 |
 |
ME 406 QuickTime Movies |
 |
This page gives access to the QuickTime movies that were presented as classroom demos. If you have Quicktime installed in your computer, then you will be able to view a movie by clicking on its title in the list below. The latest version of Quicktime for Macs can be downloaded free from Apple.
PHASE PLANE AND TIME PLOTS
Damped Linear Oscillator. The two movies here show (1) the traditional plot of displacement x versus time and (2) the phase plane plot of x versus dx/dt. In both cases the movie is made by varying the standard dimensionless damping parameter zeta. In the time plot the displacement is shown in blue. For reference each frame of the movie also shows the critically damped case (zeta = 1) in red. The zeta values run from 0 to 2. In the phase plane plot, the sequence of curves is for zeta from 0 to 1. In both cases, the initial condition is unit displacement and zero velocity.
LOCAL GLOBAL TRANSITIONS
Local to Global 1. This movie shows the scene as one rises over the phase plane with the view centered on a saddle point. At the start, the view is local and the saddle point manifolds appear to be straight lines. The view gradually becomes global as the movie progresses, and a spiral appears, connected to the saddle. The system is dx/dt = x - y, and dy/dt = 1 - x*y (from Nonlinear Ordinary Differential Equations, D.W. Jordan and P. Smith, second edition, Oxford Press, 1987, ppp. 51-52).
Local to Global 2. This movie shows the same system as in Local to Global 1, but now the view is centered on the spiral. Notice the apparent rotation of the spiral in the early stages of the upward motion of the viewpoint.
PERIODIC SOLUTIONS
Van der Pol Oscillator. These movies show the periodic motion of two phase points around the van der Pol limit cycle, for three different values of the parameter multiplying the damping term. In all cases note the acceleration of the system in the region -1 < x < 1 where the damping is negative. In the first movie, the parameter is 1, in the second movie the parameter is 2, and in the third movie the parameter is 3. For the larger values of the parameter, the system alternates periods of very slow motion with rapid bursts, a type of motion known as a relaxation oscillation. In the third movie, the region inside the vertical red dashed lines is the part of the phase plane in which the motion is amplified. Outside these lines, the motion is damped. The fourth movie shows how the shape of the limit cycle changes as the parameter increases.
Orbital Stability. The first movie illustrates the difference between Liapunov and orbital stability. The equation considered is a form of Duffing's equation: dx/dt = y and dy/dt = - x - x^3/2. The movie shows the state points moving on two neighboring orbits. They start at the same phase, but because the period is shorter on the outer orbit, they slowly but surely separate as time goes on. By the Liapunov definition of stability, these orbits are unstable. By a more appropriate definition of stability (called orbital or Poincare) which asks only that the orbits remain close and which asks nothing of the phase, these solutions are stable. The second, third and fourth movies show a stable limit cycle, namely the van der Pol cycle. The second movie shows the motion of two phase points which start close together on the cycle. Although the distance between them varies around the cycle, the variations are cyclic and the solutions never merge nor diverge, consistent with both orbital stability and Liapunov stability (although not strict Liapunov stability). The third movie shows two points initially close with one starting on the cycle and one off. In the asymptotic state (achieved in about one period in this case) both points are on the cycle with the distance between them varying cyclically.The fourth movie shows two points initially close, but both slightly off the cycle. In the asymptotic state both points are on the cycle and remain close. Because points starting off the cycle move onto the cycle, we say that the cycle is strictly (or asymptotically) orbitally stable, which is a stronger form of stability than that of solutions of Duffing's equation. By the Liapunov criterion, the orbit is stable but not strictly stable.
BIFURCATION MOVIES
These movies show simple examples of some basic bifurcations. Each movie shows how a selected set of orbits varies as the system parameter is changed. Some of the examples below are based on examples from Nonlinear Dynamics and Chaos, Steven Strogatz, Addison-Wesley, 1994.
The first movie is for a simple linear system (the system is dx/dt = ay, and dy/dt = x - 2y) in which a single fixed equilibrium changes from a saddle to a stable node to a stable spiral as the single system parameter a is changed. The bifurcations are a = 0, saddle to stable node for increasing a, and a = 1, stable node to stable spiral for increasing a. Note that there is no great change in appearance at the node-spiral transition.
The second movie is for a saddle-node bifurcation in a nonlinear system, given by dx/dt = mu - x*x and dy/dt = -y. Here two equilibria (with positions dependent on the system parameter) meet and annihilate one another. The position of the saddle is marked with a red dot, and that of the stable node with a blue dot. Note the interesting hybrid character of the equilibrium at the bifurcation mu = 0. It looks like a node on the right and a saddle on the left, and it is unstable.
The third movie shows a transcritical bifurcation in which a saddle and stable node collide and exchange roles. The system is dx/dt = mu*x - x*x and dy/dt = -y. The unstable saddles are marked with a red dot and the stable nodes with a blue dot.
The fourth movie shows a supercritical pitchfork bifurction in which a stable node bifurcates into a saddle and two stable nodes. The system is dx/dt = mu*x - x^3 and dy/dt = - y. The saddle is marked with a red dot, the stable nodes with blue dots.
