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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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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1answer
74 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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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 ...
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2answers
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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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1answer
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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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1answer
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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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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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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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1answer
27 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
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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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1answer
47 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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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
35 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
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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
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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
144 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
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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
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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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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
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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
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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
54 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
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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
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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 ...
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1answer
93 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 ...
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0answers
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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 ...
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1answer
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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 ...
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2answers
256 views
Difference between unstable fixed point and chaotic point
I am reading the Scholarpedia article on Lyapunov exponents:
Given a dynamical system
$$
\dot{\vec{x}}=\vec{F}(\vec{x})
$$
and a fixed point $\vec{x}_0$ such that $\vec{F}(\vec{x}_0)=\vec{0}$, the ...
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2answers
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Chaos implies Nonlinearity?
Why, for finite dimensions, is nonlinearity a precondition for chaos?
This article (Linear Chaos? By Nathan S. Feldman) offers an example of an infinite dimensional chaotic map, which is linear. ...
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1answer
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Is it sufficient to consider a small part of a system (without potential energy sources which can be released) to determine if it's chaotic?
The world around us abounds with chaotic systems: dripping taps (when a certain dripping rate is reached the dripping becomes irregular, which can be seen in this old but very entertaining video, ...
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1answer
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When intersections of trajectories in Poincare sections are possible?
If we get intersections of some "trajectories" in non-standard 2D Poincare sections, that have been obtained from numerical integration of Hamilton equations for autonomous system in 2D coordinate ...
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1answer
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Characteristic energy spectrum patterns in quantum chaos
https://www.scientificamerican.com/article/quantum-chaos-subatomic-worlds/
In this article, the author says that "the energy levels of a chaotic quantum system exhibit strong correlations."
I can't ...
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1answer
186 views
Issue with Bifurcation Plot for Driven Pendulum
I'm trying to create a bifurcation plot for a driven damped pendulum. In particular, I'm trying to recreate the plot found in Taylor's 'Classical Mechanics' (page 484) for a driving strength $\gamma$ ...
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1answer
72 views
Dimension of Poincaré Map
I am used to seeing bi-dimensional Poincaré maps, as the ones shown in this post:
Poincaré maps and interpretation
In that example, one manages to draw a bi-dimensional map because the number ...
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2answers
87 views
Speed of the butterfly effect
Consider the simplest abstract model of a double pendulum. What is the maximum possible amount of change in the initial conditions so that the trajectory doesn't visibly change in the first minute, i....
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2answers
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In a real life system of waving a bubble wand or baton around how do you determine when chaos has occurred?
In a real life system of waving a bubble wand or baton around how do you determine when chaos has occurred?
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2answers
237 views
How to calculate the maximal Lyapunov exponent(s) of a multidimensional system?
I've been reading up about Lyapunov exponents for a university group project on chaos theory and I'm a little confused as to how they are calculated for a system of multiple dimensions.
From what I ...
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1answer
160 views
Nonlinear dynamics and chaos theory in electrical systems and circuits [closed]
Is there a possible application of the field of chaos theory and nonlinear dynamics to electrical systems such as circuits and power?
If so, are they based on conventional nonlinear dynamics or are ...
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Simulation of multi-particle systems, randomness and chaos
The answer https://physics.stackexchange.com/a/10441/50677 for #2 (chaotic randomness) claims that the absolute knowledge (whatever that would be) of starting conditions were sufficient for a perfect ...
