SiET
EXAMPLES OF COMPLEX PHENOMENA
(VIRTUAL LAB)
IET / Mcrit / GRSC (UPC) November 2001
Examples and Models:
Created by Rodin Enchev
The logistic equation is one of the simplest examples to observe the deterministic chaos phenomenon. It consists on a discrete iterative equation:
Lorenz's attractor
Rossler's attractor
The Lorenz's attractor is a weird attractor able to describe the chaotic behavior of the simplified Lorenz's Equations that describe the weather. It's representation has a characteristic butterfly shape and it is frequently used as an example because it was the first attractor discovered.
Created by Wolfgang Christian and Shubha Tewari
Ising's Model
The Ising's model is used to study the magnetic materials formed from small magnetic spins. It is based on a cellular automata with element values either 1 or -1. The spins can change orientation due to their neighbors or to the temperature. For a critical temperature a phase transition happens. Then, the system is neither ordered (all the spins in the same direction) nor random (all the spins spin about because of the temperature).
Created by Noah Goodman and
modified by
Adrian Vajiac
The Sierpinsky's triangle is commonly used as a fractal example. It is a regular fractal based on a triangle. A first black equilateral triangle can be split into four equilateral triangles by joining the midpoints from the three sides of the triangle. The central triangles is taken away. Considering the new three triangles as the first one, the process is repeated to became the Sierpinsky's triangle.
Created by Sergei S. Maslov
The sand pile is one of the best examples of a critical self-organized system. If we imagine someone dropping regularly grains of sand in the same point, there will become a sandpile. This is a critical self-organized point because if the slope is step the system tends to have small avalanches, and if it is flat, the sand accumulates. The system self-controls in a constant critical slope.
Earthquakes have been recently described as critical self-organized phenomena. The mode that features them is simple, but captures it's essential components. It is a squared grill in which every cell is modeled as a small land portion, submitted to a certain stress. The cells accumulate this stress up to the point they reach the limit and set all the tension free to the neighbor land portions. When the neighbor portions receive this tension, they can accumulate it or set them back free, and a chain can be created to develop a small earthquake. The frequency of the earthquake magnitude follows a Zipf law: the Gutenberg-Richter law.
Created by Joeh Goldbay
The ants looking for food (foraging)
When ants look for food, specially some tropical
jungle species, they use pheromones to guide themselves. The pheromone
that the ants can detect and secrete, is used to share information
about paths that head towards food that they have found. This paths
are self-organized by the constant deposition of pheromone and the
simple rules that an ant has are:
0. Random search.
1. If you find food, take it to the hole and secrete pheromone.
2. If you find a pheromone trace, follow it and when you find food,
follow step 1.
Zipf's Laws
Zipf's laws are bundance distributions of certain phenomena. For instance, it is known that an earthquake with magnitude 2 is 10 times stronger than one with magnitude 1. And that a magnitude 3 one is 10 times stronger than a 2. This law is known as the Gutenberb-Richter law and it is used to make the Richter scale for describing earthquake magnitudes. This potential law appears in many biological and human systems and usually consist on critical point systems.
Created by Wayne Davis
The prisoners-dilema: "Tit for Tat"
The prisoner's dilema is a game on which the user and Lucifer (opponent) are accused of committing a crime. It is proposed to decide whether to cooperate with the opponent or not to cooperate. The result (penalty reduction) depends on what both players have decided, and they don't know what the other is doing. Iterative games let the user observe the best cooperation strategies.
Created by Joachim Köppen
Poincaré's three-body problem
After Newton solved the planet orbit around the sun,
the natural challenge was to solve it considering 2 planets. The
best mathematicians and physicians worked in that problem last century.
Viewing the tracetrories is easy, but when they are quasi-periodical or
chaotic it is easieer to use the Poincaré's sections. That is
when we have phase space hyper plane, we draw one point in any part of
the trajectory that crosses the plane. That means we have a graphical
representation of a plane itself (the representation of the first return
of Pointcaré). It is very attractive and by changing the representation
tools, we can understand dynamic systems.
http://www.physics.cornell.edu/sethna/teaching/sss/jupiter/Web/Rest3Bdy.htm
Created by John Conway
Conway's game of life
The game of life is a two-dimensional cellular automata created by John Conway that simulates the population growth and death. It is a totallist model, which means that it's rule doesn't depend on the precise display of the neighbors but on the number of neighbors in a certain state. The rule is simply that a life cell dies if it has not 2 or 3 neighbors and a dead cell is only born if there are 3 cells alive in the surroundings. This automata is very special in many aspects and constant research is based on it.
Axelrod's Cultural Dissemination
The complexity theory is a new paradigm to understand the dynamic processes where multiple attractors interact. A first methodology is based on agent based modeling. The agent based modeling has to do with individual agents (for instance persons, nations or organizations) that interact with the others and with the environment. The simulation is used to discover the emergent features of the model and the clues to understand the dynamic process that can be difficult to model with the standard mathematical techniques.
Created by A.K.Dewdney
Lotka-Volterra's predator-prey model
WATOR is a simulation of the interaction of a predator and a prey in the time in a finite rectangular area. It is a simple program, described in the publication Scientific American by A.K. Dewdney. The game allows to find parameters to stabilize populations when the area is very small.
Lanchester's combat model
A simple and very effective model of combat was developed by Lanchester in 1916. The significance of the model was that it helped understand the importance of the concentration of troops in battle.
Essentially the winner of the conflict is left with a number determined by the difference of the squares of the two concentrations. This is known as Lanchester's square law of combat.
Mcrit has developed a model for housing and activity attraction with the Polya Process. Given a series of alternatives for location (housing systems SH1 to SH22, p=22) with a determinate number of enterprises initially located in each of them, the non-linear stochastic model of Polya calculates the probability of each housing system to attract the next enterprise (activity A) depending on the number of activities running there in that moment. According to this activity, the model locates an activity for each housing system in every iteration (t=1,....,50). You can download the interactive model.
Model d'epidèmies de l'Èbola. Creat per The Shodor Education
Foundation (1998)
The epidemics' models follow some clear trends. The phenomenon of an epidemics consist on the transitory presence of an infectious disease in one region or zone affecting a great amount of people. So, the models must describe somehow the relationships between the individuals, the time and the space to apply possible infection conditions on them depending on the neighbor infected individuals.