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At the Azimuth Code Project, we are working to produce educational software that is relevant to the Earth sciences and the study of climate. Here we will look into a stochastic resonance demonstration program, by Allan Erskine and Glyn Adgie. Stochastic resonance is a widely studied phenomenon, which has an application to the theory of ice-age cycles. After explaining how to run the program, we’ll sketch out some of the math and science background, and then move on to dissect the program and its algorithm.

The Azimuth models are interactive web pages. Their behavior is responsive, because the code runs right in the browser, as javascript. They make use of a high-level support library (JSXGraph), which allows them to focus logic of the models rather than the presentation mechanisms of the browser.

A test drive of the model

Start by opening the stochastic resonance model web page. On the same plot it shows a sine wave, called the forcing signal, and a chaotic time-series, called the output signal. There are four sliders, which we’ll call Amplitude, Frequency, Noise and Sample-Path.

  • The Amplitude and Frequency sliders control the sine wave. Try them out.

  • The output signal depends, in a complex way, on the sine wave. Vary Amplitude and Frequency to see how they affect the output signal.

  • The amount of randomization involved in the process is controlled by the Noise slider. Verify this.

  • Change the Sample-Path slider to get a different instance of the process.

Inventory of the program

The program runs a discrete approximation for a “stochastic differential equation” (SDE), which is a specification for the derivative of the output signal as a function of time, the current signal value, and a noise process.

The program consists of the following components:

  1. Interactive controls to set parameters

  2. Plot of the forcing signal (sine curve)

  3. Plot of the output signal

  4. A function which defines a particular SDE. “Stochastic resonance” is a characteristic of the solutions to this equation.

  5. A general simulator for SDE’s, based on the Euler method

Going to the source

I would like everyone now to locate the source code, as follows:

  • Open the web page for the model. The code is now downloaded and running in your browser!

  • While there, run your browser’s view-source function. For Firefox on the Mac, it’s Apple-U, for Firefox on the PC it’s click the right mouse or touchpad button then select “View Page Source” from the drop-down menu.

  • You should see the html for the web page itself.

  • Note the following lines at the head of the file, which refer to javascript programs on the web:

   <script src=''></script>
    <script src=''></script>
    <script src='./StochasticResonanceEuler.js'></script>
    <script src='./normals.js'></script>

  • Each of these lines causes the browser to load the indicated program into its internal javascript interpreter. MathJax is an open-source formula rendering engine. JSXGraph is a cross-platform library for interactive graphics, function plotting, and data visualization. StochchasticResonanceEuler.js is the model itself, and normals.js makes a table of random numbers used in the calculation.

  • Now, click on the link within the header line for StochasticResonanceEuler.js – and you’re there!

Your next challenge is to scan through StochasticResonanceEuler.js and associate its contents with the Program Inventory listed above. Try to put a blur lens over any items that look obscure – the goal here is only to form rough hypotheses about what might be going on.

The concept of stochastic resonance

Now let’s analyze the differential equation that is used in the model. It sets the deterministic part of the derivative to a time-varying sine wave plus a bistable function of the current signal value:

DerivNonrandom(t, x) = SineWave(t, amplitude, frequency) + Bistable(x),

where Bistable(x) = x * (1 - x^2).

Now let’s analyze the effects of these terms on the output signal.

Alone, the sine wave would cause the output signal to vary sinusoidally, with a 90 degree phase difference.

The bistable polynomial has roots at -1, 0 and 1. The root at 0 is an unstable equilibrium, and -1 and 1 are stable. The basin of attraction for -1 is all the negative numbers, the basin for 1 is the positive numbers, and the point of unstable equilibrium separates the basins.

We will view each basin as one of the states of a bistable system.

Now let’s put the sine wave and the bistable polynomial together. If the wave amplitude is not too large, the system will gravitate towards one of the attractors, and then continue to oscillate around it thereafter – so it never leaves the basin. But if the wave is large enough, the system will be pulled back and forth between the two basins; its state will resonate with the driving signal.

