Conceptualizing Chaos: Continuous Flows versus Boolean Dynamics Mason Korb Thesis Advisor: Winfried Just Honors Tutorial College June 4, 2012 Contents 1 Motivations 2 2 History and Context 13 3 Some Cool Parts of Math 18 4 Independence Of the Axioms 22 5 Different Kinds of Dynamics 33 6 Chaos in Dynamical Systems 43 7 Connecting Different Dynamics 46 8 Very Few Answers and Many Questions 48 9 Appendix of Personal Contributions 50 1 Abstract In this undergraduate thesis, I focus on the properties of difficult prob- lems. I am not trying to solve these problems but focus on the notion of difficulty itself. What properties make a mathematical question hard? I explore some different concepts which capture a problem's difficulty. In par- ticular, I discuss independence, chaos, NP-Completeness, as well as some relations between these various guises of intractability. I also include some of my research work at Ohio University and show how it relates to these concepts. When relating this discussion of difficulty to applied mathemat- ics, it is crucial to discuss building multiple models of the same natural system, a process called \bi-simulation," and comparing their mathematical intractability. A difficult problem in one model may (or may not) become easier in another model. In this context, we discuss several types of models including Boolean Networks, ODEs and Markov Models. 1 Motivations 1.1 Changing our Framework Mathematics is about giving precise form to problems. When a problem is con- ceived in a rigorous form it has a certain clarity. We are now able to tinker with the problem in this form. We can see if any theorems can be proved about the questions we are asking. In this way, it is important to reconceptualize old prob- 2 lems; perhaps when a problem is cast in a new light certain features will stand out that did not before. Set theory problems can be conceived as graph theory problems and numerical questions are sometimes dynamical systems problems in disguise. The more ways problems are formulated, the more accessible they are from different areas of mathematics. Another way of gaining insight into a problem is reframing the mathematics itself. The reframing of a problem often sheds light on the problem itself and sometimes in the new context the problem becomes simpler and (if we are lucky) solvable. Of course, sometimes reframing a problem results in an equally difficult problem. One famous example of this is the notion of NP{completeness in theo- retical computer science. Mathematicians working in theoretical computer science have generated literally thousands of unsolvable and perfectly equivalent problems; if just one of them were solved we would have solved all of them and have an an- swer to the question of polynomial time reducibility over the class of NP{hard problems. Mathematics is to some extent the process of conceptualizing and reconceptu- alizing patterns. Applied mathematics focuses on patterns which exist in reality and that can be discerned by empirical science. Applied mathematicians build and study models of real world problems. These models rely on different math- ematical tools. Modeling acts as an interface between mathematics and reality. After isolating the elements of a problem, the model makers are free to focus on the mathematics. Each new mathematical model for a real world phenomenon (or 3 pattern) is a way of reconceiving reality. 1.2 Modeling with Ordinary Differential Equations and Boolean Networks A mathematical model is an application of mathematics to a real world situation. Each model specifies a set of variables which may change over time. For example, when Lorenz created his famous model of the weather he studied the interactions of the variables: temperature, air pressure, and humidity. A single state in the system is called a vector. And the collection of all the possible vectors is called the state space. Now, after we select a vector as an initial condition we would like to see how this vector changes over time. We may think of time as moving basically in two ways. In a discrete time model, time travels forward in steps. If we were only interested in the weather for one snapshot each day then we might think of time as: the first day, the second day, the third, etc. However, in a continuous time model, we imagine time moving continuously and the model makes sense when considering any moment. Let us summarize the components of a model. A dynamical system is a collec- tion of variables, their corresponding state space, a scheme for interpreting time, and a set of rules which allow us to move a vector through time. These rules for time traveling in our dynamical systems may not be explicit. In fact, it is often the case that we can relate variables in some way that does not yield a specific 4 formula. When we adopt the interpretation that time is traveling continuously, and ad- ditionally we have that the variables move continuously over the state space, we call the dynamical system a flow. For example, let us consider temperature. If at one moment we have the temperature in our model at 4◦ C and at an- other moment we have that it at 5◦ C and if the dynamic is a flow, then we are guaranteed a time when the temperature is 4:3◦ C. In fact, (although it may be hard/impossible to compute such a time) we are guaranteed that the system trav- els through every temperature between 4◦ C and 5◦ C. On the other hand, we may have a system with a discrete state space; where there are only countably many values the vectors may take. One of the most widely studied class of discrete (in both time and state space) dynamical systems are the so-called Boolean networks [33]. In these model the vectors in our state space are binary. That is, they can take on only two values. In a weather model, one node in the vector may be described as sunny or not sunny. Another node may record temperature greater than 0◦ C or not. Each node can take only one of these two values. We exclude the possibility of sunny but not too sunny days; we force ourselves to pick one. Let us move away from the weather example and think about population growth. It may not be immediately clear that an unchecked population will grow exponentially in time. C2t N(t) = C1e ; (1) 5 where N is the population, C1 and C2 are positive constant, and t is time. However, we might find it more intuitive to learn that an unchecked population's growth rate is based on the percentage of the population of reproductive capacity: _ N = C2N; (2) The equation above can be read as follows: The rate of change of a population is equal to a constant times the population. Equation (2) is an ordinary differential equation (or an ODE) and while it does not give us an explicit formula for how the population changes in time, it tells us something deeper about the underlying features of population growth. The general solution of the ODE (2) is given by equation (1). The fact that this ODE has an explicit solution is a rarity! Almost every ODE does not have an explicit solution and if we added more elements to our understanding of population growth, it is unlikely that an explicit formula would exist. In particular, if we added carrying capacity, disease dynamics, etc. to our model then we would not be able to solve for an explicit formula. When modeling a real world phenomenon we must select a paradigm in which to construct our model. Each methodology has its unique strengths and weaknesses. For example, when dealing with a small population we know that the state space is discrete: 14:5 people is not meaningful. On the other hand, in a large population we may not need to nitpick as 14:5 million people is reasonable. The same reasoning applies to discrete versus continuous time. When creating a model for a restaurant we do not have to consider anything deeper than the fiscal day. We can simply 6 think of each unit of time as a business day where we predict a certain amount expense and revenue. On the other hand, when we look at the growth of an organism every moment is potentially significant. 1.3 Chaos Imagine that we, like Lorenz did, have created a model for weather. What is next? We ask the meteorologist about the temperature, humidity and air pressure. She responds in the following way: \It is between 32:3◦ C and 32:4◦ C." We may get similar responses for the other variables. For each variable she responds with an interval rather than a specific value. We may label the state space with a range of vectors with coordinates in these intervals and select a hundred points in this interval. Next we see what our model does with these initial conditions. It looks like the weather is going to be good for the next hour. We see a cluster of points all traveling together. Next we examine what the weather will be in a week. The vectors have traveled all over the state space. Some points indicate that we might have a tornado, while some vectors indicate quiet showers. So we see the model has limited use. It may be useful for predicting the weather for today and tomor- row but it is virtually worthless when trying to predict the weather in a month. This situation is referred to as \chaos." Chaos is always marked by sensitivity to initial conditions.