Chapter 3. Chaos Theory
Butterfly Theory
Logistic Map:
What is it?
Computing the map?
Bifurcation:
Feigenbaum constant
Dynamic Systems
Experiment: Quadratic maximum
• The butterfly effect
o History
o Examples
o Double pendulum
• Nonlinear dynamic systems
• Logistic map
o What it is
o
• Bifurcation
• Experiment: Quadratic maximum + results
Chaos theory
When the coin is thrown with the same force and in the same direction, the coin will not end up on the same side every throw. The result is unpredictable in the same initial conditions. This effect is called chaotic behavior.
Chaos theory applies to systems in which each of two properties hold:
1. The systems are dynamics, meaning that the behavior of the systems at one point in time influences its behavior in the future.
2. And they are nonlinear, meaning they abide by exponential rather than additive relationships.
Chaos just means that certain types of systems are very hard to predict.
Exponential operations extract a lot more punishment when there are inaccuracies in the data than a linear operation. If the process is dynamic, meaning that our outputs at one stage of the process become our inputs in the next. The mistake keeps getting larger and larger.
Moreover, we can only observe our surroundings with a certain degree of precision. When measurements are off in even the third or the fourth decimal place, this can have a profound impact on the forecast.
Edward Lorenz, the father of chaos theory, described chaos as “when the present determines the future, but the approximate present does not approximately determine the future.”
The Butterfly Effect
The Butterfly Effect refers to the idea that small causes may have large effects. When a butterfly flaps its wings might create a tiny difference in the atmosphere that can cause ultimately a tornado. “the flap of a butterfly’s wings in Brazil can set off a tornado in Texas” 1972 by MIT’s Edward Lorenz. There are so many variables of which the weather depends so that a very small change such as the flapping of the wings of a butterfly can cause a big change. one set of conditions leads to a tornado while the other set of conditions doesn't. The Butterfly Effect shows that it is very difficult to make accurate predictions. If you use Initial Conditions to make a prediction you can never make a complete accurate prediction, because tiny differences can cause big consequences. This problem motivated the development of ensemble forecasting in which a number of forecasts are made at different initial conditions.
Non-linear dynamics systems
A Nonlinear dynamic system is a system in which the change of the output doesn’t coincide with the changes of the input, but rather changes exponentially. It’s a system in which a function describes the time dependency of a certain point in a geometric space and most nonlinear systems are chaotic.
Double pendulum
A system to examine a non-linear dynamic system, is the double pendulum. The Double Pendulum is a simple physical system that exhibits dynamic behavior and has a strong sensitivity to initial conditions. The movement of a double pendulum is chaotic and is determined by a series of differential equations. You can’t predict how the double pendulum moves. In the following analysis the double pendulum is restricted to two dimensions. A system is chaotic if it has a high sensitivity to small changes in the initial conditions. Suppose two double slings are hung next to each other. Both slings are held up at a certain angle. With pendulum 1, however, that angle is 0.0001 ° greater than with pendulum 2. The pendulums are then released and because the difference in the initial angle is very small, the pendulum will move in the same way for a few seconds. After that, the behavior will start to differ, and the orbits of both streams will no longer look alike. The small difference in the initial conditions results in the behavior becoming unpredictable after a while. If the experiment were to be repeated again, then both streamers will follow other courses over time than during the first experiment, because the initial conditions can not be set with infinite accuracy. A single pendulum is not chaotic a double pendulum because on the pivot point between the two parts of the pendulum can thus change the job drastically, because it is not predictable which side the pendulum will fall on. The existence of these kinds of points is a condition for the emergence of chaos.
Figure 1 shows the phase diagrams of two measurements on the double pendulum. The red pendulum started at a slightly different starting angle than the blue pendulum. The difference in starting angles is only 0.0001 °, but it is already clear that after 6 seconds the strings follow other tracks. This is chaotic behavior. There is a great sensitivity to initial conditions, which makes the behavior unpredictable over time.
Lyapunov coefficient
To indicate the degree of chaos, the Lyapunov coefficient λ can be used. This parameter indicates how fast paths in the phase space are separated as a function of time. To be able to say how far a path is from another path, it is necessary to look at the distance between two paths in the phase space.
