Problem Set Number 02, j/2.036j MIT (Fall 2018)

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1 Problem Set Number 0, j/.036j MIT (Fall 018) Rodolfo R. Rosales (MIT, Math. Dept., room -337, Cambridge, MA 0139) September 6, 018 Due October 4, 018. Turn it in (by 3PM) at the Math. Problem Set Boxes, right outside room There is a box/slot there for 385. Be careful to use the right box (there are many slots). Contents 1 Problem (Backwards: solutions to equation) 1 Problem Strogatz (A nonlinear resistor) 1 3 Problem Strogatz (Improved model of a laser) 4 Problem Strogatz (Find and classify bifurcations) 3 5 Problem Strogatz (Find and classify bifurcations) 4 6 Bifurcations in the circle problem #0 4 7 Excitable system with a refractory period 4 8 Perturbed pitchfork, with root preserved (bifurcation diagram) 5 9 Stability index for flows in the circle 6 1 Problem (Backwards: solutions to equation) Statement for problem (Working backwards, from solutions to equations). Find an equation ẋ = f(x) whose solutions x = x(t) are consistent with those shown in Figure 1.1. What is needed to make the consistency go beyond reproducing the topology of the phase portrait, so as to capture the shapes 1 of the plots of the solutions as functions of time? Note: The figure shows you that there are two critical points on x 1, namely: x = 0 (stable) and x = 1 (unstable). Thus you must produce ẋ = f(x) where: f has zeros at x = 0 and x = 1; f(x) < 0 for 0 < x < 1; and f is positive elsewhere. Actually, what f might do for x < 1 or x > 1.7 is not specified by the figure. Problem Strogatz (A nonlinear resistor) Statement for problem (A nonlinear resistor). Suppose the resistor in Example.. (page 0) is replaced by a nonlinear resistor. In other words, this resistor does not have a linear relation between voltage and current. Such a nonlinearity arises in certain 1 Note that the middle plot involves a single inflection point, while the others have none. 1

2 x = solution. 1 0 Figure 1.1: (Problem ). Solutions to a one dimensional equation (on the line): ẋ = f(x). Find an equation (i.e.: give f = f(x)) whose solutions look like this, including the curves shapes. Note that there is an infinite number of correct answers and wrong ones too t = time. solid-state devices. Instead of I R = V/R, suppose that we have I R = g(v ), where g(v ) has the shape shown in Figure 3 (page 38 of the book). Redo Example.. in this case. Derive the circuit equations, find all the fixed points, and analyze their stability. What qualitative effects does the nonlinearity introduce (if any)? 3 Problem Strogatz (Improved model of a laser) Statement for problem In the simple laser model considered in Section 3.3, we wrote an algebraic equation relating N, the number of excited atoms, to n, the number of laser photons. In more realistic models, this would be replaced by a differential equation. For instance, Milonni and Eberly show that after certain reasonable approximations, quantum mechanics leads to the system dn dn = GnN kn, (3.1) = GnN fn + p. (3.) Here G is the gain coefficient for stimulated emission, k is the decay rate due to loss of photons by mirror transmission, scattering, etc., f is the decay rate for spontaneous emission, and p is the pump strength. All parameters are positive, except p, which can have either sign. This two dimensional system will be analyzed in Exercise For now, let us convert it to a one dimensional system, as follows. a. Suppose that N relaxes much more rapidly than n. Then we may take the quasi-static approximation Ṅ = 0. Given this approximation, express N(t) in terms of n(t) and derive a first order system for n. This procedure is often called adiabatic elimination, and one says that the evolution of N(t) is slaved to that of n(t). See Haken; 3 also remark 3.1 below. Milonni, P. W., and Eberly, J. H. (1988) Lasers (Wiley, New York.) 3 Haken, H. (1983) Synergetics, 3rd ed. (Springer, Berlin).

