Summary of Unit 3 Chaos and the Butterfly Effect Introduction to Dynamical Systems and Chaos http://www.complexityexplorer.org The Logistic Equation ● A very simple model of a population where there is some limit to growth ● f(x) = rx(1-x) ● r is a growth parameter ● x is measured as a fraction of the “annihilation” parameter. ● f(x) gives the population next year given x, the population this year. David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos Iterating the Logistic Equation ● We used an online program to iterate the logistic equation for different r values and make time series plots ● We found attracting periodic behavior of different periods and... David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos Aperiodic Orbits ● For r=4 (and other values), the orbit is aperiodic. It never repeats. ● Applying the same function over and over again does not result in periodic behavior. David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos Comparing Initial Conditions ● We used a different online program to compare time series for two different initial conditions. ● The bottom plot is the difference between the two time series in the top plot. David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos Sensitive Dependence on Initial Conditions ● When r=4.0, two orbits that start very close together eventually end up far apart ● This is known as sensitive dependence on initial conditions, or the butterfly effect. David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos Sensitive Dependence on Initial Conditions ● For any initial condition x there is another initial condition very near to it that eventually ends up far away ● To predict the behavior of a system with SDIC requires knowing the initial condition with impossible accuracy. ● Systems with SDIC are deterministic yet unpredictable in the long run. David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos Shadowing ● Due to roundoff and finite precision, the orbit a computer gives us is not the true orbit for that initial condition. ● However, the computed orbit is arbitrarily close to some other true orbit for the dynamical system. ● This result is known as the shadowing lemma. David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos Chaos A dynamical system is chaotic if: 1. It is deterministic 2. Its orbits are bounded 3. Its orbits are aperiodic 4. It has sensitive dependence on initial conditions David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos Symbolic Dynamics for the Logistic Equation ● Convert the orbit into a symbol sequence ● If x < 0.5, L. if x > 0.5, R ● This way of converting to an orbit to symbols is known as a generating partition: the initial condition uniquely determines the symbol sequence, and vice-versa. ● Properties of the symbolic dynamical system are the same as the properties of the original system David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos As Random as a Coin Toss... ● The sequences generated by the logistic equation with r=4 are as random as a coin toss ● For both the logistic equation and the coin toss, all possible sequences of symbols occur with equal frequency ● If I gave you two long sequences, one generated by the logistic equation, the other by randomly tossing a coin, you could not tell which is which. David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos Algorithmic Randomness ● Key idea: a random sequence is one that is incompressible. ● Problem: How can we tell if a sequence is compressible or not? ● There does not exist an algorithm for finding the best way to compress a sequence. ● However, we can argue that almost all sequences are incompressible, and hence random. David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos Almost all Sequences are Random ● The number of sequences and the number of algorithms are both infinite ● However, there are infinitely many more sequences than algorithms ● So there cannot be ways to compress the vast majority of sequences. ● Hence, a randomly chosen infinite sequence is, with probability one, incompressible, and thus random. ● The logistic equation with r=4 produces all possible sequences. ● Thus, the logistic equation is producing random sequences. David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos But.... ● Any sequence can be generated by the logistic equation ● So doesn't this mean that the logistic equation serves as a compressed version of the sequence? ● The catch is that in order to generate a particular sequence, we would need to know the initial condition infinitely precisely. ● Almost all numbers between 0 and 1 are irrational and random; they are incompressible ● So in order to reproduce the sequence we need the logistic equation and an incompressible initial condition. ● And so we haven't actually compressed the sequence at all: we've traded an infinitely long incompressible sequence for an infinitely long, incompressible initial condition. David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos So the Logistic Equation Produces Random Sequences ● The logistic equation is a deterministic dynamical system that produces randomness. ● A dynamical system that is not deterministic--- that has an element of chance---is called stochastic. ● The qualities of a result can be different from the qualities of the process that made it. ● Here, a deterministic process produces a random result. David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos.