Lecture 1 - Thu 2023 09 07 Introduction, Flows on the Line. % =============================================================================== Students presentation and who is the lecturer. Mechanics of class: (1) Go through Syllabus. (2) Discuss working groups. (3) The importance of writing and notes. (4) Maybe a couple of Zoom lectures [explain why]. Questions and answers. Begin with lectures ........................................ OVERVIEW OF SUBJECT % % ----------------------------------------------------------------------------- % What is a dynamical system? Definition of dynamical system: there is a "phase space" describing the system, and a rule for evolving in time [More on this below]. We will concentrate on Driven-Damped Nonlinear Dynamical Systems. Driven: there is something forcing the system. Damped: typically, dissipation of some sort. Examples from mechanics: point mass in a potential in nD, phase space = R^{2n} Constrained mechanical systems: pendulum, point mass on the sphere [double hinge pendulum] ... more complicated phase spaces. The above are "continuous-time systems". Introduce also discrete-time systems. Later on we will see other systems from other topics. Anything in science that involves a time evolution is typically modeled by a dynamical system. Complexity diagram for continuous systems: Linear vs. Nonlinear and dimension. <--------------------------- NEXT LECT. % % ----------------------------------------------------------------------------- % Brief history (really brief); from Newton to the present. NOT COVERED IN THE LECTURE. READ IT HERE. Milestones: --- Development of the language of differential equations. --- ODE [mechanics]; PDE [fluids, elasticity, EM, etc.]; difference equations [discrete time]. --- Linearity versus nonlinearity. --- Nonlinearity spotty history. But not "neglected". People in fluids and astronomy had to face it from the start. Examples: solitons and water waves, planetary orbits, etc. --- Early tool in nonlinearity: oscillations and perturbations. Stokes ---> 1920 to now: oscillators basis of electronic industry. --- Boltzman and Ergodic systems. 1st taste of chaos. --- Poincare's GEOMETRIC APPROACH. <-- This will be a core point in 18.353. --- "3rd integral" of motion. --- Lorenz and Henon-Heiles. % % ----------------------------------------------------------------------------- % Back to dynamical systems. Concepts we will cover [partial list]. --- Discrete and continuous time. Maps and D.E. --- Linear and nonlinear. Autonomous, non-autonomous. --- ODE, reduction to 1st order and autonomous. --- Dimension (of the phase space). --- Trajectory/orbit (curve/path traced in phase space by solution ... for discrete systems: a sequence of points). --- Critical points, linearization, stability [details later]. <--- NEXT LECT. --- Bifurcations [intuitive idea, details later]. <---------------- NEXT LECT. % % ----------------------------------------------------------------------------- % Flows on the line. <--- NEXT LECT. We will begin by studying the simplest systems: phase space is the line. Examples 1-D systems --- \dot{x} = sin(x). Exact solution: Separation of variables: dt = dx/sin(x) = d/dx {ln[sinx/(1+cosx)]}. Thus t = ln{(1+cosx0)sinx/[(1+cosx) sinx0]} solves x(0) = x0. What can you say from this exact solution? Introduce Poincare's Geometric Approach [phase portrait] for eqn. --- \dot{x} = f(x) = - V'(x) and motivation [wire and goo]. --- Population grow, logistic equation. --- No oscillations in 1-D. Footnote: Check that the derivative of ln{[sin x]/[1+cos x]} is 1/sin x; and thus justify exact solution above. % % ----------------------------------------------------------------------------- % EXISTENCE AND UNIQUENESS THEOREM for ode. Lipschitz. <--- NEXT LECT. --- Example of non-uniqueness. --- Where is chaos here? Extreme example, when bound for the continuity constant, e^{K*T}, actually happens. % % ----------------------------------------------------------------------------- % EXTRA STUFF MENTIONED IN THE LECTURE (Not in original plan above). % ========== % #1 Attractors. Driven-damped system often evolve towards attractors [some sort of equilibrium, where driving and dissipation balance each other; at least on average]. Attractors range from very simple to very complicated. Examples: 1a Terminal velocity for object falling in the atmosphere, or sinking in the ocean. Example of the simplest type of attractor. 1b The oscillators that are the basis for clocks, from the old pendulum clocks to electronic clocks; as well as radio and tv transmission, etc. They work on the same principle a child uses to swing on a swing, though the physical implementations vary. In lecture: describe basic LCR circuit with feedback and swing. We will go into more of this later. This one is an "intermediate" example of complication in the attractor. 1c I did not mention this, but a slightly more complicated attractor than the one in 1b are quasi-periodic states [not one period but several]. Turns out that they are not very common for driven-damped systems [we will go into why latter], but Landau's attempted to explain turbulence with such states. 1d The most complicated type of attractors (known) are the so called "strange attractors". It is believed that turbulence is of this type. Describing them with some semblance of precision will have to wait. #2 Tipping points. This is the nomenclature used in geosciences to describe large changes in a system behavior triggered by a small change in its parameters. Generally such tipping points are examples of "bifurcations" [we will spend some time on these things]. Examples: 2a Buckling of a column at the load increases [in the lecture I used a measuring tape to illustrate this behavior]. This example also involves "spontaneous symmetry breaking". In the lecture I also sketched the "bifurcation diagram" for this. We will explain this in more detail later. Note: For some materials the phenomena can be quite dramatic. A stone column does not buckle, it fractures. 2b Mechanical switches, such as the wall light switch in the classroom. I also sketched the "bifurcation diagram". 2c Friction law. When pushing a heavy object [e.g. refrigerator] you may have noticed that it does not move at all till a critical force it is applied. And once motion starts, you need less force. A behavior caused by this phenomena is "stop and go oscillations". e.g.: the refrigerator moves in jerks. In the lecture I illustrated this by dragging a chair. You can hear the sound generated by the oscillations. An extreme case of this is the screeching sound chalk can make [I did not do this demo]. 2d Transition to turbulence. Fluid flow down a pipe is laminar a low velocity, but it becomes turbulent beyond some critical velocity. You can see a similar phenomena in many rivers. Flow is nice and smooth normally. But things change when flooding occurs. 2e Avalanches [mud, snow, etc.] Roughly, there is a maximum slope granular and similar media can support. In addition the static critical slope tends to be higher than the dynamic one, so once an avalanche starts, it goes big time. It will be triggered if for some reason the critical slope is exceeded [say, change in humidity may make a slope otherwise stable, unstable], or if motion is triggered somehow and the slope was below the critical static, but above the dynamic critical [a very dangerous situation]. All of this is related to how "weird" friction laws are [see 2c]. #3 Linear versus nonlinear. A common simplification made to model many systems is that the response will be proportional to the input [you also see this in every day life, when people assume that double the effort will cause double the effect]. In many situations this works, but it is an assumption, not a given ... and it fails in many cases [e.g.: all the examples in #2]. Engineers often design gizmos so the assumption works. This is almost the norm in electronics [this is why amplifiers have many stages]. Why? Because we understand linear systems much better than nonlinear ones, and hence we can make them do what we want, precisely. % =============================================================================== EOF