Questions tagged [chaos-theory]
Chaos theory is the study of systems that are highly sensitive to slight, even imperceptible changes in initial conditions. This is popularly known as the butterfly effect. Many natural systems exhibit chaotic behavior, including weather and electron orbitals.
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Applications or work that has been done in Non-Linear dynamics or Chaos
I have almost finished my Non-Linear Dynamics course.
I'm really interested in working on this field but first I want to see and study some of the work or application that has already been done in ...
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How to Calculate Lyapunov exponent from time series data?
I have time series data which I got from correlation function. I want to calculate maximum Lyapunov exponents but I couldn't get it?
Like this, I have computed correlation function (time series) data ...
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The “real butterfly effect”
This question stems from the confusion that I feel after reading this popular blog post by Sabine Hossenfelder. It is based on this paper which is paywalled, unfortunately.
The claim is the following:
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Why is the predictability of the solar system in the Lyapunov timescale limited to 5 million years?
Is this due to a mathematical problem that is not solved? Or could this be due to our current amount of information regarding mass and other such factors in our system?
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How to prove that volume remains zero under a state transformation
The lecture note titled "11 Strange attractors and Lyapunov dim." taken from the book of Strogatz shows in eq(2) a coordinate transformation of the volume. I want to understand how to prove ...
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Three body problem - Half Life
I did google this up but found nothing! I can't be the first to ask (the vague question)
"What is the half life of a gravitating three-body system?"
CLASSICALLY this means: Say I have three ...
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Is a satellite orbit around the Earth Lyapunov stable?
Presume there is a satellite orbiting the Earth in an orbit that follows a closed path around the planet (that is, escape orbits are not permitted here). As I understand it, there are two ...
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By what equation the future temperature predictions are made? [closed]
I'm a Ph.D student in plasma physics but interested in learning physics behind climate modeling and predictions. I want to begin with studying what equation is used for the future temperature ...
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Why do chaotic numbers improve evolutionary algorithms such as genetic algorithm?
I have implemented a genetic algorithm to solve a problem. In the process of genetic algorithm, instead of random numbers, I have used the chaotic numbers generated by the logistics map. The genetic ...
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One third of Lyapunov exponents are zero? What does it mean?
This may be quite a straightforward question, but I have a dynamical system with a high dimensional phase-space. I calculated the Lyapunov spectrum for it and saw that one third of my Lyapunov ...
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Does this quote from the TV show Devs confuse chaos theory with quantum theory?
In the dam scene in Episode 7 of Devs, one of the characters says:
A few moments from now, you climb over this rail, you stand on the other side and balance there, right on the edge of the dam. ...
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Chaos and Ergodicity in Hamiltonian Field Theory?
In classical mechanics, one intuitive formulation of chaos/ergodicity (in the loose sense) is that most trajectories should fill up phase space densely over infinite time. A classic example of such a ...
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Non-Integrable models in 1+1D
Is it possible to have a non-integrable system in (1+1)D in Classical Physics? For some reason, I get the intuition that there shouldn't be any such systems. What if we consider (1+1)D systems in ...
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Is the process of universe creation a chaotic system?
The anthropic principle says that:
The laws of nature and parameters of the universe take on values that
are consistent with conditions for life as we know it rather than a
set of values that ...
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Can an affine first-order polynomial system be chaotic?
While studying chaos theory, one of the basic principles presented to me was that chaos only occurs in deterministic nonlinear systems. This pointed me to learn more about the differences between ...
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About natural frequencies in non-excited pendulums and Poincaré sections
How can a Poincaré map be defined for a double pendulum (or Furuta pendulum) when these systems don't have external excitations?
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Destruction of integrals of motion in chaotic systems: Fermi-Pasta-Ulam (FPU) paradox
I am trying to understand behavior of system studied by Fermi, Pasta and Ulam i.e. chain of oscillators interacting via nonlinear forces. I am generally not very familiar with chaos theory and ...
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Lorenz System in reference to Astrophysics / Planetary orbits
From my research I have found that there are a system of ordinary differential equations for atmospheric convection.
What I am seeking are any Lorenz equations that apply to any areas of Astrophysics ...
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Studying Chaos in RLD circuit
We are currently working on non-linear dynamics (chaos theory) by analysing a series circuit including a diode (the 1N4004), a 100 ohm resistor and a 20 mH inductance. It is driven by an alternative ...
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Unstable sets in a chaotic attractor
I am having a hard time understanding the discussion of chaotic sets on invariant manifolds as given in Chaos in Dynamical Systems by Edward Ott.
If the invariant manifold of a particular system ...
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Is instability + sensitivity to initial conditions = Chaos?
Please correct me where wrong. I am having trouble finding answers to these specific questions.
(1) In chaotic systems, does the presence of chaos and a strange attractor indicate that there is no ...
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Why would we want to calculate the Lyapunov exponent for experimental data?
Searching Google Scholar for "Lyapunov exponent from time series" turns up multiple papers (some of them highly cited) suggesting methods for estimating the largest Lyapunov exponent or sometimes even ...
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Chaotic Hamiltonian system poincare surfaces depend on the integrator
First question on StackOverflow so go easy on me. I have a Hamiltonian system that consists on the following Hamiltonian:
$H(p,x;\textbf{P,X})=\frac{p^2}{2m}-a\frac{x^2}{2}+b\frac{x^4}{4}+x\sum_{n=1}^...
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Earth's orbit: chaotic but stable
The eccentricity of Earths orbit follows a bounded random walk-like pattern, see this chart. I presume most other planets are similar. One could think of eccentricity and argument of periapsis as "...
