Tassos Bountis

Department of Mathematics and Center for Research and Applications of Nonlinear Systems

http://www.math.upatras.gr/~crans

University of Patras, Patras GREECE

Lecture at the OUT OF THE BOX Conference

Maribor, Slovenia, May 15-17,2012

What is Complexity?

At the beginning of 21st century we have understood that:

• Complexity, is a property of large systems, consisting of a huge number of units, involving nonlinearly interacting agents, which can exhibit incredibly complex behavior.

• New structures can emerge out of non-equilibrium and order can be born out of chaos, following a process called self-organization. Complex systems in the Natural, Life and Social Sciences produce new shapes, patterns and forms that cannot be understood by studying only their individual parts.

Mathematics has already been quite helpful:

• The Theory of Chaos explores the unpredictable time evolution of nonlinear dynamical systems like the weather, the electro-cardiogram and encephalogram, mechanical, chemical and electrical oscillations, seismic activity and even stock market fluctuations.

• The Geometry of Fractals analyzes the complex spatial structure of trees and rocks, the dendritic shape of the bronchial “tree” in the lungs, the cardiac muscle network and the blood circulatory system.

• Most importantly, we can construct appropriate mathematical models that: (a) reproduce the main features of a complex system and (b) provide invaluable insight in revealing some of its fundamental properties.

Some of the main questions we face today in what is called Complexity Science are:

• How do we use Mathematics to observe, measure and understand complex phenomena in the Natural, Life and Social Sciences?

• Should we only look for universal principles and laws expressed by mathematical formulas to understand atoms, molecules, cells, trees, forests, living organisms and ultimately society?

• How can use our perception and intuition to try to construct suitable mathematical models that will help us shed some light on the remarkably complex phenomena we observe around us?

What is a tree?

• Is it what an artist would perceive?

like Mondrian (1872-1944) or van Gogh( 1853-1890)?

......or what a biologist would study?

What is it that impresses us first about a tree?

Could it be a kind of self-similarity in the way two of its branches bifurcate out of a bigger branch so that they are smaller by a scaling factor?

Why not then take advantage of this observation to construct a simple mathematical model that would describe this type of complexity?

Observe that besides shortening the branches at every bifurcation, we also apply a transformation of rotation, e.g. by 45ο…

One answer is revealed by the theory of Iterated Function Systems, introduced by the American Mathematician Michael Barnsley, in the 1980’s..

Barnsley proved that a sequence of contracting transformations applied to an original shape has always the same limit no matter what the initial shape is.

In other words, what matters is the contracting transformations and not the shapes we start with….

Could we build realistic models of trees and plants, if we follow a self-similar construction of patterns at smaller and smaller scales?

If we take an initial shape and contract it into 3 smaller ones applying a rotation to two of its parts by 90ο (one to the right the other to the left)…

…we obtain in the end a shape that looks like a christmas tree (see figure on the left)…

…if together with rotation we also shift the top piece to the left, we may design ivy-looking plants climbing the walls of our house…

What other plants can we design?

Let us use, for example, 4

such transformations, to

construct the leaf of a fern

plant, with infinitely many

smaller leaves on it:

Start with rectangle 1 for

the main leaf, 2 and 3 for

its two neighbors and 4 for

its very thin stem,….

Now observe what happens

after many iterations of this

process…

Isn’t it fascinating?

We can now start to imitate Mother Nature by drawing pictures of real – looking plants and bushes, like Barnsley’s fern shown here……

All these objects are called fractals and obey a new kind of Geometry, called Fractal Geometry!

Fractals and Chaos:

From Geometry to Dynamics!

Chaos is complexity in time, or, in other

words, the extremely sensitive dependence

of the motion on its initial conditions!

The first one who studied it was the French

Mathematician Henri Poincaré (1854 –

1912) shown here on the right.

In fact, Chaos can emerge out of a “fractal tree” of successive bifurcations as a parameter r increases in a simple model of population of rabbits living on an island!

Here is where chaos first appears in the population....

As the growth parameter r of the rabbits increases....

r

Xn

The concept of a bifurcation

is a lot more general in

nature. If you introduce

cockroaches in a dish with

two identical shelters, they

will first visit each shelter in

equal percentages, but

eventually, as the shelters’

capacity grows, they will all

end up visiting only one of the

shelters!

Note that this “collective change of behavior” occurs, without any apparent communication between the cockroaches!

J.-M. Amé, J. Halloy, C. Rivault, C. Detrain, and J.-L. Deneubourg, PNAS 103 (2006) 5835.

COLLECTIVE BEHAVIOR OF BIRDS, FISH,

TRAFFIC AND PEOPLE?

Out of chaos, patterns emerge due to self - organization...

1. We first provide the free particles with an inner steering mechanism:

+/- ∆0

Work with C. Antonopoulos, V.

Basios and A. Garcia-Cantu Ros

(Chaos, Solitons & Fractals, 2011,

Vol. 44, 8, 574-586)

How can we model this phenomenon?

