II. Nonlinear Equations of Motion

Transcription

1 II. Nonlinear Equations of Motion A. Driven, Damped Pendulum 1. Simple Pendulum a. Numerical solution decompose radial: T mg cos θ = 0 tangential: dv dω mg sin θ = m = ml dt dt dω g g = sin θ θ, if θ is small. dt l l Euler-Cromer, with h = t. g ωi+1 = ωi h θi l θ θ hω i+ 1 = i + i+1 14

2 b. Phase-space plot g For the simple pendulum, the analytical solution is known ( Ω = ). l θ = θo sin ( Ωt + φ ) ω = Ωθ cos Ωt + φ o ( ) Plotting ω versus θ gives a circle. The radius of the circle is ρ = ω + θ = θ o + Ω θ o, which is related to the total energy of the swinging mass, m. The simple pendulum traces this circle on the phase plot ever after. If there is friction, the trajectory spirals into the center the pendulum comes eventually to rest. 2. Non-simple Pendulum a. Numerical solution The first thing is not to assume that the displacement is small. Then the Euler-Cromer algorithm looks like g ωi+ 1 = ωi h sin θ i l θ θ hω. i+ 1 = i + i+1 Next, add some friction and a driving force. dω g = sin θ qω + FD sin ( Ω D t) dt l where q is the drag or damping coefficient and FD sin ( Ω D t ) is a sinusoidal driving force. Now the algorithm is g ω ( ) i+ 1 = ωi h sin θi qωi + FD sin Ω Dti l θ θ hω i+ 1 = i + i+1 15

3 b. Cases q = 0 q 0, F D = 0 q 0, F D = small q 0, F D = large The following terms arise: resonance, over-, under-, critical damping, transient and steady-state behavior. In the over driven case, q 0 and F D is large. No matter how long we wait, ω (t) never settles into a periodic, steady state. Its behavior is said to be chaotic. 16

4 B. Chaotic Behavior 1. Chaos The under-driven pendulum is predictable, since at long elapsed times its motion is periodic. It becomes so no matter what the initial conditions are. The motion of the over-driven pendulum is unpredictable at long elapsed times, despite the fact that we have a definite, deterministic rule relating ω i+ 1 to ω i. The over-driven pendulum never reaches a steady state. The effect can be illustrated by considering two identical pendula, labeled 1 and 2, with initial conditions θ and ω ω 0. Both are governed by the same equation of motion. 1, o θ 2, o 1, o = 2, o = As time goes forward, we ll track the quantity θ = θ 1 θ 2. In the Low Drive case, as elapsed time increases, the motions of the two pendula become identical. In the High Drive case, as elapsed time increases, the motions of the two pendula become increasingly dissimilar. In either case, θ e λt max. For chaotic behavior, λ > 0 ; for non-chaotic behavior, λ < 0. 17

5 2. Attractors Although there appears to be no pattern in the phase plot of an over-driven pendulum, in fact patterns in chaotic behavior can be found. Let us start once again with a simple pendulum. The phase plot is a circle. If we envision a third axis, the time axis perpendicular to the θω -plane, the trajectory becomes a helix whose axis is along the time axis. Suppose we project on the θω -plane only those points at which t satisfies some condition that nt we impose. For instance, we might plot a point only when t =, where n is an integer and T is 2π the period of the pendulum s swing. In effect, we are cutting the helix at uniformly spaced intervals along the time axis. Such a graph is called a Poincaré section. Since the motion is exactly periodic, we d obtain just one plotted point. 18

6 The trajectory in phase space for an over-driven pendulum is much messier than that of a simple pendulum. Further, the trajectories of identical over-driven pendula starting with slightly different initial conditions are completely different. However, the Poincaré sections of those over-driven pendula are the same. Those areas of the phase plot that are visited again and again by a system as it evolves in time are called attractors. We ve seen that the attractor for a simple pendulum is a single point on the Poincaré section, or a circle on a phase plot. The attractor for a chaotic system, such as the overdriven pendulum, has a weird shape, and is called a strange attractor. The significance of an attractor is that while we cannot predict the specific phase trajectory of a chaotic system on the basis of its initial conditions, the system will produce the same attractor in a Poincaré section whatever its initial conditions. 3. Lorenz Butterfly a. Coupled nonlinear differential equations dx = σ ( x y) dt dy xz rx y dt = + dz xy bz dt = Origin: Rayleigh-Binard circulation convection of a fluid between two horizontal plates kept at different temperatures. 19

7 The variables x, y, & z are related to the temperature and velocity fields in the fluid. The parameter r is related to the temperature gradient while the other two parameters, σ and b are adjustable. The behavior of the fluid depends on the temperature difference, so that if T is small the fluid does not move, if T is medium smooth convection cells are formed, or if T is large chaotic flow occurs. Of course, the terms small, medium, and large are relative terms. With h as the time-step, the numerical algorithm would look like this: x = i 1 x + i h( σ ( y + i xi)) y = i 1 y + i h( xz + + i i rx i yi) z = i 1 z + i hxy ( + i i bzi) b. Plots A variety of plots can be made, with the values of r, σ, and b specified. For instance we might plot z vs. x or y vs. t, etc. Plotting z vs. x or z vs. y yields the butterfly attractor, which is the trajectory of the system projected onto a plane. It is interesting to watch the trajectory being traced out over time. This is one of those situations in which a slow computer is an advantage. The point is, that whatever initial x, y, & z are used, the trajectory in phase space converges on the attractor if we wait long enough. On the other hand, the exact path followed by the system through phase space is very different for initial conditions that differ by very little. 20

8 4. Becoming Chaotic a. Chaotic regime Any nonlinear physical system may be chaotic or not. For instance, with the driven, damped pendulum, the motion is governed by the equation of motion dω g = sinθ qω+ FDsin ΩDt. dt l Depending on the values of g, l, q, & F D the motion will or will not be chaotic. We distinguished between low drive and high drive cases earlier. The conditions under which the motion is chaotic are called the chaotic regime. b. Transition to chaos period doubling We might simulate the motion of a driven, damped pendulum for different values of F D, leaving all other parameters unchanged. As F D is increased, we might see results such as these: [Fig in the text] 21

9 Because alternate peaks are not the same height, the period of the motion is actually The period has doubled. 2π 2 Ω D. 2π Now, the period is 4 Ω D ; it has doubled again. We can visualize increasing F D, doubling the period again and again, until the motion becomes aperiodic (i.e., the period becomes infinite) or chaotic. This process is illustrated by a bifurcation diagram. Recall that we created a Poincaré section by plotting (θ,ω ) only when 2 t = n π. For low drive, Ω that consists of a single point at θ P. Repeated with a larger F D, we once more obtain a single point, perhaps slightly different from the first, etc. At some F D, the period doubles, and the Poincaré section becomes two points. Continue increasing F D, the period doubles again, the Poincaré section has four points..... The branching occurs sooner and sooner, begins to become less distinct, finally becoming an apparently continuous smear at a critical value of F D. The period becomes effectively infinite; the motion no longer repeats itself. D 22

10 c. Transition to chaos intermittency Another phenomenon related to chaotic behavior is intermittency. This means that the system behaves chaotically for intermittent finite time intervals. Consider the Lorenz model. Any nonlinear dynamical system is potentially, but not necessarily, chaotic. 23