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Wikipedia introduces deterministic chaos as

[Small differences in initial conditions yielding widely diverging outcomes] can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution and 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

I understand that this is a physical inability to reproduce identical initial conditions. Or that the imprecision in computation between different code or computers will produce different results.

This is therefore related to imperfections in the real world (either physical or computational) but I fail to understand how this can have an effect in simulations.

If my code states that $\frac{1}{3}=0.34$, it will always be the case and subsequent simulations will lead to the same results. It is not like $\frac{1}{3}=0.34$ on Monday, and $0.3333$ on Tuesday.

My question: does the chaotic behaviour of equations happen in simulations without introducing random rounding errors or changes in initial conditions (in one code and computer)? If so, what is the fundamental reason for this?

In other words, is chaos an expression of real-world limitation more than a fundamental formal inability to predict the future?

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  • $\begingroup$ You may be interested in the Lyapunov exponent because it is a formal measure of system predictability. In that context, the maximum Lyapunov exponent being positive indicates chaos. $\endgroup$
    – mdnestor
    Jul 11, 2021 at 19:13

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I think mathematicians tend to think of things on a theoretical level. A course on deterministic models or dynamical systems would brush up on some concepts like the Lyapunov exponent (comment from M. Nestor): https://en.wikipedia.org/wiki/Lyapunov_exponent


I want to start by saying that the inability to perfectly replicate experimental conditions or numbers on a computer each time is not what is meant by "chaos". These real-world limitations exist but are often never studied because these differences in initial conditions occur on a microscopic level. Scientists who cannot repeat experiments identically each time must take averages and have error bars. Only those studying highly-controlled systems (e.g. simulations) can start talking about or quantifying chaos.

Your question of "when/where does chaotic behaviour arise" is more of a question of semantics, at least in my day-to-day experience.

In a pragmatic sense, yes, rounding errors or changes in initial conditions are how we see chaotic behaviour. When doing deterministic simulations, or at least in what I and my colleagues do, reproducability is important and expected. When we compare two simulations that have the same initial conditions and physics, we expect them to behave the same up until some point (after which, we then say some small rounding error occurred). We also both understand that if the initial conditions were changed slightly, the two simulations will behave similarly at first and then diverge wildly later.

This is the point of view that can most easily be shared with non-maths colleagues. I don't think the second question of "what is the fundamental reason" is normally answered, especially for complex systems. Often, instead, statistical properties are studied (i.e. thermodynamics, etc.).

In a theoretical sense, no, you do not need to test different initial conditions to know there is a difference. A system inherently has a degree of chaos. There will always be a question as to how big of a change in initial conditions over how long a time will be detectable. And just looking at a finite number of simulations will never give you a full picture of things. The most obvious difference is between a single and double pendulum system: https://en.wikipedia.org/wiki/Double_pendulum

As I understand, rigorously exploring the "fundamental reason" a system behaves a certain way requires quite a bit of work. See: How to study this highly nonlinear and coupled suit of ODE in network application. I am no expert in this, though.

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Quote from Exploratorium:

Deterministic chaos, often just called "chaos", refers in the world of dynamics to the generation of random, unpredictable behavior from a simple, but nonlinear rule. The rule has no "noise", randomness, or probabilities built in.

In Elementary Cellular Automata, Rule $30$ is a Class $3$ rule, displaying aperiodic and chaotic behaviour. Some ECA rules gives rise to repeatable patterns, others don't. Some produce complex patterns and others random ones.

If you scroll down this page, you see plots of different ECA rules with random initial conditions: https://www.johndcook.com/blog/automata-plots/

In some of these rules, there is no reversible process to get back to the initial conditions, because information is lost. There is a way to get back to the initial condition in Rule $30$, but you need to know how many steps it took to get there. There's a thing called Computational irreducibility, which is one of the main ideas from Stephen Wolfram. This means its almost impossible to predict what the future states of some system will be. This also makes it really difficult to do the same in reverse.

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Taking a somewhat educated guess, it seems like you misunderstood the following: Chaos is not the existence of uncertainty, but a specific kind of consequence of uncertainty.

If we are uncertain about a system’s state, this says nothing about whether the system is chaotic (or how “strong” the chaos is). Rather, chaos is about how an uncertainty evolves over time (assuming no new information). Where this uncertainty comes from does not matter. In a regular system, the uncertainty decreases or does not largely change in magnitude; in a chaotic system, the uncertainty increases exponentially with time.

Some examples to illustrate this:

  • If I excite a real pendulum with little friction, I can predict that after eight swings, it will have roughly the same position and speed as when I started it. If I know the details of the pendulum (mass, length, etc.), I can compute how long a swing will take and maybe even how much energy the pendulum loses in that time. I am as uncertain about its initial state than about the state after eight swings (zero Lyapunov exponent). The system is not chaotic.

    Moreover, if I let the pendulum swing for a long time, it will lose all its energy due to friction and I can predict that it will rest in its lowermost position. This is an accurate prediction that does not require any knowledge about the current state. Thus uncertainty decreases over time (negative Lyapunov exponent).

  • We usually cannot predict the weather for more than a few days (depending on location and current weather). The reason for this we are uncertain about the current state of the atmosphere/planet and this uncertainty increases over time (positive Lyapunov exponent). The longer we want to predict, the less accurate our predictions become. The weather/atmosphere is a chaotic system.

While I used real examples here, I can apply the same concept to simulations: If I simulate a pendulum or double pendulum, small changes in the initial conditions have different long-term effects. Where these uncertainties come from is irrelevant. Chaos is a property of the model, not of my measurement. A chaotic system is chaotic independently of whether it is measured or not. Uncertainties in measurement or realising initial conditions are just the typical reason why we are uncertain about the state of real systems and thus the reason why chaos may pose a problem.

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  • $\begingroup$ Your guess is correct, but I have the same understanding as your bolded text. My question is about making sure that my assumption (that inability to replicate physical initial conditions, or rounding errors which are different between computers/codes (and therefore lead to different results that get worse with time) drive the disparity between results across different runs/experiments). About your last sentence - chaos is a property of the model, but it is driven by measurements. $\endgroup$
    – WoJ
    Jul 13, 2021 at 13:03
  • $\begingroup$ @WoJ: The parentheses in your second sentence are broken to the extent that I cannot understand it. $\endgroup$
    – Wrzlprmft
    Jul 13, 2021 at 13:06

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