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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Is the double pendulum an example of a strange attractor?

Imagine a pendulum to which is attached another one (not necessarily the same length). Does this pendulum, when you let it go, which can be done in many ways but let's keep the total potential energy ...
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Is it possible to let a flag ``stand frozen in time``?

It´s clear that a moving flag is a chaotic system. But is it nontheless possible, under certain conditions and a uniform wind velocity, to make the flag look frozen in time, i.e., flat without ...
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Understanding this metaphor involving e-mails, chaos and phase transitions [closed]

I asked this question on the English Stack Exchange and people advised to try get the answer here. I can’t get the idea of metaphor in the last sentence of the following quote: Instead, email ...
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Understanding the Equation for the Lyapunov Exponent

I'm trying to calculate the Lyapunov exponent for a simple dynamical system, but I think I have misunderstood the equation. My calculations constantly lead to zero, although I'm varying initial ...
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Do all equillibrium points of a discrete mapping show up on the bifurcation diagram?

The question in the title is perhaps vaguely posed, so I'll include the concrete example which is bugging me. Suppose we have a mapping given by $$N_{t+1}=N_t\cdot ...
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Is the butterfly effect real?

Is the butterfly effect real? It is a well known statement that a butterfly can, by flapping her wings in a slightly different way, cause a hurricane somewhere else in the world that wouldn't occur if ...
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Does the uncertainty principle go against chaos theory?

My understanding of the uncertainty principle and quantum physics is that any given object may, without notice or explanation, spontaneously perform an action it previously was unable to do with a ...
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Randomness v. complexity

There are a few other topics I found that explore this idea from a different perspective: Is randomness deterministic? Can randomness exist? Is the universe fundamentally deterministic? My ...
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Non-Linear Behavior of Iterated Functional Maps

The universal behavior of certain iterated nonlinear function maps (ie period doubling bifurcation route to chaos): $$x_{i+1}=f(x_i)$$ have been known since Feigenbaum: (see ...
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Physical distinction between mixing and ergodicity

How can one in a very contrasting manner distinguish between the physical meaning of mixing dynamics and that of ergodic dynamics? More precisely, is one a stronger condition than the other? (which ...
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Extending the ergodic theorem to non-equilibrium systems

I try to make this as short and concise as possible. For equilibrium systems in statistical mechanics, we have the Liouville's theorem which says that the volume in phase space is conserved when the ...
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Help - calculation probabilities - new theory [closed]

I need help to calculate exact or approximated probabilities. I am interesting by the content of a new theory (view this link nokton theory for details). Currently, I studing the interactions of two ...
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How fast will a small disturbance propagate to affect all of Earth?

As far as I understand it, quantum mechanics requires that a particle's position to be not specifically determined in space, but rather be 'spread' out through space, in the sense that we can only ...
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integrability and area-preservation property of dynamical systems

Suppose we have a map defined on a plane, $x_{1}=f(x_{0})$, where $x \in \mathbb{R}^{2}$. Assume it is integable: there exists a function $I$ of the phase space variable $x$ such that $I(x)=I(f(x))$. ...
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Analytic proof that Lyapunov exponents in Hamiltonian systems pairwise sum to zero

I have read that in Hamiltonian systems, Lyapunov exponents come in pairs $(\lambda_i, \lambda_{2N-i+1})$ such that their sum is equal to zero. Is there a way of proving this analytically? EDIT: ...
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Is long-term weather forecast impossible in principle?

This question can be asked about any chaotic dynamical system, but hydrodynamics of the atmosphere makes it more concrete. Arnold describes his 1966 result as follows: I have calculated the ...
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Chaos and integrability in classical mechanics

An Liouville integrable system admits a set of action-angle variables and is by definition non-chaotic. Is the converse true however, are non-integrable systems automatically chaotic? Are there any ...
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I'm interested in the use of self-similarity in physics. Is this a reputable subject? [closed]

I'm interested in fractals, self-similarity, and chaos. Many physicists disregard these phenomena as candidates to explain the fundamental properties of the universe. However, when I read about ...
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What is the relationship between quantum physics and chaos theory?

