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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How to understand the largest Lyapunov exponent?

Some more information and answers are here: https://math.stackexchange.com/q/4451013/577710 . It is said that ..the largest Lyapunov exponent, which measures the average exponential rate of ...
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Are all turbulent flows chaotic?

I have often read that all turbulent flows are chaotic. I have come across a few explanations on web forums, but are there any authoritative references/textbooks that explicitly cover this question?
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The unlikely marriage of orbiting bodies

O.k. so we have an orbiting body such as the Earth around the sun or the moon around the Earth. The fact that they are orbiting does not fascinate me, it is my intuitive sense that there is a much ...
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Are there any symmetries in the Mixmaster model?

The Mixmaster model is usually presented as one example of chaotic models in physics and cosmology. Usually, "chaos" or "randomness" may be related to a complete lack of symmetries ...
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Estimation of number of states to be used to obtain Wigner-Dyson distribution in a chaotic coupled oscilllator

A coupled harmonic oscillator with quadratic coupling - $$ H = \frac{1}{2}(p_x^2 + p_y^2) + \frac{1}{2}(x^2 + y^2) + g x^2y^2, $$ is known to be non-integrable, hence chaotic (for reference look at ...
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Does Wick theorem hold for ensembles of random unitary matrices?

Consider ensembles of random unitary matrices with weight function $w(\phi)$, which have partition functions of the form $$ Z= \int_{U(N)} w(U)dU = \int \prod_{i=1}^N \frac{d \phi_i}{2\pi } w(\phi_i)\...
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How to understand Heisenberg time in random matrix theory?

Recently, from few papers, I have encountered the word 'Heisenberg time' $t_{\text{H}}$ which is an inverse of a mean level spacing $\Delta(\hat{\mathcal{H}})$ of a finite system Hamiltonian $\hat{\...
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Graphing the motion of a driven damped pendulum

$ \ddot{\phi}+2 \beta \dot{\phi}+\omega_{0}^{2} \sin \phi=\gamma \omega_{0}^{2} \cos \omega t$ The equation of movement is written above. I´m trying to plot the value of $\phi$ for different values of ...
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Calculating the Maximal Lyapunov Exponent for a Hamiltonian System

I am consider the following Hamiltonian: $$\mathcal{H} = \frac{1}{2}(\dot{x}^2 + \dot{y}^2) + \frac{1}{2}(x^2 + y^2) + x^2y - \frac{y^3}{3}.$$ The first step I took were to solve the equations of ...
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What is the Smaller Alignment Index (SALI)?

I was reading the following paper where the use the Smaller Alignment Index (SALI) to determine when orbits are chaotic and when they aren't. I was confused on what the deviation vectors are and how ...
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Advection Term in the Lorenz 96 Model

The Lorenz 96 Model is defined as $$\frac{dx_i}{dt}=\underbrace{(x_{i+1}-x_{i-2})x_{i-1}}_{advection}-x_i+F$$ with some forcing $F$ and periodic boundary conditions so that $x_{i+N}=x_i$ for some $N$. ...
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Can the phase portrait of SHO rotate counter-clockwise? or is it the case that there can be no physical motion corresponding to that?

Framing the question In the case for Simple Harmonic Oscillation, we have the equation: $$\ddot{x}+x=0 \tag{1} \label{1}$$ (say, we put all the coefficients to be 1) Now, if we try to solve it in ...
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Is a coin toss pseudo-random or truly random? [duplicate]

I wonder if a coin toss is pseudo-random or truly random. Sure, you could say that a coin toss is pseudo-random because you don't know the speed of the coin or its rotation, but if you were to include ...
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Calculating Lyapunov exponent (LE) for pendulum using ellipsoid growth - code yields negative LEs

Per my advisor, I have read the textbook Chaos, an introduction to dynamical systems by Alligood, Sauer, and Yorke. (side not, I have really enjoyed this book). In chapter 5, the numerical calculation ...
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Phase space portrait for dynamical system with Bifurcations

I have this dynamical system $$x'=y, y'=-x^3-y+mx$$ and I want to draw the phace space diagram for $m=-1/8, m=1/4,$ the bifurcation points. 1st of all I cant find what kind of bifruction I have( I go ...
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Is pendulum only chaotic when its force is a single sinusoidal frequency?

Chaotic systems will be chaotic under certain parameters, but when the forced, damped pendulum is discussed, the initial conditions of its position and velocity are only mentioned. Is it assumed that ...
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How many flips does a tossed macroscopic coin need to go through until the coinflip's result becomes indeterministic?