3 3 b. Show that n = 0 becomes unstable for p > p c, where p c is to be determined. c. What type of bifurcation occurs at the laser threshold p c? d. (Hard question). For what range of parameters is the approximation used in (a) valid? Remark 3.1 The idea behind any adiabatic approximation is that, when the equations are appropriately nondimensionalized, one (or maybe more, in systems with many dimensions) of the equations takes the form: n = 1 τ F n(x 1, x,... ), where 0 < τ 1 is a non dimensional (fast) time scale a measure of the speed at which x n attempts to reach equilibrium, relative to the main time scale in the problem. Then, using the fact that τ is small, the differential equation for x n can be replaced by the algebraic equation F n = 0. This adiabatic approximation idea is at the heart of most of the models in continuum mechanics (and many other subjects). For example, in writing the equations for a (compressible) fluid one assumes that the internal energy, the pressure, the density, and the temperature are related by equations of state, such as the ideal gas laws p = RρT and e = c v T. However, thermodynamical concepts such as temperature and internal energy actually only have a strict meaning at equilibrium; but, if the times scales for which this equilibrium is achieved are much shorter than the ones involved in the dynamics of the gas, the type of arguments above justify their use. Finally, note that situations where one cannot do this sort of stuff also occur. These cases, quite often, are open areas of research, where we do not have a good description for the behavior of the system (e.g.: phase transitions in dynamical situations; say: triggered by a strong compression wave.) Note: Beware of typos in the answers at the back of the book. The solution given for part (a) cannot be right. It predicts N = for n = f/g. 4 Problem Strogatz (Find and classify bifurcations) Statement for problem For the following equation, find the values of r at which bifurcations occur, and classify those as saddle node, transcritical or pitchfork (supercritical or subcritical). Finally, sketch the bifurcation diagram of fixed points, x versus r. = r x x 1 + x. (4.1) Extra question: Notice that something strange happens for r = 0 in the bifurcation diagram. Is this a bifurcation? If so, which type? Does the principle of conservation of stability apply? Hint: look at the equation satisfied by y = 1/(1 + x).

4 MIT, (Rosales) Bifurcations in the circle problem # Problem Strogatz (Find and classify bifurcations) Statement for problem For equation (5.1) below, find the values of r at which a bifurcation occurs, and classify them as saddle-node, transcritical, supercritical pitchfork, or subcritical pitchfork. Finally, sketch the bifurcation diagram of fixed points x versus r. = 5 re x. (5.1) 6 Bifurcations in the circle problem #0 Statement: Bifurcations in the circle problem #0 For equation (6.1) find the values of r at which a bifurcation occurs, and classify them as saddle-node, transcritical, supercritical pitchfork, or subcritical pitchfork. Finally, sketch the bifurcation diagram for the fixed points versus r, including the flow direction and the stability of the various branches of solutions (solid lines for stable branches and dashed ones for unstable ones). dθ = (r cos( θ)) sin(θ), (6.1) where θ is an angle (in radians). Note that the bifurcation diagram which is periodic in θ should be for a π range in θ, and a range of r that includes all the bifurcations. 7 Excitable system with a refractory period Statement: Excitable system with a refractory period The notion of an excitable system is introduced in the book by Strogatz, in problem Quoting from there Suppose you stimulate a neuron by injecting it with a pulse current. If the stimulus is small, nothing dramatic happens: the neuron increases its membrane potential slightly, and then relaxes back to its resting potential. However, if the stimulus exceeds a certain threshold, the neuron will fire and produce a large voltage spike before returning to rest. Surprisingly, the size of the spike does not depend much on the size of the stimulus anything above threshold will elicit essentially the same response. Similar phenomena are found in other types of cells and even in some chemical reactions. These systems are called excitable. The term is hard to define precisely, but roughly speaking, an excitable system is characterized by two properties: 1. It has a unique, globally attracting rest state.. A large enough stimulus can send the system on a long excursion through phase space, before it returns to the rest state. Excitable systems can also exhibit an additional property: they have a refractory period: after going through an excitation spike, there is a period of time during which it becomes harder to excite the system again.

