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.
293
questions
0
votes
1answer
21 views
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 ...
3
votes
0answers
33 views
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 ...
0
votes
0answers
34 views
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 ...
0
votes
1answer
24 views
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 ...
0
votes
1answer
28 views
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 ...
4
votes
2answers
55 views
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 ...
1
vote
1answer
52 views
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}^...
1
vote
2answers
101 views
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 "...
10
votes
2answers
367 views
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 ...
5
votes
2answers
906 views
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 ...
0
votes
1answer
47 views
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 ...
1
vote
1answer
68 views
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 ...
-4
votes
4answers
210 views
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 ...
0
votes
3answers
72 views
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?
6
votes
2answers
60 views
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 ...
1
vote
1answer
110 views
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 ...
2
votes
0answers
23 views
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 ...
1
vote
0answers
34 views
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 ...
0
votes
1answer
44 views
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 ...
-2
votes
1answer
39 views
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 ...
0
votes
1answer
135 views
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 ...
7
votes
5answers
200 views
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 ...
0
votes
1answer
32 views
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 ...
3
votes
2answers
133 views
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 ...
3
votes
2answers
146 views
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, ...
0
votes
1answer
42 views
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 ...
8
votes
1answer
200 views
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 ...
1
vote
0answers
39 views
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 ...
2
votes
3answers
44 views
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 ...
2
votes
2answers
114 views
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 '...
2
votes
2answers
117 views
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 ...
2
votes
1answer
54 views
(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 ...
3
votes
1answer
51 views
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 ...
1
vote
1answer
102 views
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 ...
1
vote
1answer
32 views
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 ...
2
votes
1answer
35 views
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 ...
1
vote
1answer
110 views
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 $\...
1
vote
0answers
31 views
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 ...
0
votes
0answers
25 views
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 (...
3
votes
0answers
59 views
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_{...
0
votes
0answers
18 views
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 ...
6
votes
1answer
231 views
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 ...
0
votes
1answer
89 views
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 ...
0
votes
1answer
43 views
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 ...
0
votes
1answer
120 views
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 ...
0
votes
2answers
42 views
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 ...
-6
votes
2answers
127 views
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 ...
2
votes
1answer
48 views
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 ...
2
votes
1answer
43 views
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 $p$ ...
1
vote
1answer
56 views
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 ...
