Would I be correct to say that chaos as a science lives in the margins of error of existing measuring instruments. For weather, we could have one input, atmospheric pressure, say 760 +/- 0.05 mm Hg (margin of error stated). So the actual pressure is between 759.95 and 760.05 mm Hg and this range just happens to be the “small difference(s) in initial value” that leads to prediction divergence. That is the weather forecast could be as opposite as bright, sunny and heavy rain, stormy depending on whether you input 759.95 instead of 760 or 760.05 instead of 760 or 759.95 instead of 760.05 for atmospheric pressure. Doesn’t this mean chaos theory says more about the accuracy of our instruments than about actual chaos in so-called chaotic systems. It possesses this subjective element (what we consider to be negligible differences) that seems to undermine its standing as a legitimate mathematical discipline.
I also call into question the divergence, at least in weather prediction. Bright and sunny, how different/divergent is it from thunderstorm? There could be something lost in translation, going from numerical outputs to natural language descriptions like sunny, rainy.. etc.
That said, I’ve seen chaotic double pendulums and those do seem to exhibit real divergence in behavior; if the location of the pendulum’s bob is our output then there definitely is a large numerical difference, especially if we consider how close to each other the initial positions were, which is the point I suppose. We could artificially amplify this divergence by making the pendulum’s arms longer, which tells its own story.
The margins of error of existing measuring instruments will tell youhow long you can expect your simulation to resemble reality, but an exponential decrease in measurement error will only buy you a linear increase in how long that simulation is good for.
I also call into question the divergence, at least in weather prediction. Bright and sunny, how different/divergent is it from thunderstorm? There could be something lost in translation, going from numerical outputs to natural language descriptions like sunny, rainy.. etc.
If you don’t like descriptive stuff like “sunny” or “thunderstorm” you could use metrics like “watts per square meter of sunlight at surface level” or “atmospheric pressure” or “rainfall rate”. You will still observe a divergence in behavior between models with arbitrarily small differences in initial state (and between your model and the behavior of the real world).
an exponential decrease in measurement error will only buy you a linear increase in how long that simulation is good for.
True, and in the real world attempts to measure with extreme precision eventually hit limits imposed by quantum mechanics. Quantum systems are unpredictable in a way that has nothing to do with chaos theory, but that cashes out to injecting tiny amounts of randomness in basically every physical system. In a chaotic system those tiny perturbations would eventually have macroscopic effects, even in the absence of any other sources of error.
I don’t know the exact values Lorenz used in his weather simulation, but Wikipedia says “so a value like 0.506127 printed as 0.506”. If this were atmospheric pressure, we’re talking about a millionth decimal place precision. I don’t know what exerts 0.000001 Pa of pressure or to what such a teeny pressure matters.
Most kind of you to reply. I couldn’t catch all that; I’m mathematically semiliterate. I was just wondering if the key idea “small differences” (in initial states) manifests at the output end (the weather forecast) too. I mean it’s quite possible (given what I know, not much) that (say) an atmospheric pressure difference of 0.01 Pa in the output could mean the difference between rain and shine. Given what you wrote I’m wrong, oui? If I were correct then the chaos resides in the weather, not the output (where the differences are as negligible as in the inputs).
I know that there’s something called the Lyapunov exponent. Could we “diminish the chaos” if we use logarithms, like with the Richter scale for earthquakes? I was told that logarithms, though they rise rapidly in the beginning, ultimately end up plateauing: log 1 million—log 100 = 4 (only)??? log 100 inches rain and log 1 inch rain = 2 (only)?
I hope you’ll forgive me if I’m talking out of my hat here. It’s an interesting topic and I tried my best to read and understand what I read.
I know that there’s something called the Lyapunov exponent. Could we “diminish the chaos” if we use logarithms, like with the Richter scale for earthquakes?
This is a neat question. I think the answer is no, and here’s my attempt to describe why.
