information Complete Partial

Structure Top down Bottom up

Search space Limited Vast

Values Simple Multivariable

View Isolated Epistatic

COMPLEX SYSTEMS VS SIMPLE MECHANISMS

COMPLEX NO LARGEST MODEL WHOLE MORE THAN SUM

OF PARTS CAUSAL RELATIONS RICH

AND INTERTWINED GENERIC ANALYTIC SYNTHETIC NON-FRAGMENTABLE NON-COMPUTABLE REAL WORLD

SIMPLE LARGEST MODEL WHOLE IS SUM OF PARTS

CAUSAL RELATIONS DISTINCT

N0N-GENERIC ANALYTIC = SYNTHETIC FRAGMENTABLE COMPUTABLE FORMAL SYSTEM

Emergence Properties are not

describable in part terms (meta-system transitions) The properties of the overall system will be expected to contain functions that do not exist at part level . These functions or properties will not be predictable using the language applicable to the parts only and are what have been called 'Meta-System Transitions' [Turchin].

Emergence properties

The unpredictability that is thus inherent in the natural evolution of complex systems then can yield results that are totally unpredictable based on knowledge of the original conditions. Such unpredictable results are called emergent properties.

Emergent properties thus show how complex systems are inherently creative ones.

Godel’s Undecidability Theorem

• Proved that the word of pure mathematics is inexhaustible.

• No finite set of axioms and rules of inference can ever encompass the whole of mathematics.

• Given any finite set of axioms, We can find meaningful mathematical questions which the axioms leave unanswered.

Kurt godel With Einstein in

Princeton in 1950

Uncertainty in Measurements

In dynamics, the presence of uncertainty in any real measurement means that in studying any system, the initial conditions cannot be specified to infinite accuracy.

The most important problem

The most important problem is we can not solve problems at the level of

thinking at which they were created.

Einstein

Initial Condition

• As dynamical laws, Newton's laws are deterministic because they imply that for any given system, the same initial conditions will always produce identically the same outcome.

Definition of Chaos

The extreme "sensitivity to initial conditions" mathematically present in the systems studied by Poincaré has come to be called dynamical instability, or simply chaos.

Chaos Theory• Aperiodic behaviour of a

given variable of a bounded deterministic system which  may appear as random behaviour.

• The chaotic system is sensitive to initial conditions, and so, is unpredictable over a large time scale since the initial conditions are rarely known with infinite precision.

Sensitivity to initial conditions. Small changes in initial conditions lead to totally different behaviour patterns after a certain time (here 14 cycles).

Butterfly Effect• This principle is

sometimes called the "butterfly effect." In terms of weather forecasts, the "butterfly effect" refers to the idea that whether or not a butterfly flaps its wings in a certain part of the world can make the difference in whether or not a storm arises one year later on the other side of the world.

Phase Space• Space in which

each point describes the state of a dynamical system as a function of the non-constant parameters of the system.

Logistic Map

• A good example of a nonlinear dynamic is what ecologists call the logistic model of logistic map, which can be used to model population dynamics. current population. Taking a certain maximum population – the "carrying capacity" – one can construct an equation that allows for a certain death rate along with a birth rate that depends on the amount of free space available. In the logistic map, s can be taken to mean the intrinsic birth rate of the population measured in rescaled time units . •xn+1 = s * xn * (1 - xn)

• Plot of xn+1 against xn for the logistic map with a particular s.

Attractor

• x0: 0.001

x1: 0.001998

x2: 0.0039880159920000005

x3: 0.007944223440895105

x4: 0.015762225509632476

...x14: 0.4999999999999971

x15: 0.5

x16: 0.5

x17: 0.5

x18: 0.5

• Logistic map, s = 2.

• In the language of dynamical systems, the value 0.5 is called an attractor for s = 2. Other initial populations with a growth rate of s = 2 will eventually settle down to the same equilibrium of 0.5 after several iterations. This term can be applied to other dynamical systems as well;

Limit Cycle Attractors

• x0: 0.001

x1: 0.0030969000000000005

x2: 0.009570658552209001

...x150: 0.5580141252026961

x151: 0.7645665199585943

x152: 0.5580141252026961

x153: 0.7645665199585943

x154: 0.5580141252026961

x155: 0.7645665199585943

x156: 0.5580141252026961

Logistic map, s = 3.1

• In dynamical system parlance, the system has arrived at a limit cycle attractor, its population going through a constant cycle of changes. Specifically, the behavior is a 2-cycle attractor, because two values are involved. Nonlinear dynamical systems can have a number of cycles.

Chaos – Strange Attractors

• x0: 0.001

x1: 0.003996

x2: 0.015920127936000002

x3: 0.06266670985000558

x4: 0.2349583733063232

x5: 0.7190117444782786

x6: 0.8081354231223248

x7: 0.6202102440689035

x8: 0.9421979888835786

x9: 0.21784375450927384

x10: 0.6815514125223083

Logistic map, s = 4.

• Chaos has appeared – not in its common usage, which can simply mean random, but in its mathematical sense indicating unpredictability. Unpredictable here does not indicate randomness, as it has been shown that the system is entirely determined by its initial conditions and its dynamic, making the sequence deterministic. This type of behavior is more precisely referred to as deterministic chaos, although just "chaos" will be used here with that understood meaning.

