What is the difference between Noise and chaos?

Question

Two terms are mixed me , I heard about noise from stochastic process phenomena and i heard about chaos from dynamical system , then Is there someone who can help me to get difference between noise and chaos ?

Answer 1

Noise refers to the random variation of values. Usually unwanted, noise causes a measurement to fluctuate over time.

Chaos happens when starting the system in a slightly different way will lead to drastically different outcomes.

• The fundamental difference between noise and chaos is that noise is stochastic while chaos is deterministic.

Stochastic means the changes in a system depend on a probability. For example, suppose you were standing on a line and flipped a coin every second. If it was heads you moved right and if it was tails you moved left. You can't predict the future because there's only a chance that you go one way or another. This property usually arises from quantum mechanics, where things are not for certain, but very likely.

Deterministic means that the system will change the same way from the same starting conditions every time. In this way one could predict the chaotic behavior if one were to know all the decimal points on a measurement. However, we cannot have perfect information (also restricted by quantum mechanics), so the tiny immeasurable differences will be amplified until the system is effectively unpredictable.

Reference:

https://ireap.umd.edu/sites/default/files/documents/trend/2016/Kevin%20Fei%20-%20MultimediaPDFout.pdf

Answer 2

The first distinction you have to make is between chaotic and deterministic dynamic systems. Even though this may be surprising at first, these attributes are best applied to model systems, and not to reality or subsystems thereof.

A model dynamics is stochastic if randomness (or noise) influences its temporal evolution. Otherwise it’s deterministic: The initial conditions completely determine the system’s future. Some deterministic dynamical systems can be chaotic, which means that their temporal evolution strongly depends on initial conditions. By contrast other deterministic dynamical systems (fixed-point dynamics or periodic dynamics) do not have this property. With other words:

• For regular deterministic systems, identical initial conditions lead to an identical outcome and similar initial conditions lead to a similar outcome.
• For chaotic (deterministic) systems, identical initial conditions lead to an identical outcome and similar initial conditions (usually) lead to a drastically different outcome.
• For stochastic systems, identical (and similar) initial conditions (usually) lead to a drastically different outcome.

Now, what about reality? Every real system is inevitably subject to phenomena like thermal noise, air turbulence, or similar, which we can only treat as noise¹. Thus the most accurate model for every real system would be stochastic² – but this model is also unreachable and infeasible. Instead, when choosing a model we have to make a trade-off between feasibility and accuracy, where we usually go for the simplest model that reproduces whatever phenomenon we are interested in with sufficient accuracy that we can understand or sufficiently predict it. Depending on the system and our needs, we may arrive at a deterministic or stochastic model.

For example, you would use a deterministic model for a pendulum in most cases, while the most successful model for reality at the microscopic level (Copenhagen-interpretation quantum mechanics) is stochastic. A real system is usually called chaotic if a reasonable model of it is chaotic, for example the double pendulum.

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^{¹ Whether these phenomena are actually deterministic or stochastic doesn’t really matter, since they are extremely high-dimensional and complex and every sufficiently complex deterministic system cannot be practically distinguished from a stochastic one.
² A prominent exception may be the entire universe, but we cannot possibly decide that one. We only know some restrictions of a possible determinism from quantum mechanics.}