The Lyapunov exponent measures the difference between the trajectories over time. If your system is the double pendulum, you need to be able to take two random states of the double pendulum and say how different they are. So it’s not like you’re measuring the speed, or the length, or something like that. And if you have this distance metric on the whole space of double-pendulum states, then you can’t “take the log” of all the distances at the same time (I think because that would break the triangle inequality).
Hopefully, not talking out of my hat, but the difference between the final states of a double pendulum can be typed:
Somewhere in the middle of the pendulum’s journey through space and time. I’ve seen this visually and true there’s divergence. This divergence is based on measurement of the pendulum’s position in space at a given time. So with initial state A, the pendulum at time Tn was at position P1 while beginning with initial stateB(|A−B|≈0), the pendulum at time Tn was at position P2. The alleged divergence is the difference |P1−P2|, oui? Take in absolute terms, |P1−P2|=106, but logarithmically, log|P1−P2|=only 6.
At the very end when the pendulum comes to rest. There’s no divergence there, oui?
Any physical system has a finite amount of mass and energy that limit its possible behaviors. If you take the log of (one variable of) the system, its full range of behaviors will use fewer numbers, but that’s all that will happen. For example, the wind is usually between 0.001 m/s (quite still) and 100 m/s (unprecedented hurricane). If you take the base-10 log, it’s usually between −3 and 2. A change of 2 can mean a change from .001 to .1 m/s (quite still to barely noticeable breeze) or a change from 1 m/s to 100 m/s (modest breeze to everything’s gone). For lots of common phenomena, log scales are too imprecise to be useful.
Chaotic systems can’t be predicted in detail, but physics and common sense still apply. Chaotic weather is just ordinary weather.
It possesses this subjective element (what we consider to be negligible differences) that seems to undermine its standing as a legitimate mathematical discipline.
I think I see what you’re getting at here, but no, “chaotic” is a mathematical property that systems (of equations) either have or don’t have. The idea behind sensitive dependence on initial conditions is that any difference, no matter how small, will eventually lead to diverging trajectories. Since it will happen for arbitrarily small differences, it will definitely happen for whatever difference exists within our ability to make measurements. But the more precisely you measure, the longer it will take for the trajectories to diverge (which is what faul_sname is referring to).
Would I be correct to say that chaos as a science lives in the margins of error of existing measuring instruments. For weather, we could have one input, atmospheric pressure, say 760 +/- 0.05 mm Hg (margin of error stated). So the actual pressure is between 759.95 and 760.05 mm Hg and this range just happens to be the “small difference(s) in initial value” that leads to prediction divergence. That is the weather forecast could be as opposite as bright, sunny and heavy rain, stormy depending on whether you input 759.95 instead of 760 or 760.05 instead of 760 or 759.95 instead of 760.05 for atmospheric pressure. Doesn’t this mean chaos theory says more about the accuracy of our instruments than about actual chaos in so-called chaotic systems. It possesses this subjective element (what we consider to be negligible differences) that seems to undermine its standing as a legitimate mathematical discipline.
I also call into question the divergence, at least in weather prediction. Bright and sunny, how different/divergent is it from thunderstorm? There could be something lost in translation, going from numerical outputs to natural language descriptions like sunny, rainy.. etc.
That said, I’ve seen chaotic double pendulums and those do seem to exhibit real divergence in behavior; if the location of the pendulum’s bob is our output then there definitely is a large numerical difference, especially if we consider how close to each other the initial positions were, which is the point I suppose. We could artificially amplify this divergence by making the pendulum’s arms longer, which tells its own story.
The margins of error of existing measuring instruments will tell you how long you can expect your simulation to resemble reality, but an exponential decrease in measurement error will only buy you a linear increase in how long that simulation is good for.
If you don’t like descriptive stuff like “sunny” or “thunderstorm” you could use metrics like “watts per square meter of sunlight at surface level” or “atmospheric pressure” or “rainfall rate”. You will still observe a divergence in behavior between models with arbitrarily small differences in initial state (and between your model and the behavior of the real world).
an exponential decrease in measurement error will only buy you a linear increase in how long that simulation is good for.
