We are still working with the Duffing Mechanical Oscillator (DMO) introduced in the last section. In this instance we will apply a Total Force made up as follows:
Applied Force = 0.2*cos(1*t+0)
Restoring Force = -x^3
Drag Force = -0.08*x'
This system is driven less powerfully and suffers smaller losses than the first example. These are fairly minor changes but as we will see the effect is considerable. Run the DMO Periodic Attractor 1a display. Notice the initial conditions shown in the bottom margin.

In the last display we have taken four different views of the Duffing oscillator under identical conditions. The attractor under these conditions was periodic with period 1, what we called a limit cycle previously. The behavior of the oscillator was pendulum like. In the next display we will use the same physical system, that is the amplitude and drag variables remain the same, but we start the oscillator with different initial position and velocity. One might expect it to settle to the same motion. That is after all the nature of an attractor. Run the DMO Periodic Attractor 1b display, where we have only changed the initial conditions to (1.05,.77).

I don't know of any way to predict ahead of time that multiple attractors exist for a particular dynamical system. I think you have to model it and run the model to see. If any of you readers know better, please let me know. The existence of multiple attractors for dynamical systems has significant implications for the builders and operators of these systems. The question often is, "Is my system going to do anything dangerous?". If the answer is, "It depends.", that is not very comforting. Especially it is not comforting if it is not clear upon what and to what extent "it depends". As you will discover, this state of affairs gets worse. Run the DMO Periodic Attractor 2a display, looking again particularly at the first four views.

Yet another change in the initial conditions causes the motion to settle on yet another attractor. This one is also a period 2 attractor. It is sort of a mirror image of the previous period 2 attractor we called 2a. It is typical of period 2 attractors to occur in pairs. Run the DMO Periodic Attractor 2b display where initial conditions are(-.46,.3).

Next we come another attractor inhabiting the phase space of our one dynamical system. This one is a period 3 attractor. It repeats every third cycle of the forcing function. In the (p,v) projection and the phase space orbit it shows three loops. Extensive investigation by a person named Ueda in 1980 indicates that these attractors are all there are for the specific dynamical system described here. This implies that any combination of initial conditions must settle on one of these attractors. Run the DMO Periodic Attractor 3 display where initial conditions are (-.43,.12).

To really understand a dynamical system we should identify all the attractors in its phase space. Then for each attractor we should find all the possible combinations of initial conditions which settle to that attractor. The set of points in the (x,x') plane which as initial conditions lead to a particular attractor are called that attractor's basin of attraction. A section of the (x,x') plane with the attractor basins marked, is called an "attractor basin (AB) map" of the system. Since each point in the (x,x') plane goes to some attractor, the map fills the entire plane.

The business of making an AB map for a dynamical system is difficult. The only approach for a personal computer is to sample a number of points in the (x,x') plane, using each as the initial conditions for the system under study. Then to let the system develop over time until it settles to an attractor. When that happens we could make a record of the attractor found for reference in subsequent samples, then color the pixel representing the starting point in the (x,x') plane a different color for each attractor. This takes too long to be of much use. A 100 MHZ pentium computer might work for several days on a high resolution AB map.

What we provide here is a quick way to identify which attractor "owns" any specific point in the (x,x') plane. Let's go back to the DMO Periodic Attractor 3 display and select the Basins of Attraction view.

Next we will get into a different kind of attractor, neither point nor periodic.

Are there any questions?