Now suppose that there is a mechanism to put energy into our dynamical system in order to make up for the losses. This is getting a lot closer to physical reality. In fact the pendulum in an ordinary grandfather's clock is exactly this arrangement. What would the phase space orbit of such a pendulum look like? To begin with let's look at a system with rather a large amount of loss such that in a single cycle the amplitude drops noticeably. Then assume we add energy equal to this loss in a single kick at the cycle's end. Run the Pendulum Driven by Kick at Peak of Swing display.

In real clockwork mechanisms the energy loss is small so that the decrease in amplitude in a single cycle is practically undetectable. Therefore the energy required to make it up is small and noticeable transients are avoided. It appears as though energy was being both lost and added on a continuous basis. In a driven pendulum, the amount of energy delivered during a cycle is approximately constant. The amount of energy lost per cycle depends on the amplitude of that cycle.

The attractor for the undriven lossy pendulum was a point in phase space called an "equilibrium point" or "point attractor". The attractor for a driven, lossy pendulum is a closed path, in this case the phase space projection view is an elliptical figure with a ding in it where the kick comes in. This sort of attractor is called a "limit cycle" or "periodic attractor".

If any external disturbance interferes with the operation of the undriven pendulum without losses it could easily be knocked into a different orbit either larger or smaller than the original. If the driven, lossy pendulum is disturbed it will tend to settle back to its original orbit after the transient associated with the disturbance dies away. The attractor in phase space of the driven, lossy pendulum is said to be stable.

Now let's look in a little more detail at the process of delivering energy to our driven pendulum. This gets right at the heart of how we model dynamical systems mathematically. Remember in the Dynamical System Background text we discussed the modeling process and numerical integration of the differential equations controlling the state variables. One of the points covered was the discontinuous nature of model time as compared to the apparently continuous nature of real time. Time in the model lurches along in little delta t increments I called chronons. One of the implications of the discontinuous nature of model time is that any applied force must be applied for at least a single chronon. Of course it could be applied for more time than that, but not less. So basically the kick we give the pendulum at the end of its cycle is an applied force of fixed magnitude and one chronon duration. The amount of energy delivered during such a kick is the magnitude of the force, times the distance through which the pendulum moved while the force was being applied. This follows (at some distance) from Newton's laws of motion. See the Physics 1 - Mechanics program for more on this.

So if I apply a force of fixed magnitude for a time of one chronon, when the pendulum is moving slowly, like at the end of a cycle, the distance covered will be rather small and the energy imparted likewise moderate. If I apply the same force for the same duration when the pendulum is moving fast, like at the bottom of a swing, more energy should be delivered. Run the Pendulum, Driven by Kick at Bottom of Swing display to see this effect. The amazing thing is that this distinction falls out automatically from the mathematical model. There is no need to write code to instruct the computer on how to adjust the energy.

Remember back at the beginning of this section we said that the solution to the differential equation describing a pendulum meeting our conditions was p=A*sin(t) , where p represented the pendulum position, measured as an angle from vertical. It is evident from this equation that the period of a cycle is independent of the amplitude A, since A does not appear anywhere in the argument of the sine function (the stuff in the parentheses following sin). If the equation holds true then a kick at a regular interval should occur at the same place in a cycle, even if the kick by adding energy changes the amplitude.

Run the Pendulum Driven by Kick at Fixed Interval display.

Finally run the Pendulum, Driven, With User Start display to play around with swings of any amplitude.

We have made extensive use of the pendulum model to introduce some of the concepts of dynamical system studies. In the next lesson we will look at another dynamical system with a more complex response than the pendulum. The next lesson is called Periodic Attractors.

Are there any questions?