Butterflies, Chaos and Feedback

Chaos and feedback are two important topics when considering climate change. The climate always behaves chaotically, regardless of what humankind does or indeed any other external influence, because of the way the climate itself works. The reason for this is feedback. It doesn't mean that what we do has no effect, or that we can't make things worse – or, in principle, better.

The climate is in fact a very complicated system, with many different feedbacks with different characteristics, but even quite simple systems can behave chaotically, if they have feedback.

System 1 shows a very simple system with no feedback.

The arrows show that the inputs affect the system, and the system affects the outputs, and that's all there is to it. If the inputs remain constant, the outputs remain constant. If the outputs vary, it's because the inputs are varying – the system doesn't generate variations itself.

System 2 shows a more general system with feedback. For simplicity, I've only shown feedback from one of the outputs to one of the inputs. Feedback can be positive or negative.

System 3 shows a system with negative feedback – the feedback from the output is inverted before it's fed into the input.

If the feedback is negative, a small increase in an input causes a small increase in the output, which feeds back to the input and reduces it. This reduces the output a little, but it's not reduced to exactly what it was without the increased input, because if it was, the feedback would have been reduced to what that was without the increased input too. Negative feedback can be strong, in which case changes in the input only have a small effect on the output, or it can be weak, in which case it doesn't make much difference – but it always reduces the changes in the output, without reducing them to nothing, and certainly not reversing them. Still no chaos unless there's external chaos coming in on the inputs.

This is how classic equilibrium works. Imagine a population of rabbits and foxes. If the rabbits increase a bit, there's more food for the foxes, so the foxes multiply – and eat the extra rabbits: negative feedback. The system settles down to a steady, equilibrium state where the number of rabbits is just enough to keep the foxes fed, and the number of foxes is just enough to keep the rabbits under control. An increase in some other input – maybe better grass for the rabbits to feed on – results in a small increase in the rabbit population, but the foxes increase too and keep the rabbits in check. Negative feedback results in stability. The stronger the feedback, the less effect there'll be on the output when any other input changes.

Positive feedback is radically different. A small increase in the input causes a small increase in the output. This feeds back to the input and increases it, which causes a further small increase in the output. This feeds back to the input and increases it...and so on. This is a runaway effect: there's no stopping it. The output, and the feedback, simply keep on going up. Positive feedback results in instability.

That can't really go on forever. In the real world, something always breaks before you get to infinity. How fast does it go up, anyway? Does it simply shoot up to breaking point immediately?

Time to think about what this system actually is. Maybe it's a field of grass. The inputs are sunshine, soil, and carbon dioxide from the air. The outputs are grass and oxygen to the air. The feedback is cow dung back into the soil, which improves it and makes the grass grow better. Put some reality into the system and we can see better what's going on!

The something that breaks is that too much cow dung doesn't improve the soil at all: some cow dung helps, but too much poisons the grass. The feedback is only positive up to a point. This is always the case with positive feedback: negative feedback always wins in the end. That's why nothing ever really goes to infinity.

The other thing is that it doesn't shoot up to breaking point immediately. The first year, there's more cow dung, and the grass is better. The next year, there's more still, and the grass is better still. This goes on for a few years, then eventually – maybe after five years, maybe after ten – there's too much cow dung one year, and the next year the grass is worse. Negative feedback is winning. But it took five or ten years to get up to that point. The reason is that there's a delay in the feedback. It's taking a whole year for an increase in the grass to affect the cow dung to affect the grass.

Really of course it may not take a year for this particular feedback – but there's still some delay.

System 4 has negative feedback and a delay.

This is more like a real system than System 3. An improvement in the pasture results in an increase in the rabbit population, which results in an increase in the fox population – but it isn't immediate, so the rabbit increase isn't halted immediately. The foxes breed a lot of young cubs, because there's plenty of food for the parents. When the cubs grow up, they eat so many rabbits the rabbit population crashes, and that causes the fox population to crash*. Then the rabbit population can soar again, and so on. Both populations oscillate. They don't go to zero, and they don't go to infinity, but they're not stable.

The exact behaviour of a system with negative feedback and a delay depends on the length of the delay, and the strength of the feedback. If the delay is short, or the feedback is weak, the system behaves in the way I described before I raised the question of delay: it's stable. If the delay is long and the feedback is strong, then it oscillates. The exact period and size of the oscillations depend on the length of the delay and strength of the feedback. If both the strength of the feedback and the delay are constant, the period and size of the oscillations will be constant too. This isn't stability, but it's not chaos either.

Real feedback is either just plain negative, or negative at the extreme (actually at both extremes) with one or more positive bits in the middle. That's called non-linear feedback – because a graph of it isn't a straight line. This results in more complicated behaviour.

Basically what's going on is that the behaviour is different depending where on the curve you are at any given moment. In some places, with positive feedback, it's unstable, and heads for the hills. This eventually takes it out of the positive feedback region. Once out of the positive feedback region, negative feedback may make it oscillate, or push it back into the positive feedback region. In general, the strength of negative feedback will vary along the curve – and so the size and period of the oscillations will vary, too. The oscillations themselves take us into different parts of the curve. This is where chaos comes from. All it takes is non-linear feedback and some delay.

There is always some delay in any feedback system; and non-linear effects are very common – indeed any system with positive feedback is bound to be non-linear. (You can't go on producing more and more cow dung, faster and faster, forever – sooner or later something has to break.)

Now where do those butterflies come in? Ah, they're one of those other inputs. We've got chaos without even having any other inputs, we don't need the butterflies at all. But that doesn't mean there aren't any butterflies. And yes, in chaotic systems tiny changes in an input can eventually make enormous changes in the output – they can push you a tiny bit further along the feedback curve, or a tiny bit less far, and that can make an oscillation bigger or smaller, and that can push you bit further still, or a bit less, and so on, building up to huge differences in the end. (This is often called sensitive dependence on initial conditions – very fancy terminology, but no more precise than simpler words. The exact moment you count as initial is arbitrary...) But you'd never know it was caused by the butterfly – or which of the zillions of “butterflies”, or which combination of them all, or whatever else. Chaos happens in these systems anyway.

The world's ecology is a chaotic system. It's a very complicated system, which makes understanding what's going on in detail more difficult, but it's the chaos that really makes it hard. Because it's chaotic, things can vary wildly without any external influences at all. Species can multiply crazily, or crash to extinction, for no apparent reason; whole ecosystems can change, thrive, or crash, just through the interactions within the system.

That's not to say that outside influences don't happen, or that they don't have an effect: they do happen, and they do have an effect. Perhaps the dinosaur era really was brought to an end by a huge meteor, but it might just have been one of those things that happens in a chaotic system, and the meteor just a coincidence (the evidence suggests that the extinction was well under way before the meteor struck). Other big meteor strikes have occurred without mass extinctions, and other mass extinctions have happened without any big meteor strike. Perhaps it was a butterfly on the other side of the world that really did it.

Within the chaos, there can be periods of near equilibrium, sometimes just locally, sometimes on a wide scale; there can be periods of quite regular oscillations, again locally or on a wide scale. There can be patterns that you can glimpse, but which are very hard to pin down, changing in some unexpected way just when you think you've understood them, and yet still seeming to follow some kind of pattern. (Take a look at pictures of the Mandelbrot Set for great examples of this – that's chaos in a very simple system indeed.)

Earth's climate is a chaotic system – closely linked with Earth's ecology, with each affecting the other profoundly. But that's the subject of another essay. Or two or three. Climate: Patterns in the Chaos

©Clive K Semmens 2008