Lecture 4 - Tue 2024 09 17 Continue with flows on the line: bifurcations. % =============================================================================== For students: read (this file end) "NUMERICAL METHODS AND CHAOS" and "WHY THERMAL CSI DOES NOT WORK. Note: the summaries often include bits like this not covered in the lecture. % % =============================================================================== BIFURCATIONS: --- Qualitative change in phase portrait. Cannot map one into the other. --- Example: tape buckling [column buckling, other bucklings, \ clickers]. Other examples: lasers, turbulence, etc. | Done in prior --- Define it mathematically. Qualitative change in phase | lectures. portrait. Cannot map one into the other. / We start with 1-D; and CLASSIFY NOW BY DEGREE OF GENERALITY. (For additional details see "baby bifurcation notes" in the Web page). CRUCIAL PICTURE for 1-D: \dot{x} = f(x, r). Picture of f "flapping" as r moves. The MOST GENERIC BEHAVIOR is: The creation, annihilation of zeros <--- Saddle node bifurcation. % % ----------------------------------------------------------------------------- % SADDLE NODE BIFURCATION [also known as turning point, fold, blue sky, ...] Canonical or NORMAL form: \dot{x} = r + x^2. BIFURCATION DIAGRAM ~ "stack" the phase portraits for each parameter value in one single picture. Doable because the phase portrait for a 1-D system is just points in the line, with arrows (flow direction) between them. Point out "ONLY ONE CASE". property: [#1C] [sign changes in r and x^2, or size changes in x^2, can be transformed out]. CRITICAL SLOW DOWN PHENOMENA [way to detect transition about to happen]. % % ----------------------------------------------------------------------------- % NORMAL FORMS Can be justified using the Inverse and Implicit theorems. The mathematical statement is that there exist a transformation (local^#, valid near the bifurcation point) reducing the ode to the normal form. ^# For \dot{x} = f(x, r), with the bifurcation for r=r_c, at the critical point x_c, local means: valid in a neighborhood of (x_c, r_c). % % ----------------------------------------------------------------------------- % Plan for SYSTEMATIC STUDY In what follows we will study "all" possible bifurcations, starting from the most generic, and adding restrictions where-by others can happen. In each case we will do this first "graphically" [flapping f] and then via the NORMAL FORM. We will also state when the bifurcations happen, in terms of the partial derivatives of f(x, r) at the critical point where the bifurcation happens [the bifurcation point (x_c, r_c)]. (1) No assumptions. This is "SADDLE NODE" already done. (2) Restrict to the case where solution is known to persist across bifurcation. This is TRANSCRITICAL (next) (3) Restrict to the case where there is a left-right symmetry. This is PITCHFORK (after next). We could keep adding restrictions, but these would lead to "weird" cases, not likely in actual applications. *ALL* these bifurcations carry over to more than 1D. OK, let us look at these LESS GENERIC SITUATIONS (with restrictions). % % ----------------------------------------------------------------------------- % TRANSCRITICAL BIFURCATION Canonical or NORMAL FORM: \dot{x} = r*x - x^2. "Simplest/generic" under "symmetry" assumption that a solution (critical point) persists across the bifurcation [often true in physical systems]. Note .............................. Only one case, property [#1C] is STILL TRUE. Do BIFURCATION diagram <========================================= Show what the "Picture of f "flapping" as r moves" means here. Point out "Conservation of stability" here and prior one. % % ----------------------------------------------------------------------------- % PITCHFORK BIFURCATION Canonical or NORMAL FORM: \dot{x} = r*x \pm x^3. "Simplest/generic" under left-right "symmetry" assumption. This bifurcation corresponds to a "spontaneous symmetry breaking" (explain). Note: unless prior ones; sign changes cannot be mapped away ....... [#1C] FAILS. TWO CASES: Subcritical (SOFT)/Supercritical (HARD). Do BIFURCATION diagrams <======================================== Point out difference in behavior across bifurcation: Continuous (subcritical/soft) versus abrupt (supercritical/hard). Again "Conservation of stability" principle holds. Applies to all three bifurcations we have seen [as well as the ones that we will see later on] % % =============================================================================== % ******************************************************************************* % =============================================================================== STUFF BELOW NOT COVERED IN THE LECTURE - STUDENTS EXPECTED TO READ IT HERE. % % ----------------------------------------------------------------------------- % NUMERICAL METHODS AND CHAOS. Numerical computations are not exact. The main sources of error are (a) fixed point arithmetic. (b) truncation errors [e.g.: derivatives approximated by difference quotients]. Effectively, the errors amount to replacing a dynamical system by a "near" one. Properties that are structurally stable (this will be defined later) should persist, if the errors are "small" enough, but if this fails the behavior (not just quantitative, but qualitative) can change dramatically ... and, in general we do not quite know what "small" enough actually is; so *caution* is a *must*. Even in the best of cases, the errors limit how accurately we can compute ... for example, they "blur" transitions [you will see this in problem set #1, with "Computer Exercises with a 1-d Map"]. Ode solvers [such as those in the MatLab ode-suite] aim at being idiot proof, and because we understand ode's quite well, they mostly succeed. But a determined enough user can, non-the-less get garbage out of them. Solvers involve numerical parameters [e.g.: error tolerances^#1], with defaults set at "reasonable" values; but what is reasonable for one problem may not be reasonable for another. NEVER "buy" an interesting result without checking that the answer does not change when the error tolerances are decreased. "Numerical chaos" (i.e.: chaos due to lack of resolution) is a notorious possibility. Example: Many years ago I bumped into a "demo" for a Nonlinear Dynamics software suite, that included a "chaotic planetary orbit example". Beautiful pictures, ... except that this was for a single (point) planet orbiting a (point) star -- i.e.: the two body problem. Do you see anything wrong with this? There have been instances of papers that had to be retracted because of numerical screw ups like this [though none on a problem were the screw up was as obvious as botching the two body problem]. #1 In MatLab these are accessible via the options [read the documentation]. % % ----------------------------------------------------------------------------- % WHY THERMAL CSI DOES NOT WORK. Someone has been murdered, and the CSI team shows up shortly after the crime, and by careful measures of the temperature in the room, they can backtrack to get the thermal signature of the criminal; which gives them info about his/her size, and other useful info. Marvelous stuff, but impossible nonsense. Why? Because the backwards heat equation is an example of an ill-posed problem. It has no solution. Sketch of why: consider what happens with a sine profile under u_t = -u_xx. The solution is u = e^{k^2 t} sine(k x). High frequencies can grow arbitrarily fast; even if you know (in advance) that the solution is band limited^#2 (there are no frequencies higher than some bound; i.e.: |k| < B), the smallest features in the data (perpetrators hairs) quickly dominate the answer, making it useless. #2 Not the case in CSI % % =============================================================================== EOF