Limit cycle Attractor1 -dimensional

attractor or limit cycle. The arrows correspond to trajectories starting outside the attractor, but ending up in a continuing cycle along the attractor.

Fixed point Attractor

a point attractor: the arrows represent trajectories starting from different points but all converging in the same equilibrium state .

Basin of Attractor

three attractors with some of the trajectories leading into them. Their respective basins are separated by a dotted line.

Fractal Objects The seemingly

chaotic behavior of noise displayed a fractal structure.

Mandelbrot recognized a self-similar pattern that the fractals formed. He then cross-linked this new geometrical idea with hundreds of examples, from cotton prices to the regularity of the flooding of the Nile River.

Attractor Landscapes• Can we apply these ideas to people issues ?

Indeed we can, we are all familiar with decisions that once made are difficult to reverse, and also perhaps with the feeling that we are being drawn into a situation against our will. Consider life then as a complex landscape full of hills and valleys. We try to navigate from attractor to attractor, using energy to climb to the top of a nearby hill - changing state, so that we can reach a better valley, a new (hopefully more rewarding) steady state – or attractor. There seems to be only one problem. We can see neither the hills nor the valleys and don't know if we are getting higher or lower on our personal quest. How is this landscape structured ?

Logistic Equation

An+1 = rAn(1 - An)

f(x) = rx (1 - x).

Source of Diversity ; Non Ergodic System

Feigenbaum's Constant• The picture shows a fraction

of the Feigenbaum tree. The vertical lines does not belong to the tree, but shows how to measure the distances d[1], d[2], ... . Feigenbaum's constant is defined to be the limit of d[i]/d[i+1] as i tends to infinity.

• Feigenbaum's constant is approximately equal to 4.66920160910299067185320382047

Logistic equation

Bifurcation diagram

BifurcationBifurcation

• In the study of chaos it is often useful to examine a bifurcation diagram of a system, with inputs (in this case, the growth rate s) on the horizontal axis and the outputs (here, the population size xn) on the vertical axis. The bifurcation diagram of the logistic map immediately shows some startling features. The first bifurcation happens at s = 1; at this point, the population shows positive growth for the first time. At s = 3 there is another bifurcation; populations with growth rates over s = 3 exhibit 2-cycle attractors. Near s = 3.45, the 2-cycle bifurcates into a 4-cycle, and at around s = 3.55 the 4-cycle changes into an 8-cycle. Further bifurcations quickly interact and plunge the system into chaotic cycles.

Edge of ChaosEdge of Chaos• This 'instability with order' is

what we call the 'Edge of Chaos', a system midway between stable and chaotic domains. It is characterized by a potential to develop structure over many different scales (the three responses above could occur simultaneously - by affecting various group members differently), and is an often found feature of those complex systems whose parts have some freedom to behave independently.

Prigogine’s three questions

1-Who will Benefit from the networked society ? Will it decrease the gap between nations ?

2-What will be the Effect of NS on individual creativity ?3-Harmony between man and nature.WhatWill be the impact of the networked society on this issue ?

At present humanity is going through a bifurcation process due to information technology

Larger role of nonlinear terms through larger fluctuations and

instability.

Rayleigh Benard Instabilities The fluid is assumed

to be Boussinesq. This means, essentially, that the density is assumed to be only a function of the temperature and that the parameters for the fluid such as viscosity and thermal diffusivity do not vary over the volume of the fluid. The system is governed by the Boussinesq equations.

Rayleigh Benard instabilities

Order out of Disorder

Benard Convection Cell

Benard convection cell up-down movement R-L rotation

Understanding distance in space .The emergence of the concept of space in a system in which space could not previously be perceived in an intrinsic manner is called symmetry breaking

In a way symmetry breaking brings us from a static, geometrical view to an “Aristotelian” view in which space is shaped or defined by the functions going on in the system.

The most remarkable feature to be stressed in the sudden transition from simple to complex behavior is the order and coherence of this system. This suggest the existence of correlations that is statistically.

Long Range Correlation The characteristic space

dimension of Benard cell in usual laboratory conditions is in the order 10-1 cm the whereas the characteristic space scale of the intermolecular forces is 108cm up to a distance equal to about one molecule, a single benard cell compromises something like 1020 molecules.

That this huge number of particles can behave in a coherent fashion, as in the case of convective flow, despite the random thermal motion of them is one of the principal properties characterizing the emergence of complex behavior .

Fathers of Chemical Oscillation

B. P. Belousov A. M. Zhabotinsky

Culture and Science

…There is tendency to forget that all science is bound up with human culture in general ,and that scientific findings, even those which at the moment appear the most advanced and esoteric and difficult to grasp are meaningless outside their culture context. Erwin Schrödinger

Mechanism of the BZ reaction

• The main substances here are HBrO2 = Bromous Acid; Br-= Bromide ion; ferroin and its oxidized form - ferriin.

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Interaction of Chemical Waves

Chemical Oscillation

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Chemical Oscillation

Spiral Wave

Spatial and Temporal Pattern

Pattern Formation

Existence = Patterned Formation in Time

Requires: Energy,Mass,Space

,Time = (Information)

The scientist does not study nature because it is useful; he studies it because he delights

in it, and he delights in it because it is beautiful. If

nature were not beautiful, it would not be worth knowing, and if nature were not worth

knowing, life would not be worth living.

Henri Poincaré