Physics 213 Wednesday 8-25-03 Notes for roughly the first lecture. COMMENTS ABOUT THESE NOTES: - These are informal notes whose purpose is to help me outline my thoughts in preparation for several lectures, and to give you a useful summary of what I covered, or, more realistically, meant to cover in my lectures. A WARNING: Based on feedback from students and from the process of trying to explain something, I often understand better what I meant to say after I give a lecture and so I will go back and try to improve the notes. For this reason, you will probably want to print out a final copy of these notes a few days after the lecture corresponding to the notes. - Note: When I say "physics" in my lectures or in these notes, I really mean inclusively the sciences in general, i.e., the testing of hypotheses by experiment. WELCOME - Welcome to Physics 213, Duke's introductory course on nonlinear dynamics. - Will begin lecture by introducing ourselves, then by discussing the syllabus and the requirements of the course. Will then finish lecture by discussing a few demos, to give you some intuition about dynamics, bifucations, and chaos. PREREQUISITES - Sophomore level background in mathematics: you need to know multivariate calculus and linear algebra. An exposure to the mathematics of ordinary differential equations will also be helpful but this course will actually show you how to think and to appreciate odes in much better ways than most introductory ode courses. - At least one year of physics at the high school or higher level. - A modest exposure to programming, it will be helpful if you know about variables, loops, arrays, functions, and parameters. If you have not programmed before, there will be others in class that you can collaborate with on the computer-oriented parts of the homework. - Because of the intedisciplinary nature of this course, drawing on science (physics), mathematics, and computer science, you will get more out of this course the stronger your background in these various areas. So course will be best taken as a senior if you are an undergraduate and probably best taken as a second-year graduate student. The math is fairly elementary at a technical level but the concepts are not. (By analogy, Euclidean geometry is not "hard" in the sense of requiring anything more than connecting points and drawing circles, but great imagination and creativity is needed to discover or prove theorems.) - Please talk to me if you have any doubts that your background may not be sufficient for this course. PLEASE ASK QUESTIONS IN CLASS AND OUT OF CLASS! It is unfortunately the nature of technical courses that they end up being a lecture format for the efficient transfer of information. Many studies have shown that while information is conveyed efficiently, insight is not and so it is critical that you think ACTIVELY about the material and ask questions when something puzzles you or bothers you. If you really want to get a lot out of this course, please ask questions! WHY YOU SHOULD TAKE THIS COURSE: CURIOSITY - Why is the world as rich and interesting as it is? How do we quantify and understand its complex structure? How does this structure change as you vary "knobs"? - Why is the weather so complex? Will it ever repeat itself? Why don't clouds form lattices like crystals? - What determines the size of raindrops? Why are you never hit by a raindrop the size of a basketball (which would probably finish you off)? - Why does a fluid become turbulent? How is transport enhanced by the turbulence? How can one understand the irregular space-time behavior of a turbulent fluid? - Did life begin spontaneously? How? Is it likely? As you may know, scientists will likely have the technology to detect oxygen in planets from remote stars within the next 20 years and so the possibility of detecting life (oxygen-producing organisms) is now possible within our lifetime. - Biological diversity: why are there as many species as we observe? How does the number depend on the size of the environment and other conditions? - How does the brain work? How did it evolve? Why are there many kinds of oscillations and synchronous dynamics? - What does "random" mean and are observed complex dynamics random? - Are economic and ecological complex the same way that a turbulent fluid or chaotic pendulum is complex? If so, can we exploit the latter to understand the former? Experience suggests that these questions are best asked and investigated from several points of view, e.g., trying to identify precise definitions (mathematics), trying to develop algorithms to evaluate the definitions from data or from equations (computer science), testing of mathematical predictions against nature (physics), and trying to design systems with given properties (engineering). SOME IMPLICATIONS OF NONLINEAR DYNAMICS FOR DIFFERENT DISCIPLINES: Note: The following I will not discuss in class and are some miscellaneous observations that may be interesting to you. PHYSICS: - Nonlinear dynamics (ND) is a natural continuation of classical mechanics, e.g., the behavior of planets under gravity, charges in a plasma, mechanical systems with many parts. - ND is also a natural continuation of statistical mechanics, the physics of behaviors that arise from many simple components that interact with each other. There is an especially close intellectual and historical link between the theory of phase transitions and several areas of nonlinear dynamics. - Nonlinear dynamics represents a major new young area of physics research: sustained nonequilibrium physics, for which no deep fundamental theory yet exists of a power comparable to thermodynamics or statistical physics. Behavior of nonequilibrium systems is profoundly different from solids, gases, liquids in equilibrium and new concepts are needed to understand this behavior. MATHEMATICS - Nonlinear dynamical systems is an important and exciting frontier in many areas of pure and appliedd of mathematics. Only recently, with widespread access of powerful computers and invention of efficient and robust numerical algorithms that general properties of nonlinear systems could be identified. - Many new mathematical ideas invented and to be invented, e.g., strange attractors, fractals, and Lyapunov exponents. - One of the million dollar Clay problem involves nonlinear dynamics, namely how to show boundedness of the Navier-Stokes