A coinflip is a macroscopic event and is deterministic in nature. A coin-flipping machine that operates at the greatest physical precision possible would be able to predict the coinflip's result (...
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Mixing for Burgers equation in 2+1D

Let us consider the following (2+1)-dimensional Burgers-like equation: $$ u_t + (u^2)_x + (u^3)_y=0. $$ Here the unknown is a function $u= u(t,x,y):(0,\infty) \times \mathbb R^2 \to \mathbb R$. Is ...
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Question about energy threshold for bounded and unbounded motion from a research paper

I was reading a research paper titled "Dynamical analysis of bounded and unbounded orbits in a generalized Hénon-Heiles system." The link to the paper is here and on arXiv. In this paper, ...
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A coupled nonlinear dynamical system in four dimensional phase space

I have come across a coupled nonlinear dynamical system given below $$ r\, \ddot{x} + \dot{x} = \sin y~,$$ $$ r\, \ddot{y} + \dot{y} = \sin x~,$$ where $r$ is some real number and $\dot{x}$ denotes $\...
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Graph Interpretation of Fold of Cyclic Bifurcation

I was studying about local bifurcations of cycles, and the basic type of it was the cyclic fold. According to the textbook, cyclic fold bifucation was the meaning of one of those that is, for two ...
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Physical reasons for why systems are chaotic?

Are there any reasons why a system would exhibit chaotic behavior? Or is this something only found through numerical modelling or experimental testing? For example, the simple forced, damped pendulum ...
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What are the conditions for RG flows to have strange attractors?

this question served as a great reference for RG flows that can end up with more complicated dynamics than fixed points, such as limit cycles. As far back as K.G. Wilson's 1971 original RG paper, ...
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What is a "classically chaotic quantum system"?

In the context of quantum mirages one can find increased probability density around paths of unstable classical periodic orbits, called quantum scarring. In this wikipedia article they use the term &...
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Terminology for scenario when energy of system $E(\theta_1,\ldots,\theta_k)$ with $k$ real parameters, is minimum whenever $\theta_1=c$ (fixed value)

Disclaimer. I'm not a physicist. Consider a physical system whose "energy" $E$ is a function of $k$ real parameters $\theta = (\theta_1,\ldots,\theta_k) \in \mathbb R^k$. Let $E_{\min}$ be ...
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The Electron at the End of the Universe

In A Passion for Science, Michael Berry's essay "The Electron at the End of the Universe" poses two scenarios. Assume that a box of gas particles obeys Newtonian mechanics and that we ...
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Why is this argument about free-will flawed/wrong? [closed]

I would really appreciate some thoughts on my argument/thought experiment. I can’t think of a logical defense of the existence of free will in the context of human action at the scale of the ...
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Time taken for a system to return to it's original state

Consider the following system: There are N particles (point-like particles) of $1$ Kg each in a Sphere of radius $R$ centered at origin in three dimensions. Randomly assign these N particles their ...
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Linearization of 1D maps about a fixed unstable point [closed]

Recently, I was going through the paper Controlling Chemical Chaos in a three variable autocatalator system, by Peng et al. Here are the references Although I have been introduced to 1D maps and the ...
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How does object move in bottom of swimming pool? [closed]

Suppose there is an object $O$ (swimming goggles) that has fallen to the bottom of a swimming pool. I have the swimming pool circulation pump turned on. Initially, the object is at some position $P_1$....
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Are chaotic systems the same as dissipative systems in inverse time?

Lyapunov exponents define whether a system expands or contracts in phase space and can be used to determine whether a dynamical system is chaotic, conservative, or dissipative. In the volume expands ...
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Is the motion of a particle in the surface of a torus always periodic?

I am trying to see if there are ballistic trajectories in the surface of the torus that are not periodic and to what extent. Maybe it is not only quasiperiodic but chaotic. I guess there are ...
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How chaotic is the double-pendulum if the arms are not perfectly rigid?

The double pendulum is a famous example of a chaotic system. It consists of one pendulum hanging from the end of another pendulum, which in turn hangs from a fixed point. In the traditional version, ...
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Is there a relationship between quantum physics and chaos theory on a classical scale?

Im a complete physics lay person and I read somewhere that chaotic systems are subject to tiny differences in initial conditions and that the brain is a chaotic system. Does that mean our thoughts are ...
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Is there a Hamiltonian system composed by three particles which is chaotic?

The Henon-Heiles system is the smallest Hamiltonian system where chaos has been observed. Smallest because it is composed by two degrees of freedom. What is a Hamiltonian system with $n=3$ degrees of ...
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Have I spotted real-world properties of chaos theory? [closed]

I know it says no financial questions but I think this is more of a physics question... I'm an artist by profession but I like to study science as a hobby. I was researching chaos theory around the ...
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What is the precise relation between level compressibility and the spectral form factor?

In the study of disordered conductors, the level number variance is defined as $$\Sigma_2 (\langle n \rangle) \equiv \langle n^2 \rangle-\langle n \rangle^2 ~, $$ where angular brackets denote ...
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Is there a general equivalence of spectral correlations between hermitian and unitary random matrix ensembles?

It is common lore that the Gaussian Unitary Ensemble (GUE) and Circular Unitary Ensemble (CUE) have the same spectral correlations as the order $N$ of the matrix goes to infinity, see e.g. section 5.9 ...
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What are the symmetries of circular billiards that makes it integrable?

I have often heard that integrability in is equivalent to extensively many conserved quantities $A_i$, i.e. the Poisson bracket $\{H,A_i\}=0$ or in quantum mechanics $[H,A_i]=0$. What are the ...
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How can one distinguish between a random process and a chaotic process? [duplicate]

Chaos is not a random process, although it may look like one. If I am given a set of observations, is it possible to determine if the observations are generated by a random process or if they are ...
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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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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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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 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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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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