An example of stochastic resonance

This post is a response to some discussion that has been going on on the American Scientific Affiliation list, which is archived here
The specific discussion this refers to is here Also see my previous post on this blog. The ASA list does not currently support uploading graphics, so I will post this part of the discussion here.

Last month I posted an example of a resonant system in which an input excitation that varied only by 0.1 % caused the system output to grow without limit, to refute the argument that solar input could not cause the warming experienced in the 20th century because TSI only varies by 0.1 %. Rich Blinne responded with

Even though resonance is theoretically possible before you go there you need to answer the question of why wasn’t there resonance for thousands of years and suddenly in the mid-20th Century things just changed?

The example I posted was a simple, second order linear system. The resonant behavior shown was constant -- it just caused the response to grow and grow forever. If the earth/sun system were a second order linear system, Rich's comment would be right on. Nonlinear systems, including the sun/earth system, are far more complicated. To illustrate I programmed the example given by Tobias and Weiss in [1].

By fiddling with the system parameters I was able to produce the plot shown below, which demonstrates one of the characteristics of nonlinear systems: the ability to show little activity for long periods of time followed by a burst of activity. (The plot is of the variable Tobias and Weiss label T.)



Not to say that this is an accurate model of interactions between the sun and climate, although Tobias and Weiss use it to illustrate sun/climate interactions — only that in nonlinear systems like the sun/climate system you can have long periods of little activity followed by a burst of activity. The sun/earth system is highly nonlinear and probably chaotic. This sort of behavior is common in chaotic systems. So the answer to Rich's comment is that the resonance was always there -- it just wasn't as active before the 20th century.

In case anyone is interested, I will be glad to make the mathematica notebook I used to generate the plot available.

Reference
[1] Tobias, S. M., Weiss, N. O. “Resonant interactions between solar activity and climate”, Journal of Climate, V 13, 1 Nov 2000, pp3745-3759