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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Interpretation of Poincare Map

I have been trying to interpret a Poincare Map. The Hamiltonian for the system is $$H=\frac{1}{4m}\left(p_r^2+p_z^2\right)+m\omega_\perp^2 r^2 +m\omega_z^2 z ^2+ ...
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Does a simple double pendulum have transients?

Suppose, we have the most simple double pendulum: Both masses are equal. Both limbs are equal. No friction. No driver. Arbitrary initial conditions (no restriction to low energies) Does this ...
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What does unfolding of attractor mean?

What does unfolding of attractor mean? Effect of time scales on the unfolding of neural attractors paper talks about Takens embedding theorum. It says that the embedding dimension should be large ...
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How classical chaos can be described quantum mechanically?

How can we describe the chaotic properties of classical systems using quantum mechanics when the Schrodinger equation that describes quantum dynamics is linear? How can we use quantum mechanics that ...
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511 views

Why is the computer useful if a chaotic system is sensitive to numeric error?

In every textbook on chaos, there are a lot of numerical simulations. A typical example is the Poincare section. But why is numerical simulation still meaningful if the system is very sensitive to ...
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Conceptual questions about chaotic signals

Are chaotic signals stationary or non stationary signals? They are ergodic and ergodic signals are a kind of stationary signals. But, from statistics point of view, are chaotic signals classified as ...
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Dynamical localisation in delta-kicked rotor

The quantum delta-kicked rotor is a common tool for studying quantum chaos. The energy of the rotor increases ballistically when kicking at the Talbot time (resonance) and jumps between zero and some ...
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Help explore a self-feedback camera-monitor chaotic system

We are trying to emulate the chaotic system Jim Al-Khalili demonstrate (3 min video). In our chaos lab, we are trying to research the chaotic system shown in the video. We are using just a webcam and ...
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Logistic map and attractors

Does the logistic map have a strange attractor for some "chaotic" values of the parameter?
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891 views

Will glass always break in the same way?

This question has had me thinking for a while. If I have two large panes of glass and a rock or similar item is thrown in exactly the same place on the glass, would the two panes break in the same ...
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Relationship between dynamical entropy and chaotic noise

I am interested to know if Kolmogorov Sinai Entropy of a system also known as Kolmogorov Complexity in information theory or popularly known as metric entropy increases with increase in dynamical ...
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35 views

On the relationship between entropy and chaotic noise

I have few conceptual questions related to application of chaos in communications. Kolmogorov-Sinai Entropy1 , Kolmogorov-Sinai Entropy2 and Kolmogorov-Sinai Entropy3 KS is an entropy metric for ...
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Chaos theory deterministic or non-deterministic?

While i was studying about chaos theory, i stumbled upon this, When a ball confined in a square, and at the center is located a circle, is to bounce elastically, the path of the object deviates ...
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What do matrices in the Gaussian orthogonal ensemble look like?

I've been reading a fair amount about quantum chaos, and random matrix theory comes up a lot. I get that they're looking at the distribution of eigenvalues from an ensemble of random matrices, but I ...
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337 views

Non-linear dynamics vs Chaos

I am confusing between non linear dynamics and chaos. Chaos is also a non-linear dynamics right? then what is the difference between chaos and non-linear dynamics? What I understood about chaos is ...
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40 views

Minimum amount of fluid to experience turbulence?

Turbulence is a challenge to model and simulate: "When I meet God, I am going to ask him two questions: Why relativity? And why turbulence? I really believe he will have an answer for the ...
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36 views

Meaning/picture of the statement: “Turbulent flow is chaotic. However, not all chaotic flows are turbulent”

Wikipedia states that Turbulent flow is chaotic. However, not all chaotic flows are turbulent. Someone give a picture for that?
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143 views

Does Lyapunov exponent equate to exponential inflation?

Physics can be modeled by dynamical systems $f^t(x)$ as well as by PDEs. The most common dynamical system has hyperbolic fixed point and can be an attractor or a repellor. The dynamics at repellors ...
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42 views

Meaning of Smooth Dynamical System?

What does smooth dynamical system mean? It is the title of a paper I am supposed to read in non linear systems.
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What is the event in history where iterated functions became appropriate for modeling physics?

Wolfram as well as Aldrovandi and Freitas 1 maintain that iterated functions $f^t(x)$ are a valid alternative to PDEs for modelling physics. Instead of just citing 1, I want to be able to cite the ...
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What creates the chaotic motion on a double pendulum?

As we know, The double pendulum has a chaotic motion. But, why is this? I mean, the mass of the two pendulums are the same and they have the same length. But, what makes its motion random? I'm just ...
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237 views

Calculating Lyapunov exponents from a multi-dimensional experimental time series

Wolf's paper Determining Lyapunov Exponents from a Time Series states that: Experimental data typically consist of discrete measurements of a single observable. The well-known technique of phase ...
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Reference for the Landau-Lifshitz system

I'm interested in understanding the dynamics of the discrete Landau-Lifshitz system. It's solutions to equations like $$\frac{\partial X_n}{\partial t} = X_n\times (X_{n-1}+X_{n+1})$$ where the $X_n$ ...
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Kolmogorov entropy and noise

There are various ways of calculating the Kolmogorov entropy (KE)of dynamical system. According to Pesin's theorum, it is the summation of the lyapunov exponents; frominformation theoretic concet, it ...
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Can a small change in the Earth temperature give rise to large-scale climate changes?