Now, let’s add the noise into the picture. Suppose the sine wave was large enough to periodically pull the system close to zero – but not enough for it to cross over. Then a well-timed noise event could push the system over to the other side. So there will be a phase-dependent probability of the noise triggering a state change. More noise will lead to transitions on more of the cycles. In the flip-flopping between states, we will see a stochastic reflection of the driving sine wave. Further noise will lead to multiple transitions within the cycles, thereby drowning out the signal.

Moral: under the right conditions, the noise may amplify the effect of the input signal.

Relevance to the theory of the ice ages

The theory of the timing of the ice ages is a fascinating, challenging and open problem in science.

Note: here we are using the term ice age in the colloquial sense of being a period of maximal glaciation. The scientific term Ice Age refers to a major period of time that spans thousands of such “ice ages” and the warm periods between them. There have only been four such Ice Ages in the history of the Earth. Each one is characterized by a different timing pattern for the many ice ages that it contains.

A current hypothesis for the timing of the ice ages uses a stochastic resonance model. The climate is modeled as a multistable system, and the forcing results from certain slowly varying, cyclical changes in astronomical variables (such as the tilt of the Earth’s axis). These are known as Milankovitch cycles, and their periods are on the scale of 10,000 to 100,000 years. Note that in our current major Ice Age, the minor ice ages occur roughly every 100,000 years.

In the simplest model, the climate has two stable states: a cold, snowball Earth, and a hot, iceless Earth. Each state is self-reinforcing. A frozen Earth is white, so it doesn’t absorb much solar energy, which keeps it cold and frozen. A hot Earth is dark, so it absorbs a lot of solar energy, which keeps it hot and melted. Included in the model is the fact that the glaciers are concentrated in the northern latitudes – and hence the northern temperatures can trigger a change in the state of the climate.

There are three astronomical cycles that contribute to the forcing function:

  • Changing of the eccentricity (ovalness) of the Earth’s orbit, with a period of 100 thousand years

  • Changing of the obliquity (tilt) of the Earth’s axis, with a period of 41 thousand years

  • Precession (rotary wobbling) of the Earth’s axis, with a period of 23 thousand years

These effects sum to produce a multi-frequency variation in the amount of solar energy reaching the northern latitudes. The induced temperature changes, however, are not large enough to trigger a state change. According to the stochastic resonance hypothesis, it is other, random variations in the heat received up north that may trigger the climate to change states. One such source of variation may be changes in the amount of heat-trapping gases in the atmosphere.

Note that the boost in atmospheric carbon dioxide introduced by human industry is pushing us further away from the tipping point to the next ice age. This issue was discussed on the Azimuth blog, on Dec. 4, 2012:

Arrow said:

Hopefully the extra CO2 is enough to avert the next ice age.

John Baez replied:

There’s been a lot of work on this, discussed very nicely here:

• Andrew Revkin, The next age and the Anthropocene, Dot Earth, 8 January 2012.

The consensus seems to be that we have put enough CO2 into the air to postpone the next glacial period, perhaps for the next 100,000 years. And this paper:

• Gary Schaffer, Long time management of fossil fuel resources to limit global warming and avoid ice age onsets, Geophys. Res. Lett. 36 (2009), L03704.

suggests that we if we save our remaining fossil fuels, we could head off the next few glacial cycles by burning them at appropriately chosen times. Another possibility would be to deliberately release more potent greenhouse gases than CO2 whenever a glacial period was imminent.

If we were smart, we might be able to manage the Earth’s temperature for quite a while, avoiding glacial periods without turning up the heat full blast as we’re doing now.

For further information, see:

Milankovitch cycle

Stochastic resonance

Purposes of a stochastic modeling program

Besides serving an educational function, such programs also have applications within research: (1) they can be used to perform computational experiments in the pursuit of purely theoretical questions, and (2) they can be used to test empirical theories, by generating their predictions and comparing them to observed and historical data. For an example of concept exploration, suppose we asked how the effectiveness of a forcing function depends on its frequency. With the current program, we could vary the frequency parameter, and observe how it affects the output. On a more systematic basis, we could write a driver program that varies the parameters and applies measures to the output signal. For an example of theory testing, one could imagine, a program that implements a model of the Milankovitch astronomical cycles, then outputs this signal into the state-changing model of a particular theory of climate, and then finally compares the output signal of the climate model with historical data.

