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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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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Chaos synchronization & Desynchronization- terms and concepts

I am new to the world of chaos theory & control. I am reading the paper, "Synchronization of chaotic systems " https://aip.scitation.org/doi/full/10.1063/1.4917383 I have some basic questions ...
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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 ...
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Minimal dynamical system with quasiperiodic oscillations

What is a minimal, explicit dynamical system (as in, a series of coupled ordinary differential equations) that exhibits quasiperiodic oscillations for some region of parameter space? Two coupled Van ...
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Sensitivity to initial conditions and predictability

Are there examples of chaotic systems that are predictable and at the same time sensible to initial conditions? or would that violate the notion of sensibility to initial conditions? Lets imaginge A ...
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Discontinuities in a Poincare map for a double pendulum

I'm generating poincare sections of a double pendulum, and they mostly look okay, but some of them have weird discontinuities that seem wrong. The condition for these sections is the standard $\...
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Are my voltmeter readings expected for a correct chua's circuit?

I am following this article Robust OP Amp Realization of Chua's Circuit by Kennedy to implement a Chua's circuit. I use exactly the same design as the article; the only difference is that I use a 15mH ...
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Can the onset of atrial fibrillation be compared to the onset of chaos in a dripping tap?

Atrial fibrillation, roughly speaking, starts when the end of a complete cycle of a heartbeat overlaps with the beginning of the next heartbeat, which makes the heart behave in a chaotic way (...
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Chua's Circuit: an inequality ensuring that the equilibrium is not stable

According to Kennedy's Robust op-amp realization of Chua's circuit(1992), the differential equations satisfied by several physical quantities in Chua's circuit are $$\begin{aligned} C_{1} \frac{d v_{...
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Why is the angular momentum of a 3D kicked rotor non-negative?

We know that for a 2D kicked rotor the angular momentum quantum number can be any integer from minus infinity to infinity. However, for a 3D kicked rotor this is not the case: it can only be positive ...
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What does the Ostrogradsky instability have to do with stability?

Ostrogradsky's instability theorem says that under some conditions, a system governed by a Lagrangian which depends on time derivatives beyond the first is "unstable". In the proof, one computes the ...
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Double pendulum Poincaré section issues

I'm generating high-resolution Poincaré sections of a double pendulum, and I'm running into some issues in creating the initial conditions for a given section. In general, I describe my pendulum with ...
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In what sense do bifurcations concern change in quality?

I've heard such vague statements several times and also read: Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family. (From ...
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Master equation for the mechanical modes

Consider the standard model of optomechanical systems with a single optical cavity mode coupled to a mechanical oscillator, which is canonically modeled as a FP cavity with one fixed mirror and one ...
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Can a perfectly symmetrical round bead dropped into a perfectly level Galton Board indefinitely balance on a peg?

Probability implies that "The Galton Board consists of a vertical board with interleaved rows of pegs. Beads are dropped from the top and, when the device is level, bounce either left or right as they ...
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Can someone show me the chain of events which will lead to a faraway hurricane when I clap my hands (leaving everything else the same)? [duplicate]

I've been thinking about the butterfly effect since I asked my first question on this site, which was different because here I ask for showing me the mechanism for the "how". I even gave one (the ...
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To see if a system behaves chaotically does one have to vary (in a tiny way) the initial momenta of ALL constituents of the system?

Consider a deterministic system (a gas, a liquid, or a solid, each of which can have an arbitrary form; for example, the atmosphere, a waterfall, or a double pendulum) which consists of a huge number ...
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Why are marginal eigenvalues of Jacobian of a periodic orbit related to the symmetry?

In ChaosBook, at page 61 of the unstable version of the book, it is stated that $$J_p (x) \mu (x) = \mu (x,)$$ i.e the velocity vector is an eigenvector of the Jacobian along periodic orbit $p$ ...
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What is the general definition of thickness of a strange attractor?

Disclaimer: This question is cross posted on Math.SE because I don't know which site is more appropriate for this question. In Chaosbook, at page 56, it is asked to find the thickness of Rössler ...

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