Earth's atmosphere is a chaotic system. In such systems arbitrarily small changes the conditions can give rise to very large effects. There are many rumors about the physical and large scale ...
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242 views

About Poincare section for the double pendulum

I am reading Prof. Louis N. Hand's Analytical Mechanics. In the chapter about chaos, it introduces the concepts of Poincare section based on the example of double pendulum. Also, it plot the section ...
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220 views

Physical interpretation of the Lorenz system

The Lorenz equations $$ \frac{dx}{dt} = \sigma(y-x);\\ \frac{dy}{dt} = x(\rho-z)-y;\\ \frac{dz}{dt} = xy - \beta z $$ were (I believe) the first set of nonlinear equations known to exhibit chaotic ...
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Stability theory [closed]

I'm studying stability theory recently and met a lot of phrases like linear stability and nonlinear instability. After searching on Google, I became more confused. Thus I wonder if there is any ...
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183 views

Does the logistic map have an attractor for a particular value of the parameter?

Background: Currently I am studying a course on non-linear dynamics. We have been studying about attractors only intuitively, so I do not have a definition for an attractor. Let me give you a couple ...
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104 views

Is there a one-to-one mapping between numeric sequences and symbolic representations?

I am not a physicist but applying symbolic dynamics for information coding in signal processing. I have represented a chaotic signal obtained from Lorenz system into {0,1}. Then this symbolic sequence ...
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386 views

“windows of order” in the Bifurcation diagram

When looking at the bifurcation diagram of a chaotic system, one observes "windows of order", namely short intervals where the system briefly leaves its chaotic state and then rapidly returns to ...
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226 views

What are the *necessary* conditions to deterministic chaos?

What are the necessary conditions (not saying sufficient conditions) in mathematical terms that a deterministic dynamic system can transit to deterministic chaos? We collected yet: A positive ...
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352 views

What are the principles of deterministic chaos?

I see in literature very different (and chaotic) descriptions of what is deterministic chaos. Can you explain to me based in a type of formal definition, which principles need to be exactly fulfilled ...
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435 views

Calculate/Estimate the fractal dimention of the logistic map

This is the logistic map:. It is a fractal, as some might know here. It has a Hausdorff fractal dimension of 0.538. Is it possible to calculate/measure its fractal dimension using the box counting ...
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168 views

Phase Space dimension of Lorenz Strange Attractor

It is often discussed in 3 spatial dimensions and the need for third dimension to prevent self intersection is mentioned. But shouldn't the phase space of the Lorenz system be 6 dimensional, i.e., the ...
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Are there real life applications for Hausdorff dimensions, specifically crack formations?

I was curios about Hausdorff dimensions. They seem to neatly describe rough surfaces. So I was wondering if there are common applications of Hausdorff dimensions in things like complicated friction ...
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What is a physical example of a Saddle-Node Bifurcation?

I am doing a presentation on bifurcations and would like physical examples to go along with each type of bifurcation but I am unable to find or think of any good example of a simple Saddle Node ...
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Universal Sequence and relationship of mathematics and reality [closed]

In "The Special and General Theory of Relativity" Einstein says: How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably ...
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358 views

Ljapunov exponent of driven damped pendulum

I have written a computer simulation of the driven damped pendulum, pretty much as the one shown here, only that I did it Python. Next, I have found some parameters for which the pendulum behaves ...
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What can be the smallest chaotic system?

As I am talking about 'smallest' can I expect that it should be a quantum system? I understand that we use quantum chaos theory instead of perturbation theory when the perturbation is not small. For ...
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Is that true that real quantum chaos doesn't exist?

I read several books and papers on quantum chaos, to my understanding they all emphases that the quantum chaos does not really exist because the linearity of the Schrodinger equation. Some works were ...
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Staying in orbit - but doesn't any perturbation start a positive feedback?

I am not a physicist; I am a software engineer. While trying to fall asleep recently, I started thinking about the following. There are many explanations online of how any object stays in orbit. The ...
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776 views

Does chaos theory occur in quantum mechanics? Or in any non-newtonian physics?

Does chaos theory occur in quantum mechanics? Or in any non-newtonian physics? Apart from perhaps thermodynamics?
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244 views

Morse potential and chaos

I have heard that the Morse potential equation $$\tag{1} -\frac{\hbar ^{2}}{2m} \frac{d^{2}}{dx^{2}}y(x)+ae^{bx}y(x)-E_{n}y(x)=0 $$ is related to the two dimensional equation on the Poincare half ...
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Fractal Cosmology and Misner's Chaotic Cosmology

I have a question pertaining to the ideas behind the considered homogeneity and isotropic nature of the universe (at a grand scale) versus the theory of a chaotic and anisotropy structure of the ...
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294 views

Are a quantum mechanical system a chaotic (yet deterministic) system?

The title is slightly misleading. I really want to know if the randomness and probabilities observed in quantum mechanics is really just the result of a chaotic (yet deterministic) system. If it is ...
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Reference for the predictability of rigid body dynamics

I'm looking for a reference, journal article, paper, etc. that supports the idea that classical mechanics, in particular rigid body dynamics, is largely predictable. A view coming from the background ...
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212 views

Chaos is predictable?

I'm reading a book of computational physics [1] where the driven nonlinear pendulum is studied in deep. This is the equation used in the book: $$ \frac{d^2\theta}{dt^2} = -\frac{g}{l}\sin\theta - ...
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734 views

A good, concrete example of using “chaos theory” to solve an easily understood engineering problem?

Can anyone suggest a good, concrete example of using "chaos theory" to solve an easily understood engineering problem? I'm wondering if there is a an answer of the following sort: "We have a high ...
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The ideal trampoline

Suppose we have a mass attached to the top of an ideal (linear and massless) spring oriented vertically in a uniform gravitational field, and on top of that mass there is another mass resting on it. ...