Thursday, 17 December 2009

Crisis induced sudden death of chaotic attractors ... Contemplating.

I read the abstract of the paper "Multiple attractors and crisis route to chaos in a model food-chain" by Ranjit Kumar Upadhyay, of the Department of Applied Mathematics, Indian School of Mines ,Dhanbad 826 004, Jharkhand, India published in the journal Chaos, Solitons & Fractals, intrigued my senses, contemplated upon its contents, and the thoughts sparked in my mind, I laid them down as they occurred, as I read through the abstract, line-by-line.

"An attempt has been made to identify the mechanism, which is responsible for the existence of chaos in narrow parameter range ..

narrow parameter range .. the existence of chaos in narrow parameter range .. narrow signifying the small change necessary for chaos to exist .. existence of chaos.

The notion of existence of chaos, is new to me. Is it a way to describe chaos, occurring? It seems as it is extending the properties of chaos, or may be it is better to refer to as extending the concept of chaos.

Whatever, in that particular mention, significant is the reference to a mechanism. The mechanism that is responsible for bringing into existence after, well narrowly manipulating a given connected or attached parameter range. Or, maybe just, parameter. The existence of chaos, a result of manipulating or adjusting the parameter within a narrow range. Values given of the parameter fall into a narrow stretch of the parameter's total possible range of values.

in a realistic ecological model food-chain. Analytical and numerical studies of a three species food-chain model similar to a situation likely to be seen in terrestrial ecosystems has been carried out.

ecological model food-chain .. a situation likely to be seen in terrestrial ecosystems

The study of the model food chain suggests that the existence of chaos in narrow parameter ranges is caused by the crisis-induced sudden death of chaotic attractors.

'chaos existence caused' .. 'by crisis-induced sudden death of chaotic attractors' .. profound notion. First, is the statement, the best way I can put it, sudden death of chaotic attractors. Which, come to think of it, it accurately puts in the right perspective. Attractors, which, weight added, they are chaotic, their origin is chaos too, suddenly die. Suddenly die. Cease to exist. For chaos being dynamic, it postulates, if that term is correct, the necessity of what brought them into existence, in the first place. For the attractors to die, meaning, they lack of what they were fed with, what sustained them. What was necessary for their existence. Once, the condition that was keeping them alive, cease to be, they ceased to be, too.

That what was brought, was induced by a crisis. And their sudden death caused chaos. The existence of chaos, in narrow parameter range. Minute changes in the parameters involved brought chaos into existence. One would assume that this fresh bout of chaos would lead to new chaotic attractors. Chaos, let's say, methodology.

Varying one of the critical parameters in its range while keeping all the others constant, one can monitor the changes in the dynamical behaviour of the system, thereby fixing the regimes in which the system exhibits chaotic dynamics.

'critical parameters' .. implying sensitive dependence situations. In order to identify the result, each of the crucial parameters have, in the overall state of the system, the referred to, as dynamical behaviour, you vary one at a time. Monitoring the changes, varying one of the critical parameters, brings forth. 'Fixing the regimes', what combinations of critical parameters, combined values within their ranges, that exhibit chaotic dynamics, in the sense that certain regimes are more chaotic than others. As, they would be regimes, where the system exhibits no, behaviour at all, by taking behaviour to mean, the tendency for changes, among the members of a population. If, there are no changes, there is no behaviour to speak of, and with the sudden death of the already existing chaotic attractors, there is nothing to keep the system alive. Extinction follows.

Which points towards the realisation that its submersion into chaos, is vital, in order to keep the system alive, to avoid extinction.

The computed bifurcation diagrams and basin boundary calculations indicate that crisis is the underlying factor which generates chaotic dynamics in this model food-chain.

'computed bifurcation diagrams' .. 'basin boundary calculations' .. bifurcations induced by crisis, test the boundaries of the attracting basins. Basins, within which, the chaotic attractors reside, and once these boundaries are surpassed, chaotic attractors can sustain themselves no-more. Their purpose ends, the system requires to spring up, rejuvenate itself, if not, the members of the population, in crisis, will perish along with the death of their attractors that was keeping them alive. Chaos is necessary for their survival.

We investigate sudden qualitative changes in chaotic dynamical behaviour, which occur at a parameter value a1=1.7804 at which the chaotic attractor destroyed by boundary crisis with an unstable periodic orbit created by the saddle-node bifurcation.

'saddle-node bifurcation', 'created an unstable periodic orbit' .. unstable, there is no consistency, its trajectory changes between runs, cause a crisis manifesting at the boundaries, the chaotic attractors disintegrates. It is destroyed. As it occurs when a parameter acquires a specific value. An instability embedded in the system as the parameter fluctuates beyond its allowed range. The range that was necessary to keep the system stable, by keeping the periodic orbits stable.

Multiple attractors with riddled basins and fractal boundaries are also observed.

riddled basins, no clear-cut basins with fractal boundaries, many attractors, chaotic states, unstable

If ecological systems of interacting species do indeed exhibit multiple attractors etc., the long term dynamics of such systems may undergo vast qualitative changes following epidemics or environmental catastrophes due to the system being pushed into the basin of a new attractor by the perturbation.

Perturbations push the systems into the basin of a new attractor. Perturbations brought forth by epidemics or environmental catastrophes.

Coupled with stochasticity, such complex behaviours may render such systems practically unpredictable."

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