The system dynamics are given by
x'' + 2.0*delta*omegan*x' + (omegan^2)*x =b*omegan^2 +a*(omegan^2)*Sin[omega*t]
where x'' is the second derivative of x with respect to tim, and x' is the first.
Two runs were made. For both runs
omega = omegan = 10.0
delta = 0.001
b = 1.0
x' = 0
x = 1.0
For the first run a = 0.0. Thus the excitation is just a constant 1.0.
Here is a plot of the results of the first run
Note that since the excitation is the constant value 1.0, the output is the constant value 1.0
For the next run I set a = 0.001 - 0.1 percent of 1.0. Here are the results of the second run:
The addition of a 0.1 percent variation of the excitation makes a huge difference in the output.
A few comments are in order:
- This is the simplest possible example. In no way do I intend to claim that this is what's going on in global warming. The point of the example is that a resonance in a system can lead to large swings of the system variables
- Obviously for the resonance to occur, energy storage is required (the spring in our simple example) In the earth/sun system the energy storage may be in the oceans, the atmosphere, or eaven the earth itself
- Scafetta suggests that one possible source of cycling of the earth' temperature is the wobble of the sun's position caused by the gravitational pull of the Jovian planets (Jupiter, Saturn)
- Scafetta Talk at EPA:
- Various publications on Scafetta's web site: