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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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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1answer
43 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
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1answer
35 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 $p$ ...
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1answer
45 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 ...
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4answers
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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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1answer
48 views
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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0answers
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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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1answer
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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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1answer
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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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1answer
61 views
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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2answers
106 views
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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1answer
38 views
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?
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0answers
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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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0answers
30 views
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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1answer
48 views
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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0answers
43 views
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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1answer
92 views
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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47 views
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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1answer
1k views
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 ...
2
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2answers
113 views
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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1answer
51 views
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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1answer
43 views
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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2answers
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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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5answers
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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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1answer
327 views
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 ...
4
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1answer
111 views
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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1answer
34 views
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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1answer
31 views
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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0answers
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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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1answer
37 views
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 ...
2
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1answer
59 views
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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0answers
34 views
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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1answer
51 views
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:$$
\begin{cases}p_{n+1}=...
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3answers
87 views
Is Lyapunov function the ultimate method to assess the stability analysis of a system?
I have migrated from physics to electrical engineering and I'm seeing people in control admire Lyapunov methodology and control designs as if there is no other solutions and they consider it very sane ...
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1answer
77 views
Unpredictability, per definitions of chaotic behavior
Apparently I've been confused about the meaning(s) of "chaotic behavior". I always thought it meant that infinitesimal perturbations of a system parameter would lead to large changes in the system's ...
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1answer
156 views
Can a linear system be chaotic?
A chaotic system is a system in which infinitesimal perturbations of a parameter can result in large changes in the behavior of the system. I thought it is not possible for a linear system to exhibit ...
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1answer
30 views
Extreme values in dynamical systems
The unpredictability of chaotic systems can lead to values of physical quantities that peak up to an extremely high value for a short time. This holds also e.g. for economic dynamic systems in Terms ...
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1answer
126 views
A question over Liouville’s Theorem
I have some doubts about Liouville theorem, probably its just something conceptual.
So: I know that for a system in which Liouville’s theorem holds, the volume in the phase space is conserved.
But ...
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0answers
27 views
Proof of factorization at late times for chaotic systems
While reading the paper "A bound on Chaos - Maldacena et. al", https://arxiv.org/abs/1503.01409 in equation (23) of the paper they factorize a correlator of the form,
$$ Tr [\rho^{1/2} W(t) V \rho^{1/...
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3answers
240 views
Physical intuition behind Poincaré–Bendixson theorem
The Poincaré–Bendixson theorem states that: In continuous systems, chaotic behaviour can only arise in systems that have 3 or more dimensions. What is the best way to understand this criteria ...
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1answer
67 views
Apparent emergence of conserved quantities in non-integrable systems
This question arises from the comments relevant to the post When is the ergodic hypothesis reasonable?
Consider a Hamiltonian system having more effective degrees of freedom than conserved quantities....
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2answers
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When is the ergodic hypothesis reasonable?
Consider an Hamiltonian system.
In which circumstances is it possible to assume that all the states belonging to the hypersurface $H=E_0$ are equally visited?
Is it necessary to have a very high ...
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1answer
58 views
Physical significance of orbital stability
I saw the orbital stability in Wiki, I just understand it from mathematics angle. But in physical, what is its mean? Since I saw many paper talk about the stability of Schrödinger equation, I think ...
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1answer
133 views
Why do the laws of physics fail to predict the behavior of frustrators? [closed]
This is my attempt to make an earlier question less broad. This question takes the form of a thought experiment, and is based on this video.
Suppose you are given:
The positions, velocities, ...
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0answers
104 views
Wigner 's unreasonable effectiveness of mathematics in natural sciences [closed]
This question is related to Wigner's problem, related to the unreasonable effectiveness of mathematics in natural sciences. Understanding a phenomenon means constructing a mathematical model , and ...
2
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1answer
98 views
Why do punctured balloons fly around chaotically?
If an inflated balloon is punctured, it can fly around wildly like in this cartoon @18:07.
Why is this motion so chaotic as opposed to being like a straight line or parabola as with rockets? Is ...
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1answer
60 views
Why do my calculation of Lyapunov exponents strongly depend on the number of iterations?
I have a project in my school so I have to calculate my arranged double pendulum system's Lyapunov exponents, I refer to this method.
http://sprott.physics.wisc.edu/chaos/lyapexp.htm
As the title ...
2
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0answers
126 views
Why is information conservation not restricted by the uncertainty principle?
The idea of information conservation seems to be: if all field equations/states of all particles/matter/waves at a certain time are known, all trajectories/waves can be backpropagated to retrieve all ...
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2answers
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Is the trajectory of a particle with constant velocity (though its direction can change by collisions) always non-chaotic?
Suppose we have a particle that travels with constant velocity, without heat losses by friction, and no forces acting on it except for occasionally collisions with much bigger wall-like masses than ...
6
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1answer
91 views
Simplest model of chaos with time-independent smooth Hamiltonian and trivial topology?
What is the simplest model of chaos governed by a time-independent smooth Hamiltonian on a phase-space with trivial topology?
We know that...
With trivial topology, the minimal number of dimension ...