The fifth movie shows a subcritical pitchfork bifurcation in which a stable node and two saddles coalesce and form a saddle. The system is dx/dt = mu*x + x^3 and dy/dy = -y. The saddles are marked with red dots, the stable node with a blue dot.
The sixth movie shows a supercritical Hopf bifurcation in which a stable spiral becomes unstable and throws off a stable limit cycle at the bifurcation. The cycle is born with zero amplitude, and the amplitude increases as the parameter is increased. In this particular example the limit cycle is circular. The system is dx/dt = y - x*(x^2 + y^2 - mu) and dy/dt = -x -y*(x^2 + y^2 - mu).
The seventh and eighth movies show a subcritical Hopf bifurcation. Initially the system has two limit cycles -- one stable and one unstable. As the bifurcation parameter is increased, the stable and unstable cycles coalesce in a transcritical exchange of stabilities. As the parameter is increased further, the unstable cycle shrinks toward the equilibrium point. The cycle engulfs the equilibrium and changes it from a stable to an unstable spiral. The first of the two movies for this example shows only the limit cycles and the equilibrium as the bifurcation parameter changes. The second movie shows the same sequence with the addition of two orbits connecting to the equilibrium point. The system is dx/dt = -y + x(x^2 + y^2 + mu)(1-x^2-y^2) and dy/dt = x + y(x^2 + y^2 + mu)(1 - x^2 - y^2).
The ninth and tenth movies show bifurcations for a predator-prey model. The model is analyzed in detail in the Mathematica notebooks predpt1.nb and predpt2.nb. There are also pdf versions of the notebook: predpt1.pdf and predpt2.pdf. The first movie shows the three system equilibria as the food supply for the prey is varied. In the phase plane, the horizontal axis is the prey population, the vertical axis the predator population. There are three equilibria. The first is the origin, which is always unstable -- it is a saddle point. The second equilibrium is a state in which the predators are absent. In that case, the prey population equation reduces to the logistic model. The equilibrium state corresponds to the prey population equal to the carrying capacity in the logistic model. This state is stable for prey food supply less than a certain critical value. At the critical value of prey food supply, a third equilibrium state with both prey and predator moves from the fourth quadrant into the first quadrant, becoming both relevant and the only stable equilibrium at the same time. As the prey food supply is increased further, the equilibria continue to move, and at a second critical value of prey food supply, all three equilibria become unstable. This is a Hopf bifurcation, and the further development of the limit cycle is shown in the second movie. The movie illustrates an interesting and well-known paradox in ecology. As the prey food supply is increased, the limit cycle grows larger, giving an increased average value of the prey population. The paradox is that the danger of prey extinction increases with increasing food supply because the larger limit cycle approaches more closely the states of zero prey population on the vertical axis.
LORENZ EQUATIONS
The Lorenz equations depend on three parameters, conventionally called sigma, r and b. In the movies shown here, sigma is fixed at 10, and b is fixed at 8/3. We show examples of different behaviors for different values of r.
Stable Equilibrium. For r = 0.5, the attractor for the system is a stable equilibrium at the origin. The movie shows 8 selected integral curves converging on the origin.
Chaotic Attractor. For r = 28, the solution is chaotic. The movie shows a spatial rotation of the chaotic attractor produced by two solutions (red and blue) initially very close together. As time goes on the solutions become widely separated.
A Large-r Limit Cycle. One might think that the larger r is, the more chaotic the system, but it is more complicated than that. The movie here shows a spatial rotation of a stable limit cycle which occurs for r = 100.
Sensitive Dependence on Initial Conditions. In this movie we return to the chaotic solution with r = 28. The movie shows the developing time plot of two solutions which start with initial conditions {0,1,0} and {0,1.001,0}. For a long time the solutions stay together and appear to be one solution. Then around t = 20, they start to diverge. The divergence becomes rapid at around t = 25, and they remain well separated beyond that time. This illustrates the sensitive dependence on initial conditions which makes long-time prediction impossible for chaotic solutions.
The Lorenz Map. As Lorenz first showed, chaotic solutions have a short-term regularity which shows up in the map named after him. In this map, one plots the n+1 peak in z versus the nth peak in z, where the peaks are obtained from a numerical solution. Surprisingly, there is a strong relation between two successive peaks, as demonstrated by the grouping of the points along a well-defined curve. This curve is shown in the movie for a chaotic solution with r = 28. The movie shows the build-up of the curve as the time of integration increases from 25 s to 800 s in increments of 25 s. Notice how stable the shape of the curve is. Integrating for longer times simply fills in the curve more solidly.
DRIVEN PENDULUM AND POINCARE MAP
The problem considered here is the sinusoidally driven pendulum. Depending on the parameters of the system, the pendulum response can be periodic or chaotic. The periodic response can have the frequency of the driver, or can be a subharmonic. A powerful tool for examining the system response is the Poincare map, which is in effect a strobe of the system once each driver period. A periodic response at the driver frequency appears as a single dot on the Poincare map. A response at, for example, a frequency of 1/2 that of the driver appears as an oscillation between two dots. The first movie shows the strobe for a case in which the response frequency is 1/2 the driver frequency. The system starts off the periodic orbit, so there is a brief transient. The second movie shows a chaotic response. In this case, the system is strobed but all points from past strobes are kept. What the movie shows is how the pattern changes with phase angle.