Notes on Dynamical Systems (continued) 2. Maps The surprisingly complicated behavior of the physical pendulum, and many other physical systems as well, can be more readily understood by examining their discrete time versions. The fact is that observations of change are always recorded by sampling systems at discrete moments. Thus, while we imagine that we can approximate the actual behavior of a system evolving continuously in time by sampling very rapidly, we often are stuck with a fastest sampling rate fixed by the clock speed or memory limitations of the computer doing the sampling. So let s reexamine some of the examples given previously in terms of discrete time. In all of the following we assume that the instants at which the system we are interested in are sampled are given by t n = ndt, where Dt is the sampling interval the time between samples. (We assume that we sample at a constant rate.) Example #5: Exponential dynamics: First let s return to the example of a constant force applied to a viscously damped mass. The velocity of this mass at any instant was previously stated to be v(t) = v T + (v - v T )exp(-bt / m). Let v(t n ) be denoted by v n. The quantity m b has the dimensions of time and when t increases by m b, v decreases by a factor of /e. Thus, m b is called the e-folding time. Let s assume that our sampling rate is fixed so that N samples are acquired every m b. In other words, we assume that Dt = (m b) N. If we insert this in our expression for v we get v n = v T + (v - v T )exp(-n / N ). Now, let s find how the next v is related to the present one: v n+ = v T + (v - v T )exp(-[n +]/ N ) = v T + (v - v T )exp(-n / N )exp(- N ) = v T + (v n - v T )exp(- N ) = av n + b where a = exp(- N ) and b = v T (- exp[- N ]). In the language of discrete dynamical systems, an equation relating future values of the dynamical state at discrete times to past values is called a map. The map v n+ = av n + b is said to be one-dimensional because the next value of v depends only on the current value and no others. Example #6: Periodic dynamics: Consider the periodic state v(t) = v max cos(wt +f). Let Dt = 2p wn. Then and v n± = v max cos(2p[n ±]/ N +f) v n = v max cos(2pn N +f) = v max [cos(2pn N +f)cos(2p N ) m sin(2pn N +f)sin(2p N )] Adding v n+ to v n- and rearranging terms lead to Physics 455, Fall 23 Dynamical Systems 6
v n+ = av n - v n- where a = 2cos(2p N ). The map v n+ = av n - v n- is two-dimensional because the next value of v depends on two earlier ones (not one, as in the previous example). 3. Graphical iteration and fixed points Irrespective of how successive values of a dynamical process are related, a plot of v n as a function of n is called a time series plot and a plot of v n+ as a function of v n is called a first return plot. The first return plot of one-dimensional dynamics is sufficiently simple that graphical iteration can be used to get a sense of what the dynamics is capable of producing. Let s look at exponential dynamics as a specific example. Make a graph with the v n+ along the y-axis and v n along the x. Draw on the graph the dynamical rule v n+ = av n + b 45 line as well as the 45 line v n+ = v n. Graphical iteration is a way of v(n+) 2 determining successive values of v by drawing. Pick a starting value (v, v ). Draw a vertical line from that point to the curve (line) representing the dynamical rule. The y-value of that point is v. Draw a horizontal line from the latter point until it hits the 45 line. The new point on the 45 line has coordinates (v, v ). Repeat v(n) (iterate). The figure to the right shows an example, where a is positive and less than, and b is positive. (In the language of dynamical systems a and b are called control parameters.) The point of intersection between the dynamics and 45 lines is special. If the starting value were exactly there, there would be on place to iterate to. Such a constant value is called a fixed point. For exponential dynamics, the fixed point, v f, is easy to calculate: v f = av f + b fi v f = b -a = v T (- exp(- N ) = v T. - exp(- N ) The fixed point is the terminal velocity. That s reassuring. You can see from the figure above that the starting velocity ( ) is less than v f and as time goes on it gradually increases to it. Try graphically iterating for a starting value greater than v f. Do you get what you expect? It is instructive to allow a and b in the dynamics rule to have any values. Contrast the situation shown in the dynamics line 45 line previous figure with the case to the right where a > and b <. Obviously, the starting value now runs away from v(n+) the fixed point. Thus, in the previous case, the fixed point is an attractor while in the case shown to the right the fixed 2 point is a repellor. These cases are also sometimes said to represent stable and unstable equilibria, respectively. (The bottom of a bowl is a stable equilibrium point for a marble if v(n) the bowl is open side up and the marble is inside. Small displacements of the marble to either side of the bottom lead to returns. On the other hand, if the open side is down and the marble is perched on top that spot is an unstable equilibrium. Small displacements now run away from the top.) The technical difference between stable and unstable dynamics line Physics 455, Fall 23 Dynamical Systems 7