This For is an a good example of scientific concept programming, exploration, being suppose used we in asked how the service effectiveness of our a understanding forcing of function depends on its frequency. With the Earth. current It program, is we Azimuth could programming. vary the frequency parameter, and observe how it affects the output. On a more systematic basis, we could write a driver program that varies the parameters and applies measures to the output signal.

For an example of theory testing, one could imagine a program that implements a model of the Milankovitch astronomical cycles, then outputs this signal into the state-changing model of a particular theory of climate, and then finally compares the output signal of the climate model with historical data.

Here we see a nice example of scientific programming, in the service of our understanding of the Earth. And it is a natural part of the Azimuth project, which follows the North Star of the science that matters most for human survival.

Implementation of the algorithm

Our scientific program consists of seven functions. The top-level function is initCharts. It dispatches to initControls, which builds the sliders, and initSrBoard, which builds the curve objects for the forcing function and the output signal (called the “position curve” in the program). Each curve object has a method that is responsible for computing the (x,t) values for the displayed time series. These calculation methods get called whenever the defining input parameters are changed. The calculation method for the forcing curve is set to a function that computes the sine wave time series. This function reads the amplitude and frequency values from the sliders.

The calculation method for the output signal is set to the function mkSrPlot, which carries out the actual work of the simulation. Its first action is to define a function for the deterministic part of the derivative:

deriv = Deriv(t,x) = SineCurve(t) + BiStable(x),

Then it constructs a “stepper” function, through the call Euler(deriv, tStep). In general, a stepper function maps the current point (t,x) and a noise sample to the next point (t’,x’). The Euler stepper maps ((t,x), noiseSample) to (t + tStep, x + tStep * Deriv(t,x) + noiseSample).

The simulation loop is performed by the function sdeLoop, which is given:

  • The stepper function

  • The noise amplitude (“dither”)

  • The initial point (t0,x0)

  • A randomization offset

  • The number of points to generate

The current point is initialized to (t0,x0), and then the stepper is repeatedly applied to the current point and the current noise sample. The output returned is the sequence of (t,x) values.

The noise samples are read from an array normals[i] and scaled by the noise amplitude. The contains many more data points than are needed by the calculation. The randomization offset controls the starting point in the array, which leads to different instances of the random process.

Changing the program

Now that we’ve tried out the program, downloaded its source code, and understood how it works, it’s time to roll up our sleeves and start tweaking it to do new things!

We’ll proceed by a series of “baby steps.” First let’s get a local copy of the program to run on your machine. Copy the html file and the main java script to a folder on your local machine. I’ll suppose that you’ve stored them into the following folder on your machine: c:\pkg\webmodels.

Now check that the html file is active, by

(TODO: finish this section)

Problems and challenges

(TODO: rewrite for clarity)

  • Effect of frequency

  • Design a study of the effectiveness of signal transmission, as a function of noise amplitude and signal frequency. How you define the effectiveness measure?

  • How would you restructure the code for general, statistical studies of the output time series?

  • When the sliders are moved, an event must be fired, which causes the recalculation to take place. How is this mechanism implemented in the javascript / JSXGraph application library?

  • Modify to add an exponent slider

  • Modify to show graph of expected value (add slider for nTrials) (Not enough random numbers.)

  • Add a standard deviation plot

  • If you are a climate scientist, let us know of next steps

  • Begin to study this book —-, and think of how to write programs for some of the models. Simplify! The hierarchy of models. All models that you post here will be considered as candidates for the Azimuth Code Project page. This may be a way for programmers, ultimately, to give back to the Earth.

Open invitation to the Azimuth Code Project

We’re just beginning to organize for a new round of development. Programmers, we would welcome your participation, on many levels:

  • Discussion and planning on the forum

  • Writing specs for new modelling programs

  • Coding

  • Testing

  • Documentation and blogging

Check out the forum. If you’re interested in participating, just send a request to John Baez for a login. No earnest request will be denied!

(TODO: scale this back just a bit. We’re more in a brainstorming phase, so e.g. there is no testing to be done.)

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