equations. ENGINEERING: - Engineering traditionally concerns the design and control of systems. Very difficult to achieve these goals without a basic understanding of what is going on: - Architecture: Tacoma Narrows bridge near Seattle, see the movie http://www.enm.bris.ac.uk/research/nonlinear/tacoma/tacoma.html - Biomedical engineering: fibrillation of the heart: how to stop or control? Similarly for epilepsy of the brain: why does it commence, how to stop it? - Powerful lasers: single large laser, grid of small lasers, either case you are in trouble. - Fluctuations in the American power grid. - Fusion plasma reactors: main problem is instability. - These systems all have in common a complex evolution in both time and space. - There has been some significant useful progress in how to control complex systems, and this progress was made by leading members of the nonlinear community. Will discuss this later in the course. COMPUTER SCIENCE: - One important aspect of nonlinear dynamics for computer science is new insight regarding when the answer to some problem can be computed. It was Alan Turing's great discovery in the 1940's that not all problems could be solved by a digital computer, e.g., the halting problem. (See Alan Biermann's book, "Great Ideas in Computer Science" for a nice elementary discussion of this). Nonlinear dynamics raises similar profound issues that roughly can be stated as saying that not all well-defined mathematical problems can be simulated to arbitrary accuracy with a digital computer. - Idea of randomness: what is a random number, how can a non-random determinstic algorithm produce random-like output? - Numerical analysis: how does one compute certain quantities like fixed points or linear stability? When does a numerical integration follow an analytical orbit? - Course will teach you powerful and practical computational tools for analyzing time series, as well as conceptual mathematical tools to suggest new experiments or theory. A COMMON LANGUAGE: - Although many people claim that scientific knowledge is becoming ever more specialized and esoteric, in fact there are many fields where people are working on nearly identical problems without realizing it. This similarity is a deep and interesting mystery, and is partly understood from the geometric global methods we discuss later this semester. In this course, we will discuss papers by physicists, ecologists, mathematicians, meteorologists, and economists. We will see that Nature is suprisingly restricted in how many ways it can accomplish certain tasks such as develop a time dependence or change from one kind of time dependence to another. - The common language that chemists, biologists, physicists, economists, and mathematicians use is the language of nonlinear dynamics, an active area of mathematics. You can't understand issues or achievements without a minimal introduction to the common ideas - Course will teach you a vocabulary of about 40 words, which appear in most papers in the literature. - This language is essential in nearly all the "Grand Challenge" problems identified by the US Government as important for near term solution: - Vision - Conformation of biomolecules - Climate Modeling - Fluid Turbulence - Human Genome - Ocean circulation - Semiconductor modeling - Superconductor modeling - Quantum Chromodynamics FUN - Nonlinear dynamics has many novel and neat ideas that are weird and exciting, e.g., fractals, strange attractors, Feigenbaum bifurcations, and universality. This is simply fun to learn about. - Field is still relatively unexplored, a great area to get involved for future research, either mathematical, experimental, or computational. Not mature like particle physics, where only a few extremely difficult questions remain. STYLE OF COURSE - This course will be decidedly scientific, i.e., mathematics will be useful when it helps to explain nature. We will give some derivations from time to time and the mathematics will not always be easy but we will rarely prove theorems. We will also be looking at real laboratory experiments and see what can be learned. Such experiments are often shocking in that they contradict basic mathematical intution, and then nonlinear dynamics becomes that much more interesting. - Formal mathematical style sometimes a hindrance in studying nature. You may be familiar with the saying "Physicists are mathematicians in a hurry, engineers are physicists in a hurry". SOME GOALS OF THIS COURSE - Explain and apply basic concepts of a geometric global way of thinking about driven dissipative systems. - How to _classify_ complex dynamical states - power spectra - fractal dimensions - Lyapunov spectra - How to study _transitions_ between different dynamical states - Linear stability theory - Bifurcation theory - Renormalization and universality - Determine when ideas of nonlinear dynamics are applicable and useful - When do simple nonlinear models describe complex systems? - Compare theory with experiments and with simulation. - Understand limits to mathematical analysis. ROUGH OUTLINE - Quick overview of conepts via the logistic equation. 1 lecture. - Properties and kinds of nonlinear dynamical systems. 2 lectures - First way of classifying complex data: power spectra and correlation functions; periodic, quasiperiodic, and broad-band signals. - Application to and discussion of paper of Gollub and Benson; experimental bifurcations 2-3 lectures - Definition and discussion of dissipative and non-dissipative nonlinear systems in terms of average contraction of phase-space volumes. - Concept of attractors, why dissipative systems are easier than non-dissipative ones. 3 lectures - Stability of fixed points and periodic states, Poincare maps and relation between maps and flows. 3 lectures - Application to neural networks and associative memory. 1 lecture - The elementary bifurcations, application to 3-species ecological model of May-Leonard. 2 lectures - Ruelle-Takens theorem, experimental and numerical confrontations. - Fractals and fractal dimensions. - Strange attractors, sensitivity to initial conditions, and Lyapunov exponents. - Reconstruction of attractors from time series and algorithms for estimating fractal dimensions. - Discussion of Swinney et al's Couette flow data, measles data. - Conclusion and future ideas: spatial structure and high-dimensional systems. PLEASE REMEMBER: ASK QUESTIONS!!