fixed points is the slope of the dynamics rule at the fixed point. If the slope has a magnitude less than, the fixed point is stable. If the slope has magnitude greater than, the fixed point is unstable. (What happens if the slope has a magnitude exactly equal to?) 4. One-dimensional, nonlinear dynamics The most famous one-dimensional, nonlinear dynamical system is the logistic map: v n+ = a(- v n )v n, (5) where usually one uses values a 4 and v n. If the stuff inside the parentheses in (5) were equal to, then (5) would just describe simple exponential dynamics (with b = ). The addition of the factor (- v n ) makes an amazing difference. A first return plot derived from (5) is a parabola that is zero at both v n = and =, and that has a maximum at v n = /2. The maximum height of this parabola is a 4. (Thus, to keep v n between and, a has to be between and 4. Should v n ever be greater than, then v n+ would be <, and so would all successive values. Indeed, v n Æ -, once this occurs.) The fixed points of (5) are v f = a(- v f )v f fi v f = and v f = a - a The slopes of the tangent lines to the parabola at the fixed points determine their stability. The slope at the fixed point is a, and at the nonzero fixed point it is (2 -a ) a. When a is less than, the slope at is positive and less than, so the fixed point at is stable. On the other hand, the nonzero fixed point is unstable because the slope there is greater than. The situation reverses as a increases beyond. For a slightly larger than, the slope at is greater than, but the slope at the nonzero fixed point is less than. The dynamics is said to bifurcate at a =, meaning that it makes a transition from having an attractor at to having one at some positive value. Once a is greater than 2, the slope at the nonzero fixed point goes negative. Its magnitude is still less than until a = 3. For values of a between 3 and 4, however, both fixed points are unstable. So what happens to the dynamics then? To see what occurs it is useful to consider higher order return plots. For example, v n+2 = a(- v n+ )v n+ = a[-a(- v n )v n ][a(- v n )v n ] This expression is quartic in v n. Because of this it has four fixed points. These fixed points correspond to states whose values repeat every two samples. They are roots of the polynomial equation v f = a[-a(- v f )v f ][a(- v f )v f ]. As the fixed points of the first order map repeat every time, they obviously also repeat every two times. So two of the four fixed points are and (a -) a. The other two can be obtained by dividing v f -a[-a(- v f )v f ][a(- v f )v f ] = by v f [v f - (a -) a]. The result is v f ± = 2 + 2a ± 2a (a - 3)(a +). (6) If we plug v f + into the first order logistic map, (5), we get v f - out, and vice versa. In other words, the two roots (6) form a two-cycle two values that alternate back-and-forth between each other. You can see immediately that if a < 3, v f ± are complex. That is, they do not exist as real values between and for Physics 455, Fall 23 Dynamical Systems 8
a < 3. The logistic map has no two-cycle until a > 3. The slope of the tangent line to the curve vn+2 versus vn at the fixed points determines the stability of the fixed points. These slopes are: a 2 at, (2 - a )2 at (a -) a, and -a 2 + 2a + 4 at both v f + and v f -. The magnitude of the latter is less than (and hence the two-cycle is an attractor) up to a = + 6 = 3.449489743.... For larger values of a, the two-cycle is no longer stable. The attractors that replace the fixed points and two-cycle can be determined analytically by arguments similar to those leading to (6), but the algebra gets pretty scary. Instead, let s look at numerical results. The figure to the right shows a superposition of two plots one (the parabola) of vn+ versus vn, the other of vn+2 versus vn. Both plots are for a = 3.5. Note that both curves intersect the 45 line at and at (a -) a =.7428.... Those are the two fixed points of (5). The vn+2 versus the 45 line at vn curve also intersects.857 andat.428. Those are the two-cycle values..9.8.7.6.5.4.3.2...2.3.4.5.6.7.8.9 The figures to the right versus v and v.9 show vn+3 n n+4.8.7 versus vn. The former curve only.6 intersects the 45 line in two places,.5.4 at the fixed point values of (5)..3 (Repeat every time also repeats.2 every 3 times.) The latter curve. intersects the 45 line in 8 places,..2.3.4.5.6.7.8.9 however. These correspond to the two fixed points of (5) (repeat every time repeats every 4 times), the two values (6) (repeat every 2 times also repeats every 4 times), and 4 more values. These constitute a 4-cycle. This 4-cycle is an attractor for 3.449... < a < 3.554.... As a is allowed to increase further, we find that the 4-cycle attractor gives way to an 8-cycle attractor (at about a = 3.554... ), which in turn gives way to a 6-cycle attractor (at about a = 3.5644... ), and so on. The periods of successive attractors keep doubling at values of a that get closer and closer. Eventually, at about a = 3.57... we find that the period of the attractor goes to infinity! That is, the attractor at this value of a never repeats. Such a deterministic, aperiodic behavior is called chain of events (at successively higher values a ) preceding the appearance of deterministic chaos. The chaos is called a period doubling cascade. It is instructive to make an attractor diagram a plot of what all the attractors look like as a function of a. This is shown in the figure to the right. The horizontal axis is a running from on the left to 4 on the right. The vertical axis is attractor values. Plotted above each value of a is 4 values on the attractor. Where you see just one point above a given a, there are actually 4 identical values. That s a fixed point. Two points is a 2-cycle, and so on. For a above 3.57 we see a mess of points. The mess indicates chaos..9.8.7.6.5.4.3.2...2.3 Physics 455, Fall 23 Dynamical Systems 9.4.5.6.7.8.9
Surprisingly, if the mess is magnified, it isn t as messy as it first seems. To the right is shown a portion of the attractor diagram for 3.45 < a < 4. There s a big hole at about a = 3.83.... Close examination shows that this region corresponds to the appearance of a stable 3-cycle. As a is varied in this region, we find a cascade of period doublings; 3-cycle to 6-cycle to 2-cycle, etc. See below to the left. Magnification of central part of this 3-cycle window (below right) shows a picture that looks very similar to the whole plot. The attractor diagram of the logistic map is filled with shrunken copies of itself. It s a fractal! Physics 455, Fall 23 Dynamical Systems