Physics 213 Wednesday 8-27-03 Notes for roughly lectures 2-4. ANY QUESTIONS SO FAR? - Has everyone read the syllabus carefully? - Does everyone have a copy of Strogatz now? READING: - Various pages in Gleick's book on "Chaos", e.g., pages 69-80 which cover some history on discoveries made with the logistic map. - Chapter 10 of the Strogatz book "Nonlinear Dynamics and Chaos", especially Sections 10.0 through 10.4. - A more definitive discussion of the logistic-like maps and their relevance to biology is given in J. D. Murray's book "Mathematical Biology, Second Edition". - For those who can't sleep at night because they need to know what the carrying capacity K is for the human race, I would recommend your reading the recent book "How Many People Can the Earth Support?" by Joel E. Cohen (Norton, New York, 1995). To cut to the quick, the answer is not known nor even well estimated by the experts for reasons that are quite interesting. DEMO: - Will show a simple demo of a subcritical bifurcation, namely the flopping over of a curtain wire loop when made sufficiently large. The buckling of a yard stick would be an example of a supercritical bifurcation. Because subcritical bifurcations involve a sudden _finite_ change in the nontransient properties of a system, they are difficult to analyze mathematically; one can not understand the final state as some how being close to the starting state. Because supercritical bifurcations involve gradual _continuous_ changes in some property of the system as a parameter is varied, e.g., the extent of lateral buckling of a meter stick, one can often make substantial theoretical progress by using Taylor series or perturbative expansions in some small quantity. Thus the class of supercritical bifurcations are much better understood theoretically than the class of subcritical bifurcations. A SPECIFIC EXAMPLE OF A DYNAMICAL SYSTEM: THE LOGISTIC MAP - In the last lecture and in my notes for the last lecture, we discussed broadly and rather vaguely how various disciplines were interested in nonlinear dynamics. In today's lecture, I would like to go to the opposite extreme and discuss a specific example of a dynamical system, the logistic map. With this example, I can show you qualitatively many of the ideas that we will be concerned with for the rest of the semester: - modeling - numerical simulation - dynamics, - transients - attractors - linear stability - bifurcations - chaos. My goal here is to give you a quick glimpse of ideas and issues so that you can get a sense of what is coming later in the course and to give you a sense for what some of these words mean. We will then start discussing these ideas in greater detail and more precisely in following lectures. DEFINITION: NONLINEAR DYNAMICS - Before continuing and possibly losing you in details, we should at least try to define what this course will be talking about: what is the subject of nonlinear dynamics? I am afraid there is no simple answer since there are multiple levels of thinking about the subject. Here is one definition: Nonlinear dynamics is the study of systems that evolve in time. This definition is informal because I haven't defined yet what I mean by a "system" or by "evolution in time" or even what I mean by "study". We will discuss this in detail later on. For now, you should think of a dynamical system as being either of two things: - A MATHEMATICAL model that describes the evolution with respect to time of some quantity, e.g., the simple equation dx/dt = - x is a simple dynamical system that tells us how the quantity x(t) changes with time (it decays exponentially). - A PHYSICAL system that has observable properties that can evolve in time, e.g., the position theta(t) of a mechanical pendulum, the voltage V(t) of an electronic circuit, or the x-velocity Vx(t,x,y,z) at a particular point (x,y,z) in space at time t of a fluid flowing in a pipe. Physical systems are more tricky than mathematical systems because it is never completely clear what mathematical description is needed for a full understanding. Most systems of interest consist of so many particles that one can never give a complete description. Fortunately, people have found fairly practical ways to characterize many physical systems and we will see some examples later in the course. - CHALLENGE: From time to time, I will give you upper-case CHALLENGES in these notes that you should take the time to think about as important issues related to the lecture or high-probability questions that might appear in a future quiz. So here is a challenge: If nonlinear dynamics is the study of systems that evolve in time, what is meant by TIME? How does one define time in a practical or scientific way? Can we define time without referring in a circular way to the concept of some dynamical system? Note: There is a nice discussion of "time" in Section 5-2 in Volume I of the "Lectures in Physics" by Richard Feynman. BACK TO THE LOGISTIC MAP - Let's consider the question of modeling a population of insects that have the property that they lay eggs once a year and die (non-overlapping generations). Examples would be spiders, flies, and mosquitoes, but perhaps especially cicadas that appear in large numbers once every 19 years or so. The specific question is to understand how the number of insects of a given type change from year to year. For example, we might like to know if there will be a large increase in the number of mosquitoes next summer or whether there might be wild oscillations in successive years that could lead to extinction. - A first step in trying to invent a model is trying to identify appropriate variables that can describe the insect population, what we will call later the "state" of the insect population. Let's take a simple route and hope that, as a first guess, we can characterize the insect population in any given year by a single number, by how many insects there are. So let's define the variable: N[t] = number of insects in year t This is an non-negative integer-valued variable and is known at consecutive integer years, say year t=0, year t=1, year t=2, etc. In principle, you can even measure N[t], e.g, you could count all the spiders in a given sealed room of a house. The assumption that the evolution of the population of insects is determined just by a knowledge of N[t] is a MAJOR and dubious assumption. E.g., it may turn out that we need to know the health, age, sex, and location in space of insects to obtain a quantitative model. We may even have to know the genetic structure of each insect, e.g., what genes or alleles are available which is NOT easily measured even in the early 21st century. We may need to know details of the environment and what diseases or parasites or predators act on these insects. These complications are a good example of how it is difficult to model a system unless you know how to define its state and there is no simple recipe for this. - We will make another simplification: that the number of insects N[t] is so large in any given year that we can treat N[t] as a _continuous_ real variable. This assumption is incredibly useful since we can then possibly treat N[t] as a differentiable function and bring in the machinery of calculus. - So let's ask: if N[t] is the number of insects in the current year, what will be the number of insects in the next year, N[t+1]? If we think biologically, we can reason that each insect next year has to come from an egg laid by an insect this year and the number of eggs laid should be a simple function of the number of insects N[t]. So we can make the hypothesis that the number of insect N[t+1] in year t+1 depends only on the number of insects N[t] in the present year, by some function g(N) which is so far unknown: N[t+1] = g( N[t] ) . This is a substantial simplification and idealization but is a good beginning. MAPS AS DYNAMICAL SYSTEMS - A dynamical equation or evolution equation that involves discrete integer times is called a MAP. The equation N[t+1] = g(N[t]) is a simple example of a map in that it involves only one variable N. Dynamical equations that involve a continuous time variable such as ordinary differential equations or partial differential equations are called FLOWS. We will see later in the course that most flows can be reduced to maps so, in a technical sense, a mathematical theory of maps lies at the heart of all nonlinear dynamics. BIOLOGY OF THE FUNCTION G(N): - Questions for the class: - Can you deduce anything general about the shape of the function g(N) from biological common sense? - What must g(N) look like for small insect populations, N[t] small? - What must it look like for large insect populations, N[t] big? - What do we mean by "small" or "big", i.e., is there a characteristic or natural unit of size for the insect population?