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At the Azimuth Code Project, we are aiming to produce educational software that is relevant to the Earth sciences and the study of climate. Our software takes the form of interactive web pages, which allow you to experiment with the parameters of a model and view its outputs. But in order to fully understand the meaning of a program, we need to know about the concepts and theories that inform it. So we will be writing articles to explain the science, the math, and the programming behind these models.

In this two-part series, I will cover the Azimuth stochastic resonance example program, by Allan Erskine and Glyn Adgie . Here Today I we’ll will look outline at some of the math and science behind the program, and next time I’ll we’ll dissect the program. By way of introduction, I am a software developer with research training in computer science, so this is a new field area of investigation for me. Any amendments or clarifications are welcome!

The concept of stochastic resonance

Stochastic resonance is a phenomenon in which, under certain conditions, a noise source may amplify the effect of a weak signal. This concept was used in an early hypothesis about the timing of ice-age cycles, but and that has since been applied to a wide range of phenomena, including neuronal detection mechanisms, and patterns of traffic congestion.

Suppose that we have a signal detector whose internal, analog state is driven by an input signal, and suppose the analog states are classified partitioned into a two region regions, of called the “on” states and a region of “off” states. state This is so the we have a digital state, state which is abstracted from the analog state. With a light switch, we could take the force as the input signal, the angle as the analog state, and the up/down classification of the angle as the digital state.

Let’s consider the effect of a periodic input signal on the digital state. Suppose that the wave amplitude is not big enough to change the digital state, yet large enough to drive the detector’s analog state close to the digital state boundary. Then, a bit of random noise, occurring near the peak of an input cycle, may “tap” the system over to the other digital state. So, We there will be therefore see a phase-dependent probability of transitions state-transitions between that digital is states. synchronized This with relationship between signal phase and state-transition probabilities bears the stamp of the input frequency. signal. The In a complex way, the noise hasamplified the input signal.

But it’s a pretty funky amplifier! Here is a picture from the Azimuth library article on stochastic resonance:

Stochastic resonance has been found in the signal detection mechanisms of neurons. There are, for example, cells in the tails of crayfish which that are tuned to low-frequency signals in the movement water of arising from the water, generated by the motions of predators. These signals alone are do too not weak to cross the firing threshold for the neurons, but with the right amount of noise, the neurons do are respond triggered to by the these signals.

See:

Stochastic resonance, Azimuth Library

Stochastic resonance in neurobiology , David Lyttle. Lyttle

Bistable stochastic resonance and Milankovitch theories of ice-age cycles

Stochastic resonance was originally defined for the systems special that case are of a bistable system – where each digital state is the basin of attraction for a stable point of stable equilibrium.

An early application of stochastic resonance was to a hypothesis, within the framework of bistable climate dynamics, about the timing of the ice-age cycles. Although it has not been confirmed, it remains of interest (1) historically, (2) because the the timing of the ice-age cycles remains an open problem, and (3) because the Milankovitch hypothesis upon which it rests is an active part of the current scientific research. research agenda.

In the bistable model, the two climate states are a cold, “snowball” Earth and a hot, iceless Earth. The snowball Earth is stable because it is white, and hence reflects solar energy, which keeps it frozen. The iceless Earth is stable because it is dark, and hence absorbs solar energy, which keeps it melted.

The Milankovitch hypothesis states that the drivers of climate state change are long-duration cycles in the insolation the solar energy received in the northern latitudes (called the “insolation”) that are caused by periodic changes in the Earth’s orbital parameters. The significance of the north is significant because that is where the glaciers are concentrated concentrated, there, and so a sufficient “pulse” in the northern temperatures could trigger initiate a state change.

Three such astronomical cycles have been identified:

• Changing of the eccentricity of the Earth’s elliptical orbit, with a period of 100 kiloyears

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

• Precession (swiveling) of the Earth’s axis, with a period of 23 kiloyears

In the stochastic resonance hypothesis, the Milankovitch signal is amplified by random events to produce climate state changes. More In more recent forms of Milankovitch theories theories, invoke a deterministic forcing mechanism. mechanism is used. In a theory by Didier Paillard, the climate is modeled as with having three states, called interglacial, mild glacial and full glacial, and the state changes depend on the volume of ice as well as the insolation.

See:

Milankovitch cycle, Azimuth Library

Mathematics of the environment (part 10), John Baez. This gives an exposition of Paillard’s theory.

Increasing the Signal-to-Noise Ratio with More Noise, Glyn Adgie and Tim van Beek, Azimuth Blog. Subtitle: Are the Milankovitch Cycles Causing the Ice Ages?

Bistable systems defined by a potential function

Anytime Any we smooth have a function that with has two local minima, minima we can use be that used as a function to define a bistable system. For instance, consider the function V(x) = x4V(x)=x 4/4x 2/2V(x) = x^4/4 - x^2/2 /4 : - x2/2:

To define a bistable system, construct a differential equation in where which the time derivative of x is set to the negative of the derivative of the potential at x:

dx/dt=V(x)=x 3+x=x(1x 2)dx/dt = -V'(x) = -x^3 + x = x(1 - x^2)

So, for instance, at a place where the potential graph is sloping upward as x increases, the -V’(x) time derivative X’(t) is negative, which sends X(t) “downhill” towards the potential minimum. As it approaches the minimum, the slope of the potential graph goes to zero, which means the motion of X(t) slows down. It asymptotically approaches rest at the minimum.

The roots of V’(x) yield stable equilibria at 1 and -1, and an unstable equilibrium at 0. The unstable equilibrium separates the basins of attraction for the stable equilibria.

Discrete stochastic resonance

We Here now we talk describe about a the discrete-time discrete model model, that which exhibits stochastic resonance this is implemented what is used in the Azimuth demo program.

The We Azimuth construct demo program implements a discrete discrete-time model, derivative, which by exhibits combining stochastic the resonance. negative of the derivative of the potential function, a sampled sine wave, and a normally distributed random number:

We use the potential function just described, but using discrete time; the derivative is assumed to be constant over the interval between time points. In addition, a random number is added to the discretely sampled derivative. This gives us a discrete-time difference equation:

ΔX t=V(X t)*Δt+SineWave(t)+RandomSample(t)=X t(1X t 2)Δt+α*sin(ωt)+β*GaussianSample(t)\Delta X_t = -V'(X_t) *\Delta t + SineWave(t) + RandomSample(t) = X_t (1 - X_t^2) \Delta t + \alpha * sin(\omega t) + \beta * GaussianSample(t)

where Δt\Delta t is a constant, constant andtt is restricted to multiples of Δt\Delta t , . and GaussianSample(t) is a sampling from a normal distribution with zero mean and unit variance.

Note that this differential difference equation is the discrete discrete-time version counterpart of to a correspondingStochastic differential equation.

In Next the time next article, we will take a look into at the Azimuth demo program: how to use it, how it works, and how to modify change it to develop make new programs.

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