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Dynamicity inside a stationary water drop
I was doing some experiments with water drops on lampblack when I saw this.
You can see the full video here.
Inside a water drop which is perfectly still from the outside, you can see some moving ...
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How are jerk equations connected to chaos theory?
I read in this Wikipedia article:
It has been shown that a jerk equation, which is equivalent to a system of three first-order, ordinary non-linear differential equations, is the minimal setting ...
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Path between fixed points in logistic map
I have a question about period doubling and fixed points in the logistic map. Let's say I have a basic logistic map, $$f(x) = 4\lambda x(1-x).$$ Let me then compare 1,2 and 4 iterations of this map on ...
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A subtlety about Lyapunov stability of stationary rotations of rigid body
On Page 145 of Arnold's mechanics book there is the intermediate axis theorem:
"The stationary solutions of the Euler equations corresponding to the largest and smallest principal axes [of the ...
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What is the chain of cause(s) and effect(s) which does a butterfly's wing-flapping cause a hurricane 1000's of kilometers further away?
I once asked if the butterfly effect in the weather system is a real effect: a butterfly flaps her wings, which can cause a hurricane thousands of kilometers further.
But how will this happen? To be ...
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Closed form description of a chaotic system
What's the simplest (or at least, a simple) chaotic system which can be described in closed form?
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What effect stabilizes chaos by randomness?
Reading the book Antifragile-Nassim Nicholas Taleb, I encountered the following paragraph.
And ironically, the so-called chaotic systems, those experiencing a brand of variations called chaos, can ...
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The Physics behind “The Wall” Game show ball drop
In "The Wall" game show, the slots and also the diverters are designed symmetrically and also identically. So when a ball is dropped from a particular slot number it should end up in a particular ...
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Thermalization in non-disordered systems
The eigenstate thermalization hypothesis explains the mechanism of the thermalization of generic many-body quantum systems. The presence of disorder, on the other hand, provides an elegant example of ...
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What can one conclude about the stability of limit cycles without the use of numerical methods?
Let's assume one asserts the existence of a closed orbit by applyling the Poincaré-Bendixson theorem to a trapping region $R$ that is constructed such that all phase vectors on its boundary point ...
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Is the motion of a spinning top chaotic?
When you spin a top, while it first wobbles, it will eventually reach a period when it spin 'smoothly', then finally falls to the ground. I'm curious about whether its motion (or maybe the path that a ...
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Lorentz Equation Symmetry
I was going via Lorentz equation & learning the topic on Symmetry, what I couldn't understand is how did they performed this type of substitution & what is the philosophy behind this way of ...
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Is the three-body problem always chaotic?
I was reading this recent article in Forbes about the fact that relativistic problems can't be solved exactly. In it the author makes the argument "the two body problem has an exact solution, so all ...
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On understanding the relation between Heisenberg Uncertanity Principle, Indeterminism and classical chaos
I recently read Emperor's New Mind by Roger Penrose. In it he talks at length on the notion of determinism in science. In this context how does the Heisenberg Uncertanity Principle bring about the ...
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What is the classical counterpart of ultra-cold atoms?
I'm studying quantum chaos in ultra-cold atoms. However, quantum chaos denotes the quantum mechanics of classically chaotic systems and it is not clear to me what is the classical counterpart of ...
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Is it possible to quantify how chaotic a system is?
In relation to this other question that I asked:
Is there anything more chaotic than fluid turbulence?
I had assumed that there are methods by which the level of 'chaotic-ness' of a system could be ...
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Is there anything more chaotic than fluid turbulence?
Fluid turbulence is a highly complex and non-linear chaotic phenomenon. Great difficulties and complications are encountered when trying to accurately and robustly calculate or simulate fluid flows, ...
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What is a “Doppler instability”?
In the paper "Flow-induced control of chemical turbulence" by Berenstein and Beta, the term "Doppler instability" is mentioned in the context of the Belousov-Zhabotinsky reaction.
I am looking for a ...
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What do physicists mean by an “integrable system”?
The notion of "integrability" is everywhere in physics these days. It's a hot topic in high energy theory, atomic physics, and condensed matter. I hear the word at least once a week, and every time, I ...
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Solar system and $n$ body problem [duplicate]
How is possible for the solar system to be stable if has been proved that even the trhee body problem is chaotic with a very small amount of stable solutions?
Do we need to consider the solar ...
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Does the butterfly effect apply to models intended to be long-term?
We know that complex models, especially for the atmosphere, are likely to be subject to the butterfly effect, meaning that small variations in initial conditions may result in very different states in ...
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Normal diffusion and dynamical chaos
Are there any central results/theorems which concern the implication that a dynamical system which is chaotic (in the sense of a largest positive Lyapunov exponent) will exhibit normal diffusion? By '...
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Arnold's Mathematical Methods of Classical Mechanics and Lyapunov stability
In Arnold's Classical Mechanics of Classical Mechanics, he refers to Lyapunov stability in many of the problems in the second chapter.
E.g. on page 20: "Problem: Consider a periodic motion along the ...
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(Why) Is there only one Lyapunov exponent?
Lyapunov exponents describe how two (infinitesimally) close initial conditions behave (exponentially) in the long run. If a system is chaotic, the largest Lyapunov exponent is positive. However, as ...
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Why is chaos a common property of thermal systems?
https://arxiv.org/abs/1811.06949 pg 3 mentions that chaos is a common property of thermal systems. Can someone please explain why that is?
While looking at [1], I found that indeed most examples ...
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Difference between time series and trajectory terminology
What is the difference between trajectory and time series? To me both seem the same thing.
In the 3D diagram (cube picture on left of Fig.2 from the paper titled “Review and comparative evaluation of ...
