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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Calculating the Lyapunov exponents for the seir epidemic model
I am trying to numerically calculate the Lyapunov exponents for the seir epidemic model given as:
$$
s^{'} = b - bs - \beta si \\
e^{'} = \beta si - (\alpha + b)e \\
i^{'} = \alpha e - (\gamma + b) i ...
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Chaos of the Duffing oscillator: Where's the third dimension?
It's often said that all continuous chaotic systems must have at least three dimensions of phase space. The Lorenz system has three explicitly, the double pendulum has four (two angles and two angular ...
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Deterministic and stochastic chaos
I have a question about chaos, but first a foreword of what I understood.
Noise refers to the random variation of values. Usually unwanted, noise causes a measurement to fluctuate over time.
Chaos ...
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Eigenvalues of system with 5 or more degrees of freedom?
When finding eigenvalues for a system consisting of a single particle, its position and velocity are used when making the system of equations. So that there is an equation like
$\dot{x} =\begin{...
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1answer
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Analytical expression for density of random matrix level ratios
Consider a hermitian matrix $H$ with eigenvalues $E_{i-1}<E_i$. The level spacings are defined as $s_i=E_i-E_{i-1}$ and the level ratios as $r_i = s_i/s_{i-1}$. To make the support of an underlying ...
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Relative Phase distribution for two coupled van der Pol oscillators where one has a Drive?
I was solving the dynamics for a driven coupled (inertial) van der Pol oscillators, where only one oscillator is driven. I started with the complex amplitudes $\alpha$ for both of the systems which ...
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Computing correlation between two time series: confusion regarding nonlinear relationship and nonlinear data
I am trying to understand if correlation can be computed between two time series generated from two different initial conditions for chaotic dynamical systems. In general, correlation is applicable ...
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What is a uniform measure over a unit square?
Going through some reading material on two-dimensional maps, I came across the following text in reference to the Baker's map:
"Consider the map:
$$
x_{n+1} =
\begin{cases}
\lambda_a x_n & \...
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What are the implications of deterministic chaos: useful or detrimental? [closed]
I am new to the concept of chaos theory and as a layman I am struggling to understand what is the significance and implication of chaos in ecological systems such as the chaotic predator prey model. I ...
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Are chaotic systems examples of complex systems?
I am struggling to find a proper source or reference where examples of complex systems which are chaotic are given. Based on my understanding, complex systems consist of interacting components, each ...
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Poincaré maps and how they are used in dynamic systems [duplicate]
I am trying to find a solid definition for what a Poincaré map is and how they are used but the ones online are very complicated. Can someone explain with a simpler definition of what a Poincaré map ...
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Thermalization of perfect incompressible fluid
Suppose we have an perfect incompressible fluid (no viscosity) in a box. Lets assume we shook the box and let the fluid sit for a long time. Since there is no dissipation, the energy is conserved. Is ...
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Is consciousness likely to be a result of chaotic, or random processes? [closed]
Given that just about everything in the universe at a non quantum level is deterministic, what does this say about free will and consciousness? Are these likely to be a result of randomness at the ...
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Lyapunov Exponent of the Logistic map [closed]
My dynamical system professor (and the textbooks we use) all claim that the Lyapunov exponent for the Logistic map with $r=4$ ($x_{n+1} = 4x_n(1-x_n)$) is $\log(2)$. Would someone be able to sketch ...
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1answer
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“Non-analytic interaction”…what does it mean?
Reading an article about Hamiltonian chaos, I found this passage:
Importantly, the few Hamiltonian systems for which the KAM theorem
does not apply, and for which one can prove ergodicity and the
...
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Chaos Bound in BCFT
What is the Chaos bound in boundary conformal field theory (In 2d or in higher dimension)? can one derive much restricted bound in these settings?
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When is molecular chaos dynamical chaos?
It is very common to have uncorrelated velocities in chaotic dynamical systems. Yet, we should be wary in equating the two quite different meanings of chaos.
Instead of matching dynamical chaos to ...
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Uncertainty propagation in dynamical systems
I'm not a physicist, my training is in math and CS. If anything in this question is ill defined or doesn't make sense, say so in the comments and Ill try to fix it.
Suppose I have a discrete dynamical ...
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Non-Analytic Equations and Chaos
Could anyone please tell me an example of an equation with no analytic solution(s) that is not a chaotic one? And what is the physical meaning of having analytic solution? For instance, the three body ...
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Generalizations of Feigenbaum universality for multidimensional maps and ones with multiple order parameters
Feigenbaum showed that for discrete 1D dynamical systems with a (smooth) unimodal evolution function, the route to chaos is universal, and depends only on the order of the map's maximum. (I'm told ...
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2answers
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Can phase trajectories intersect for non-autonomous system?
There has been enough discussion about intersection of phase trajectories in autonomous system,where the system wasn't time dependent. And we came to the conclusion that, at a point in space there can'...
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2answers
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Is the universe's Kolmogorov complexity growing over time?
The Kolmogorov complexity of a deterministic universe is constant.
The Kolmogorov complexity of a nondeterministic universe grows over time. It grows whenever something happens that is not ...
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Solutions to Chen System/Attractors
I have a problem involving the new chaotic system dubbed as the Chen System. This involves a system of coupled nonlinear ordinary differential equations. My problem is to determine for which ...
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3answers
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What causes water droplets to drop in periodic, but not uniform time intervals?
A little while ago I noticed water droplets forming from a slightly overflowing reserve in my sink. They dropped in a special periodic time pattern, which was not uniform. Instead two droplets would ...
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Level spacing distribution
I have started learning level spacing distribution and it says that level spacing distribution for classically regular system show poison curve. But is it valid for the integrable system as well, ...
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1answer
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What prevents chaos theory from being the principle of a deterministic universe?
One could potentially suggest observations of atomic locations partly with chaos theory by suggesting that the seemingly random pattern simply results from a sufficiently small Lyapunov time that ...
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Semiclassical quantization of chaotic classical system
So far in the introduction of quantum chaos, I have read that in the early day's physicists quantized classically chaotic systems but could not find any signature of chaos in quantized systems. My ...
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Can we have chaotic motion due to the finite precision of our calculations? [duplicate]
I understand chaotic motion to mean that very small perturbations in the initial starting condition can lead to very different trajectories in phase space. For this reason, we can never predict the ...
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3answers
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How do we know chaotic systems are actually chaotic and not periodic?
The usual way to calculate how chaotic a system is would be to measure the divergence rate using the Maximal Lyapunov exponent, but it requires you to wait until infinity, measure the divergence, then ...
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How to compute the connected spectral form factor for integrable models?
Given a spectrum of $N$ real eigenvalues, $\{E_m \}$ of some Hermitian operator, the connected spectral form factor is defined as follows:
\begin{align}
K_c(t) = \langle \sum_{m,n=1}^N e^{it (E_m-E_n)}...
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How does a discrete map (logistic map being only one such map) relate to a Poincare section?
Basically the title. I am wondering if there is a relationship between the two, and if so, what is it?
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Is there a mathematical way to determine if a force, phenomena or physical entity is in a state of chaos?
We often talk about chaos, but is chaos an objective term or a subjective term? If it's an objective term, is there a mathematical way to determine it? Is it possible there's a threshold where ...
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Probabilities for quantum random number generators
Consider a quantum random number generator (QRNG) X, which generates integers at random.
(Apparently, due to quantum statistical properties, this type of generation is truly at random, see e.g. "...
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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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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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1answer
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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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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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 ...
