OBC | Complexity science and the role of mathematical modeling
Tassos BountisDepartment 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 ofnonlinear dynamical systems like the weather, the electro-cardiogram and encephalogram, mechanical, chemical and electricaloscillations, seismic activity and even stock market fluctuations.• The Geometry of Fractals analyzes the complex spatial structureof trees and rocks, the dendritic shape of the bronchial “tree” inthe lungs, the cardiac muscle network and the blood circulatorysystem.• Most importantly, we can construct appropriate mathematicalmodels that: (a) reproduce the main features of a complex systemand (b) provide invaluable insight in revealing some of itsfundamental 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 itsbranches bifurcate out of abigger branch so that they aresmaller by a scaling factor?Observe that besides shorteningthe branches at every bifurcation,we also apply a transformation ofrotation, e.g. by 45ο…Why not then take advantageof this observation to constructa simple mathematical modelthat would describe this typeof complexity?
Could we build realistic models of trees and plants, if wefollow a self-similar construction of patterns at smaller andsmaller scales? 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….
If we take aninitial shape andcontract it into 3smaller onesapplying a rotationto two of itsparts by 90ο (oneto the right theother to theleft)……we obtain inthe end a shapethat looks like achristmas tree(see figure onthe 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 otherplants can wedesign?
Let us use, for example, 4such transformations, toconstruct the leaf of a fernplant, with infinitely manysmaller leaves on it:Start with rectangle 1 forthe main leaf, 2 and 3 forits two neighbors and 4 forits very thin stem,….Now observe what happensafter many iterations of thisprocess…
Isn’t it fascinating?We can now start toimitate MotherNature by drawingpictures of real –looking plants andbushes, likeBarnsley’s fernshown here……All these objects arecalled fractals and obeya new kind of Geometry,called Fractal Geometry!
Fractals and Chaos: From Geometry to Dynamics!Chaos is complexity in time, or, in otherwords, the extremely sensitive dependenceof the motion on its initial conditions!The first one who studied it was the FrenchMathematician Henri Poincaré (1854 –1912) shown here on the right.In fact, Chaos can emerge out of a “fractal tree” of successivebifurcations as a parameter r increases in a simple model ofpopulation of rabbits living on an island!
As the growth parameter r of the rabbits increases....Xn Here is where chaos first appears in the population.... r
The concept of a bifurcationis a lot more general innature. If you introducecockroaches in a dish withtwo identical shelters, theywill first visit each shelter inequal percentages, buteventually, as the shelters’capacity grows, they will allend up visiting only one of theshelters!Note that this “collectivechange of behavior” occurs,without any apparentcommunication between thecockroaches!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...
Work with C. Antonopoulos, V.Basios and A. Garcia-Cantu Ros How can we model(Chaos, Solitons & Fractals, 2011,Vol. 44, 8, 574-586) this phenomenon? 1. We first provide the free particles with an inner steering mechanism: +/- ∆0
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 0 φti 1
We find the following patterns of motion:(a) Chaotic flight, (b) synchronized rotation or (c) “flocking”,depending on whether φit 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 ofcongestion” or “clustering”of cars we commonlyobserve on the highways,moving backward in time:Martin Treiber, Ansgar Hennecke, and Dirk Helbing, “Congested Traﬃc States in Empirical Observationsand Microscopic Simulations”, Phys. Rev. E 62, 1805–1824 (2000)
Recent work of our group in Patras with Prof. Ko van derWeele connects Granular Transport and…… Traffic Flow !Q in Q outThe dynamics of the grains involves a certain Fluxfunction F(nk), which must be specified in advance!
As a model we used the Eggers flux function: 2 BR , L nk FR, L (nk ) 2 Ank e ...which follows the reasonable argument that for few particles in the k- Here box the flux increases but BR = 0.1 beyond a certain maximum the flux will have to decrease! i.e., hL = 2hR BL = 0.2 J. Eggers, PRL 83 (1999); KvdW, G. Kanellopoulos, Ch. Tsiavos, D. van der Meer, PRE 80 (2009)
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
Increasing the inflow rate Q, a“backward” wave develops... outflow = inflow Q = 1.80
… leading to a critical value: Qcrit = 1.8740 outflow vanishes….where clustering occurs at the top of the staircase!
Traffic flow: Unidirectional version of the staircase problem [veh/ h per lane]Δx = 500 m with ρk(t) = car density in cell k [veh/km per lane] A similar equation is obeyed here as with granular transport: d k x F ( k 1 ) F ( k ) Qk (t ) dt time step dt = 12 s (= Δx/vmax) in- and outflow (only in certain cells k) Now the Flux function F(ρk) is measured by induction loops at periodic locations in the asphalt of the highway!
Measurements on the A58 in the Netherlands:(b) 3000 Traffic flow (veh/h/lane) 2500 2000 1500 Provide evidence for a flow function of the 1000 form… 500 0 0 10 20 30 40 50 60 70 80 90 100 Car density (veh/km/lane)
Observe the waves of congestion traveling backward! The frontlane is the slow on (90 km/hr) and the back the fast one (100 km)
Finally, about Biology: How can we model diseases like ischemia or cardiac infarction of the heart? Work of Dr. Adi Cimponeriu, T. Bezerianos, F. Starmer and T. Bountis at the Department of Medicine of the University of Patras
We can model electrical pulse propagation through ion channelsby a one-dimensional array of electrical oscillators..... ....obeying the well- known Kirchoff laws:
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.
Of course, Hamlet may well advise us here: «There are many more things on earth and heaven, Horatio, than are dreamt in your philosophy....» 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…..
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).