2. Next, we include interactions with nearby flock mates, so that two particles interact (avoiding collisions)

3. Finally, we introduce a time-dependent coupling parameter φti from..

Periodic domain

Weakly chaotic domain

Strongly chaotic domain

φti 0 1

We find the following patterns of motion: (a) Chaotic flight, (b) synchronized rotation or (c) “flocking”, depending on whether φi

t belongs to: (a) The strongly chaotic, (b) periodic or (c) weakly chaotic regimes.

with random initial conditions and FREE boundary conditions

100 birds starting in the chaotic region, as time passes, gather near the domain of weakly chaotic motion

Birds starting with parameters only in the chaotic region tend towards the flocking (weak chaos) region!

Do pedestrians behave as individuals or social beings? Observe how lanes of uniform walking direction

emerge due to self-organization.

Taken from: Dirk Helbing, Chair of Sociology, in particular of Modeling and Simulation, ETH Zürich www.soms.ethz.ch

Helbing’s Intelligent Driver Model (IDM)

....produces the “waves of congestion” or “clustering” of cars we commonly observe on the highways, moving backward in time:

Martin Treiber, Ansgar Hennecke, and Dirk Helbing, “Congested Traffic States in Empirical Observations

and Microscopic Simulations”, Phys. Rev. E 62, 1805–1824 (2000)

The dynamics of the grains involves a certain Flux function F(nk), which must be specified in advance!

Recent work of our group in Patras with Prof. Ko van der Weele connects Granular Transport and…… Traffic Flow !

Q in

Q out

i.e., hL = 2hR

As a model we used the Eggers flux function: 2

,2, ( )

R L kB n

R L k kF n An e

Here BR = 0.1

BL = 0.2

J. Eggers, PRL 83 (1999); KvdW, G. Kanellopoulos, Ch. Tsiavos, D. van der Meer, PRE 80 (2009)

...which follows the reasonable argument that for few particles in the k-box the flux increases but beyond a certain maximum the flux will have to decrease!

Watch how the grain density along a 25-step staircase becomes unstable as Q grows!

Q = 1.00 (relatively small)

Stable dynamic equilibrium: outflow = inflow

Q = 1.80

Increasing the inflow rate Q, a “backward” wave develops...

outflow = inflow

….where clustering occurs at the top of the staircase!

… leading to a critical value: Qcrit = 1.8740

outflow vanishes

Traffic flow: Unidirectional version of the staircase problem

500 m

Now the Flux function F(ρk) is measured by induction loops at periodic locations in the asphalt of the highway!

1( ) ( ) ( )kk k k

dx F F Q t

dt

[veh/ h per lane]

Δx =

with ρk(t) = car density in cell k [veh/km per lane]

in- and outflow (only in certain cells k)

time step dt = 12 s (= Δx/vmax)

A similar equation is obeyed here as with granular transport:

Measurements on the A58 in the Netherlands:

0 10 20 30 40 50 60 70 80 90 1000

20

40

60

80

100

120

140

Ve

locit

y (k

m/h

)

Car density (veh/km/lane)

(a)

0 10 20 30 40 50 60 70 80 90 1000

500

1000

1500

2000

2500

3000

Car density (veh/km/lane)

Tra

ffic

flo

w (

veh

/h/la

ne

)

(b)

Provide evidence for a flow function of the form…

Observe the waves of congestion traveling backward! The front lane is the slow on (90 km/hr) and the back the fast one (100 km)

Work of Dr. Adi Cimponeriu, T. Bezerianos, F. Starmer and T. Bountis at

the Department of Medicine of the University of Patras

Finally, about Biology: How can we model diseases like ischemia or cardiac infarction of the heart?

We can model electrical pulse propagation through ion channels by a one-dimensional array of electrical oscillators.....

....obeying the well-known Kirchoff laws:

Normal (healthy) behavior

Non-normal (ischemic) propagation

The action potential “breaks” at the necrotic region and may develop spiral waves that lead to arrythmia.....

In conclusion:

Complexity Science: Offers a unified methodology to study complex

physical, biological and social system.

Familiarizes us with Mathematics, the common language of all sciences, through the use of models.

Proposes new concepts, principles and techniques to better understand and perhaps predict and control complex phenomena.

Makes young people enjoy science, because it excites their curiosity and imagination and make them appreciate the interdisciplinary connections between different scientific fields.

Still, Complexity Science through the use of mathematical modeling opened a new “window” of communication with nature,

through which we have begun to glimpse the “global picture” of ourselves and the world

that surrounds us…..

«There are many more things on earth and heaven, Horatio, than are dreamt in your

philosophy....»

Of course, Hamlet may well advise us here:

References

• G. Nicolis & I. Prigogine, “Exploring Complexity” Freeman, New York (1989)

• T. Bountis, “The Wonderful World of Fractals” (in Greek), Leader Books, Athens (2004).

• G. Nicolis and C. Nicolis, “Foundations of Complex Systems”, World Scientific, Singapore, 2007

• C. Tsallis, “Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World”, Springer, New York (2009).

• T. Bountis and H. Skokos, “Complex Hamiltonian Dynamics”, Synergetic Series, Springer (April, 2012).