Now we can look how the distance between two paths develops over a certain time. It turns out that if two double pendulums start with roughly the same initial conditions, then the difference between their paths in the phase space (Δ) increases exponentially. How strong that exponential growth is is given by the following formula:
Δ (0) is the difference between the paths of the two double pendulums. The parameter λ is the Lyapunov coefficient. The larger λ, the faster the two paths will move away from each other. On the basis of the Lyapunov coefficient, the degree of chaos can be read. When the Lyapunov coefficient is negative, there is no chaos. If the Lyapunov coefficient is positive, then the system is chaotic. If you want to measure the Lyapunov coefficient, you have to perform a lot of measurements because the coefficient is different for each measurement. An average of all measurements is taken to determine the Lyapunov coefficient.
Logistic map
The logistic map is a polynomial function with a maximum exponent of two. In a logistic map chaotic behavior can arise from very simple non-linear dynamic equations.
The Logistic map was promoted in a 1976 paper by biologist Robert May. The logistic map is written as:
Formule
A rough description of chaos is that chaotic systems exhibit a great sensitivity to initial conditions. A property of the logistic map for most values of r between about 3.57 and 4. A common source of such sensitivity to initial conditions is that the map represents a repeated folding and stretching of the space on which it is defined. In the case of the logistic map. By varying the parameter r:
With r increasing beyond 3.54409, from almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, etc. The lengths of the parameter intervals that yield oscillations of a given length decrease rapidly; the ratio between the lengths of two successive bifurcation intervals approaches the Feigenbaum constant δ ≈ 4.66920.
At r ≈ 3.56995 is the onset of chaos, at the end of the period-doubling cascade. From almost all initial conditions, we no longer see oscillations of finite period. Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos.
a small error in the supposed initial state of the system will tend to correspond to a large error later in its evolution.
Bifurcation diagram for the logistic map:
Bifurcation theory and the Feigenbaum constant
A bifurcation occurs when a small change made to the parameter values of a system causes a sudden change in its behavior.
In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the mathematician Mitchell Feigenbaum.
Formule
The first constant is the limiting ratio of each bifurcation interval to the next between every period doubling of a one-parameter map. δ = 4.669201609102990671853203821578…
Feigenbaum constant δ expresses the limit of the ratio of distances between consecutive bifurcation diagram on Li / Li + 1
The second constant is the ratio between the width of a tine and the width of one of its two subtines (except the tine closest to the fold). α = 2.502907875095892822283902873218…
Overfitting
Overfitting is the name given to the act of mistaking noise for a signal. Overfitting leads to worse predictions. The name overfitting comes from the way that statistical models are “fit” to match past observations. The fit can be too loose—this is called underfitting—in which case you will not be capturing as much of the signal as you could. Or it can be too tight—an overfit model—which means that you’re fitting the noise in the data rather than discovering its underlying structure.
You are most likely to overfit a model when the data is limited and noisy and when your understanding of the fundamental relationships is poor; both circumstances apply in earthquake forecasting.
Our tendenct to mistake noise for signal can occasionally produce some dire real-world consequences.
Japan being extremely seismically active, was largely unprepared for its devastating 2011 earthquake. A magnitude 9.1 earthquake is an incredibly rare event in any part of the world: nobody should have been predicting it to the exact decate, let alone the exact date. In Japan, however, some scientist and central planners dismissed the possibility out of hand. This may reflect a case of overfitting.
The relationship almost follows the straight-line pattern that Gutenberg and Richters method predicts. However, at about magnitude 7,5 there is a kink in the graph. There had been no earthquakes as large as a magnitude 8.0 in the region since 1964, and so the curve seems to bend down accordingly. If you go strictly by the Gutenberg-Richter law, ignoring the kink in the graph, you should still follow the straight-line, as in the figure above. Alternatively, you could go by what seismologists call a characteristic fit, as in the figure beneath. Which just means that it is descriptive of the historical frequencies of the earthquake area. The characteristic fit suggests that a earthquake of a magnitude 9 was nearly impossible, it implies that one might occur about every 13000 years. The Gutenberg-Richter estimate, on the other hand, was that you’d get one such earthquake every three hundred years. The characteristic fit could imply an overfit model. In this case, an overfit model would dramatically underestimate the likelihood of a catastrophic earthquake in the area.
The characteristic fit (overfitted line) and the Gutenberg-Richter law fit (accurate line).