True, and in the real world attempts to measure with extreme precision eventually hit limits imposed by quantum mechanics. Quantum systems are unpredictable in a way that has nothing to do with chaos theory, but that cashes out to injecting tiny amounts of randomness in basically every physical system. In a chaotic system those tiny perturbations would eventually have macroscopic effects, even in the absence of any other sources of error.
I don’t know the exact values Lorenz used in his weather simulation, but Wikipedia says “so a value like 0.506127 printed as 0.506”. If this were atmospheric pressure, we’re talking about a millionth decimal place precision. I don’t know what exerts 0.000001 Pa of pressure or to what such a teeny pressure matters.
That’s the point. Nobody thought such tiny variations would matter. The fact that they can matter, a lot, was the discovery that led to chaos theory.
Most kind of you to reply. I couldn’t catch all that; I’m mathematically semiliterate. I was just wondering if the key idea “small differences” (in initial states) manifests at the output end (the weather forecast) too. I mean it’s quite possible (given what I know, not much) that (say) an atmospheric pressure difference of 0.01 Pa in the output could mean the difference between rain and shine. Given what you wrote I’m wrong, oui? If I were correct then the chaos resides in the weather, not the output (where the differences are as negligible as in the inputs).
I know that there’s something called the Lyapunov exponent. Could we “diminish the chaos” if we use logarithms, like with the Richter scale for earthquakes? I was told that logarithms, though they rise rapidly in the beginning, ultimately end up plateauing: log 1 million—log 100 = 4 (only)??? log 100 inches rain and log 1 inch rain = 2 (only)?
I hope you’ll forgive me if I’m talking out of my hat here. It’s an interesting topic and I tried my best to read and understand what I read.
Gracias, have an awesome day.
This is a neat question. I think the answer is no, and here’s my attempt to describe why.
The Lyapunov exponent measures the difference between the trajectories over time. If your system is the double pendulum, you need to be able to take two random states of the double pendulum and say how different they are. So it’s not like you’re measuring the speed, or the length, or something like that. And if you have this distance metric on the whole space of double-pendulum states, then you can’t “take the log” of all the distances at the same time (I think because that would break the triangle inequality).
Hopefully, not talking out of my hat, but the difference between the final states of a double pendulum can be typed:
Somewhere in the middle of the pendulum’s journey through space and time. I’ve seen this visually and true there’s divergence. This divergence is based on measurement of the pendulum’s position in space at a given time. So with initial state A, the pendulum at time Tn was at position P1 while beginning with initial stateB(|A−B|≈0), the pendulum at time Tn was at position P2. The alleged divergence is the difference |P1−P2|, oui? Take in absolute terms, |P1−P2|=106, but logarithmically, log|P1−P2|=only 6.
At the very end when the pendulum comes to rest. There’s no divergence there, oui?
Any physical system has a finite amount of mass and energy that limit its possible behaviors. If you take the log of (one variable of) the system, its full range of behaviors will use fewer numbers, but that’s all that will happen. For example, the wind is usually between 0.001 m/s (quite still) and 100 m/s (unprecedented hurricane). If you take the base-10 log, it’s usually between −3 and 2. A change of 2 can mean a change from .001 to .1 m/s (quite still to barely noticeable breeze) or a change from 1 m/s to 100 m/s (modest breeze to everything’s gone). For lots of common phenomena, log scales are too imprecise to be useful.
Chaotic systems can’t be predicted in detail, but physics and common sense still apply. Chaotic weather is just ordinary weather.
I think I see what you’re getting at here, but no, “chaotic” is a mathematical property that systems (of equations) either have or don’t have. The idea behind sensitive dependence on initial conditions is that any difference, no matter how small, will eventually lead to diverging trajectories. Since it will happen for arbitrarily small differences, it will definitely happen for whatever difference exists within our ability to make measurements. But the more precisely you measure, the longer it will take for the trajectories to diverge (which is what faul_sname is referring to).