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I read that the mathematical definition of bifurcation is that, at a critical value of a parameter that governs the dynamical system, the system changes to a topologically different system than the previous.

I recognize that the scope of this question is large, but still:

Why is this phenomenon relevant in atmospheric sciences?

One can say that since the topology of the systems orbit changed, so different, previously unaccounted for, phenomena will show up - but that is obvious.

Why, if at all, should a climatologist trying to predict a long term time series keep bifurcation in mind, given they can also compute the results numerically within the limits of precision, without explicitly worrying about bifurcation?

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I don't have enough points here to make tiny little edits yet. Can someone fix that first sentence to read I read that the mathematical definition of bifurcation is that, ... – David Hammen Apr 29 '14 at 21:55

Bifurcation is relevant to the atmosphere because it effects the predictability of the evolution of the atmosphere. I can't speak to its merits with climate models, but with weather it is quite important. As Lorenz discovered, the atmosphere is chaotic and numerical prediction is very sensitive to the initial conditions. If the initial conditions are close to the bifurcation point, the solution can be wildly different and this leads to poor predictability.

In particular, weather models assimilate observations and each observation is a value and an uncertainty. When the uncertainty contains the bifurcation point that essentially means for that particular scenario the instrument error is such that the instrument can provide an observation on either side of the bifurcation. This can be explored with ensemble modeling where many models are run with slightly different initial conditions (or slightly different models, or both) and leads to a large spread in the ensemble and low confidence in the forecast. Our atmosphere can be more sensitive than the instruments we use to measure it can detect.

I'll try to dig up some references to presentations I've seen that show hurricane dissapation / rapid strengthening determined by differences in gulf of Mexico observations smaller than the instrument error reporting them. For more NWP-centric studies, you can search for articles on predictability. I know there should be a couple by Dr. Fuqing Zhang (PSU) and I'll edit when I can point to some specific articles.

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I am waiting for your references, while I read articles of predictability and bifurcation. As for the ensemble situation I personally note, when I do that, that I can run the initial model ensemble, and either my parameter spectrum will include the bifurcation point or not - if a bifurcation does not appear in the long term evolution, then I ask myself: "the system is not supposed to remain this way so long, so go look for bifurcation" - so basically it is a phenomenon which one has to guarantee as happening / not happening - for sake of the merit of the prediction. Is it only that? – Sean Apr 30 '14 at 12:16

Bifurcation is a characteristic of chaotic systems. To quote the Wikipedia article on chaos theory

Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions—a paradigm popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable.

The climate of the Earth is one of these chaotic systems. What happens at a bifurcation point in a chaotic system is that a tiny change of the input parameters to the model results in an entirely different outcome. For this reason it is important to understand the bifurcation points of the system in order to understand when your model has no predictive power.

The most classic example of a chaotic system which exhibits bifurcation is the logistic map. This equation is used to model demographics or population growth in biological systems.
$$ x_{n+1}=r x_n(1-x_n) $$ The outcomes of this equation for different values of the parameter $r$ are shown in the plot below (taken from the Wikipedia article cited above). You can see that after $r$ goes over $3$ the outcomes of each successive iteration become completely unpredictable. The first two bifurcations in this example are at $r=3$ and $r=3.5$.

enter image description here

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