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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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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Gravitational effects of a single human body on the motion of planets

(This is going to be a strange question.) How big a difference does the existence (or positioning) of a single human body make on the motion of planets in our solar system, millions of years in the ...
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Are there chaotic maps that commute?

My question is in the title. You can imagine 1D or 2D maps, the simpler the better. Let us say we have chaotic map $T$ and chaotic map $R$. We need that $RT(x(n))=TR(x(n))$.
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Logistic Map on unbounded domains

A lot of natural phenomena can be modeled with Logistic Map or a similar map as they universally show transition to chaos. Logistic Map maps bounded [0,1] -> [0,1] intervals. Is there an analog with ...
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Is it possible for a system to be chaotic but not ergodic? If so, how?

In a recent lecture on ergodicity and many-body localization, the presenter, Dmitry Abanin, mentioned that it is possible for a classical dynamical system to be chaotic but still fail to obey the ...
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How to calculate the parameter values for which the Lorenz system is chaotic?

I was recently going via a book (Strogatz), that mentions Lorenz's attractor, and that it was found out that for values such as $a=10$, $b=\tfrac{8}{3}$, $c=21$, the system behavior is chaotic. How ...
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Unexpected wavelength jumps in a laser observed: a fundamental noise source in VCSELs?

I observed an unexpected signal while developing an interferometric sensor using a VCSEL laser. In order to debug this noise I designed the following device that converts minute changes in the laser's ...
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Shocking examples of chaos theory at work [closed]

Completely out of curiosity: what's the most shocking example of chaos you know? Something that shows (to a non-expert audience) how quickly errors grow in a chaotic system. For example, I was ...
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Poincaré Map (Quasi-periodicity; Stability)

In a Poincaré map, when quasi-periodicity is exhibited by the dynamical system, what does it mean in terms of stability for the dynamical system?. Why is it so that as Maximum Lyapunov exponent (MLE) ...
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Spontaneous synchronization references

Can someone suggest references for an introduction on spontaneous synchronization, theory/examples. I am trying to understand it so I can test it for some problems I am working on. I have no prior ...
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Classical chaos at finite temperature

Is there any finite temperature generalization of classical chaos? In quantum chaos, at least with regards to out-of-time-order correlators, the generalization is clear - one simply takes a thermal ...
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Lieb-Robinson bound and spin chain

I am trying to understand the paper Localized shocks better. There is Lieb-Robinson bound on the page 6. How does formula (7) imply that: the radius of the operator can grow no faster than linearly ...
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Topological shape of the equilibrium point

"All dynamical system possess topological shapes that characteristics it's equilibrium point"-so my question is what is the topological shape of the equilibrium point for a cart and Inverted pendulum ...
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Intuition behind the meaning of Lyapunov exponents

Can anyone help me in understanding the contraction and the expansion of the phase space? what are Lyapunov exponents? and how come one understand this concept intuitively?
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Infinite series vs compact representation

I understand the attractiveness and usefulness of infinite-series expansions such as Taylor expansions, but I wonder if they sometimes hide important aspects of the described system. For example, ...
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Are all aperiodic systems chaotic?

So I understand that a chaotic system is a deterministic system, which produces aperiodic long-term behaviour and is hyper-sensitive to initial conditions. So are all aperiodic systems chaotic? Are ...
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Statistical Mechanics & Dynamical Systems

As a (theoretical) physics student I've taken (advanced) undergrad courses in both statistical mechanics and dynamical systems (which was purely mathematical, treatment of nonlinear differential ...
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Gaining intuition about summing over random basis vectors in random matrix theory

I'm currently reading the following reference on eigenstate thermalization and chaos in quantum mechanics: https://arxiv.org/abs/1509.06411 I'm confused by a derivation that I think is very important ...
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Is the Lorenz attractor a cyclotron?

By using a plotter to output a computer generated strange attractor solution to the Lorenz equation, that draws a line corresponding to the same fixed interval for every time step, it was found that ...
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Are there any known models with limit cycles in their RG flow?

The text-book presentation of the renormalization group (RG) leaves one with the impression that all systems will eventually flow to a fixed point. This is somewhat enforced by the phenomenological ...
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How do computers “solve” the three-body-problem?

I've done a bit of research, and have learned that computers "solve" the three-body-problem by using "Numerical methods for ordinary differential equations", but I can't really find anything about it ...
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Why can't we humans solve the three-body-problem? and why can computers solve it?

Why is it that we humans can't solve the three-body-problem? (calculate the positions of the 3 bodies in a dynamical system) And why can computers do it? My thoughts: Computers do it in way smaller ...
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Bound on Quantum Chaos

I am currently reading the paper A Bound on Chaos. In this paper, they evaluate the quantity C(t), which is an out-of-time-order correlator (OTOC), and use very clever arguments to show that there ...
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Chirikov standard map derivation

This might be a stupid question, but I am having trouble understanding the derivation of Standard map by integrating Hamilton's equation of motion over one period. I am going through this dissertation ...
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If oscillatory motion is not simple (or chaotic), is it then by definition complex?

I'm trying to logically deduce or show that a specific type of motion is complex. It is two-dimensional oscillatory motion that can be expressed by coupled second order non-linear differential ...
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Euler three body problem, what exactly is it? [closed]

I have a question about the 'Euler three-body' problem. I have to write an essay about this subject for the course 'chaos theory', which is about dynamical systems and chaos. Does anybody know what ...
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Application of Correlation Dimension to Fractals other than Sets of Points

In Chaos Theory, the Correlation Dimension is defined to calculate the dimension of fractals. At least in the context where I've learnt it, it is applied to fractals made up of sets of points. Is it ...
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Poincaré plane and Logistic Map

How can we draw Poincaré plane and phase portrait for the Logistic Map for different parameter values?
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Mixing and Entropy in Dynamical Systems

I'm writting a short introductory report about chaos theory, and one of the conditions for a dynamical system to be chaotic seems to be the presence of topological mixing. Now, the document I'm using ...
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Inverse of the standard map

I'm trying to plot the homoclinic tangle that can be observed following the evolution of the unstable and stable manifolds of the standard map. The map I am using is defined as:$$ \begin{cases}p_{n+1}=...