Transcript 2 - K.N.Toosi University of Technology

THE PARADIGM OF COMPLEX SYSTEMS
• M.G.Mahjani
• K.N.Toosi University of Technology
• [email protected]
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
transformation
A set of rules, a mapping or
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 inputoutput 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
complexity
Complex System
Uncontrolled
Unpredictability
Soft Computing
Fuzzy sets
Nonlinear
Neural network
genetic alogorithm Genetic Programing
Non-Equilibrium
non-standard
instabilaty
vast new horizons have been opened up for
our imaginations requiring new
conceptualizations and innovative research
Non-Uniform
complex system
Emergence
attractor
self-Modification self-Reproduction self-Organization
Chaos
Coevolution
Strange Attractor
Fractal
Type of Complexity
Complex System
Static Complexity
Fixed structures, frozen in time
Dynamic Complexity
Systems with time regularities
Evolving Complexity
Open ended mutation, innovation
Self-Organising Complexity
Self-maintaining systems, aware
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
Organic
Construction
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 Attractor
1 -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 selfsimilar 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
4.66920160910299067
185320382047
Logistic equation
Bifurcation diagram
Bifurcation
•
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 2cycle attractors. Near s = 3.45,
the 2-cycle bifurcates into a 4cycle, 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 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.What
Will 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 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.
Benard
Convection Cell
 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 .
Long Range
Correlation
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.
Chemical Oscillation
Interaction of Chemical Waves
7/20/2015
53
Chemical Oscillation
Spiral Wave
7/20/2015
54
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