Lecture 18 - Tue 2020 11 03 - Virtual % ============================================================================== Example of a Hopf bifurcation; with details left to the students to read on their own [see the notes "Hopf Bifurcation Using Complex Notation" --- same section of the web page where this Lecture is posted]. The notes "Hopf Bifurcations" in the "Notes" section of the web page provide more examples. Consider a system (in the plane) where we assume that there are no quadratic terms. At the bifurcation the linear system has a matrix A0, with a pair of pure imaginary eigenvalues: \pm i*omega \neq 0]. Let v be the eigenvector A0*v = i*omega*v. Let v = v_1 - i*v_2, so that Av_1 = omega*v_2 and Av_2 = - omega*v_1. Then; assuming that the critical point is the origin through-out bifurcation, and writing vectors using v_1 and v_2 as a base: Y = x*v_1 + y*v_2, the system takes the complex form [z=x+i*y] \dot{z} = i omega z + delta (a z + b z^*) + (c z^3 + d z^2 z* + e z (z^*)^2 + f (z^*)^3) + O(delta^2, delta*z^2, z^4) where a, b, c, ... are complex constants, and delta is the bifurcation parameter --- note that delta appears in the linear term in the expansion. Then write delta = sigma epsilon^2, sigma = \pm 1, 0 < epsilon << 1, and carry a two time expansion. Under generic assumptions this yields a Hopf bifurcation. The final answer is, modulo some scaling, yields the same equations as in [A] of the prior lecture: \dot{r} = a*r-r^3 and \dot{theta} = 1; except for some higher order frequency corrections. Adding quadratic terms makes things messier, but the answer has the same form. % % ============================================================================== % GLOBAL BIFURCATION OF CYCLES. Note: next lecture includes a "summary" of this topic. READ IT. Assume a situation where there is a stable limit cycle, so that there is an annulus where the flow gets trapped [or a torus in 3-D, etc.] Then, as we vary the bifurcation parameter, this object will "generally" persist. That is: a very special set-up is needed to have both the limit cycle, and this object, act "in sync". This is similar to the argument for Hopfs: the critical point may switch from a stable spiral to an unstable spiral, but the nonlinear effects will [generally] not switch simultaneously ... hence producing a limit cycle on one side of the bifurcation point. Case 1 % ----------------------------------------------------------------------- Angular flow around cycle persists. Start by Poincare map near cycle: [A] below. This leads to *Saddle node bifurcation of cycles* --- Explain why this is not as weird as it sounds; how do two cycles get close together and cancel? Is this not a very unusual thing? No #1: via Poincare map. No #2: because once the cycles are close some place, they will be close everywhere! --- Examples: easy to do with polar coordinates. Note: at the bifurcation the limit cycles created have period and size O(1). Compare this with Hopf, where the period is O(1), but the size is O(\sqrt{epsilon}). Later on we will deal with the bifurcations where not only the Poincare map persists, but the limit cycle survives as well. These give rise to an analog of the trans-critical bifurcation of critical points and, in more than 2-D, a new type of bifurcation (period doubling) which turns out to be fundamental. In addition, an "analog" of the Hopf also can arise [at least for conservative systems]. --- On the other hand, the analog of the pitchfork is not generic, because it requieres a symmetry which is hard to get for limit cycles. [A] Saddle node bifurcation, using the Poincare Map. In 2-D the map reduces to a map from a line to a line [transversal direction to cycle], which "moves around" as the bifurcation parameter varies. The limit cycle(s) are the intersections of this line with the diagonal x = y. The same argument used for 1-D bifurcations of Critical Points shows that a "likely" bifurcation is a saddle-node bifurcation of limit cycles. Note: Because solutions cannot cross, the map is an increasing function [Imagine the limit cycle is the unit circle, and the Poincare map is the return map to the positive real axis. Then if R_i is the orbit starting at r_i, and r_1 < r_2, then it must be R_1 < R_2 everywhere, in particular after one trip around the origin]. Note: the stability of the cycles follows from size of the derivative of the map at the cycle [size < 1 stable; size > 1 unstable]. In more than 2-D, in a situation where the cycle looses stability in only one direction, the same picture applies. Case 2 % ----------------------------------------------------------------------- Flow around the cycle destroyed; no Poincare map. Need critical point to produce this! Then: 2.1 *Infinite period* % ---------------- Critical point pops up in the cycle. Note that, right at the bifurcation, the critical point on the limit cycle is asymptotically stable but not Liapunov. A bit beyond it, we get an excitable system. Compute how period behaves as bifurcation approaches. --> To next lecture. Example using polar coordinates [as in book]. --> To next lecture. 2.2 *Homoclinic* % --------------------- Critical point approaches cycle. Example: picture a spiral point near a saddle. As their distance varies, or the strength of the spiraling changes, there will be a sweet spot with a saddle connection. On one side of it get a limit cycle, on the other, not one. Compute how period behaves as bifurcation approaches. --> To next lecture. Example using "phase plane surgery" --> To next lecture. TABLE OF PERIODS AND AMPLITUDES. % --------------------------------------------- Table with the 4 types of limit cycle bifurcations, indicating how they can be distinguished by the behavior of the amplitude and period of the limit cycles (as a function of the bifurcation parameter). EXAMPLES. % -------------------------------------------------------------------- Note the profusion of examples in the book from chemistry and biology. You should read them. % % ============================================================================== EOF