5 MIT, (Rosales) Perturbed pitchfork, with root preserved (bifurcation diagram). 5 This exercise deals with an extremely simple caricature of an excitable system with a refractory period. Let dθ = f(θ, r, z) (7.1) where θ is an angle (in radians), and r, z are parameters satisfying 0 < r < and r < z < π. The function f is periodic of period π in θ, and it is defined as follows for 0 θ π where f(θ) = θ (θ r) for 0 x r, f(θ) = b (θ r) (z θ) for r x a, f(θ) = c ( π θ) for a x π, a = z + r + z r The choice of b is so that f achieves a maximum value f 1 r 4, b = 4 (z r), and c = r 8 π 4 a. (7.) ( ) z + r = 1. ( The choice of a is so that f(a) = r r ) 4 note that the minimum value achieved by f is f = r 4. The choice of c guarantees continuity at θ = a. Note that f is Lipschitz continuous in fact, it is piece-wise smooth. Let the signal put out by equation (7.1) be σ = θ, and assume that r is small (0 < r 1) and z is somewhere in the center of the interval 0 < θ < π, say 4 z π. Then show that (7.1) models an excitable system with a refractory period. That is, show that: A. The system has a unique, globally attracting, 5 critical point. Note that, while there, σ 0. B. A small perturbation (to the attracting critical point) results in a very small signal: σ never exceeds r /. C. Above a certain threshold, an O(1) signal can result from a perturbation to the globally attracting critical point. The signal then reaches a maximum value σ = 1, (almost) independently of the size of the perturbation beyond the threshold. What is the threshold? D. In case C, after the signal becomes small again (σ = O(r )), there is a time period (in fact, a long time period for this example), where a small perturbation cannot trigger another spike the refractory period. Hint: investigate the properties of f when 0 < r 1. It is OK to do this graphically: that is, plot the function using, for example, MatLab. 8 Perturbed pitchfork, with root preserved (bifurcation diagram) Statement: Perturbed pitchfork, with root preserved (bifurcation diagram) Consider the structural stability for a (soft) pitchfork bifurcation, with the restriction that the main solution branch is preserved across the bifurcation. Specifically, consider the situation where: 4 The exact value of z does not matter, as long as it is not close to 0 or to π. 5 Globally attracting means that all the solutions approach the critical point as t. = g(x, r) (g odd in x), (8.1)

6 MIT, (Rosales) Stability index for flows in the circle. 6 has a (soft) pitchfork bifurcation at (x, r) = (0, 0). Assume that the problem depends on a hidden parameter h i.e. let g(x, r) = f(x, r, h) h=0, where you only know that h is small (but it may not be zero). Assume also that you know that f(0, r, h) = 0, though f may not be odd for h 0. Provided that f is reasonably smooth, and f is generic, it can be shown that the canonical equation 6 describing this situation is = r x + h x x 3. (8.) Tasks: Assume h 0 small (say, h = 0.05), and draw the bifurcation diagram for (8.), including the flow lines recall that the bifurcation diagram is, basically, all the phase portraits (one for each r) stacked in one single -D plot. What happens to the pitchfork? Furthermore: estimate the level of noise (in x) under which the distinction between the pitchfork and the new behavior will be hidden do this in terms of h. 9 Stability index for flows in the circle Statement: Stability index for flows in the circle Show that the stability index S for any flow in the circle vanishes. To be precise, consider an equation of the form dθ = f(θ), (9.1) where θ is an angle (in radians), and f is periodic of period π and Lipschitz continuous. Assume also that the equation has a finite number of critical points: 7 θ 1 < θ < < θ N < θ 1 + π. Now assign a weight w = 1 to each stable critical point, a weight w = 1 to each unstable critical point, and a weight w = 0 to each semi-stable critical point. Then show that N S = w n = 0. (9.) n=1 Hint 9.1 Consider the intervals I n, 1 n N, where I n is the interval θ n < θ < θ n+1 here θ N+1 = θ 1 + π, which is the same point as θ 1 because we are in the circle. Then in each such interval either 8 f > 0 or f < 0. Define σ n = 1 if f > 0 in I n, and σ n = 1 if f < 0 in I n. Then relate the w n to the σ n to show (9.). What information do the σ n capture? THE END. 6 That is, near the bifurcation, the full problem can be mapped into equation (8.). 7 The critical points are the zeros of f. 8 If f were to switch sign in I n, then (since it is continuous) it would have a zero in I n. This zero would no be one of the θ n, which are supposed to be all the zeros.

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