I am not a physicist, I am looking for a non-technical explanation. Articles such as this one seem to hint at the fact that "macro reality" regulated by classical mechanics is somehow a pattern ...
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What are the differences between logistic map, poincaré map, attractor, phase portrait, bifurcation diagram? [closed]

What are the differences between Logistic map, Poincaré map, Attractor, Phase portrait, Bifurcation diagram Currently I became interested in chaos theory and non-linear dynamics. While ...
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Is there any book about chaos theory and nonlinear dynamics? [duplicate]

I'm interested in chaos theory and nonlinear dynamics. I learned some knowledge about phase space, attractors, bifurcation diagram, etc from Wikipedia. But I want to study more comprehensive about ...
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Why is my Lyapunov exponent similar for single and double pendulum?

This is my first question here on stackexchange. I hope that I can be understood. If not, tell me and I will reformulate and fill in with details. I have simulated a single pendulum and a double ...
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Can binary sequences generated from ergodic maps be chaotic?

Briefly, the way symbols are generated is: Consider a one-dimensional chaotic map $T: [0,1]→[0,1]$ and a time series $\{x_n\}_{n=1}^N$ generated with this map. Define a threshold $A$ and a ...
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Theory of chaotic systems : Bijective mapping

Considering a one-dimensional chaotic map $T$. The domain of the map (the closed interval $[0,\,1]=I_1 \cup I_1$) is partitioned into two regions (i.e. $I_0\cap I_1=\emptyset$): $[0,\,A] = I_0$ and ...
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Origin of chaos in Chua's circuit

I am doing a project on Chua's circuit, but I can't seem to find anything that explains where the chaotic nature of the system comes from. Does anyone know of articles that explain it well on an ...
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Ratio between power of chaotic and regular airflow

Turbulent field is created as a result of an impact of an airjet on an edge (the flow velocity is high enough). The field of velocities have a regular and a chaotic component. What I need is to ...
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Change of variables to apply Melnikov method

Supposing there is a system of non-autonomous non-linear differential equations with small damping and small forcing. The unperturbed system (zero damping and forcing) is Hamiltonian but neither has a ...
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Does big bang have really any justification while we are living within a huge chaos?

All the physicists already know that the n-body problem reveals chaos, so that the planets around the sun should undergo deterministic chaos with a Lyapunov exponent of the order of, say, ...
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Hysteresis in the Lorenz Equations

I was going through Strogatz's wonderful book on nonlinear dynamics and while reading through one problem he posed at the end of the chapter, I did not really understand what was going on. So I hope ...
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75 views

Can quantum fluctuations affect the double pendulum?

The double pendulum is a simple example of a chaotic system which is extremely sensitive to tiny perturbations in its initial conditions. If we set off two identical double pendulum systems from ...
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Determining the geometry of the phase space of a system [closed]

How do we check the geometry of the phase space ? I mean in classical mechanics we use position and conjugate momenta as a space of all possible states of the particle. How do we know that this phase ...
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What is the relation between Rössler attractor and thin accretion discs (like in the movie Interstellar)?

Is there any relationship between the Rössler attractor and thin accretion disks, like the accretion disk(s) in the movie Interstellar?
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What are resonant tori?

What is the definition of a resonant/invariant torus (in the phase space of a Hamiltonian system)? Are there non-resonant tori?
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Time it takes for a mass in a linked pendulum to flip?

I have created Mathematica code that simulates a double pendulum. So I've numerically solved for $\theta_{1}(t)$ and $\theta_{2}(t)$. I have also found the momentum from the Lagrangian as well. My ...
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Importance of periodic orbits

In the study of dynamical systems, one often talks about solutions that repeat themselves after a certain time, hence their name of "periodic orbits". Then one moves to the distinction of "stable" ...
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What is discrete phase space?

I've been reading a little about the usual, continuous Wigner functions and phase space quasi-distributions in general, and I believe I understand the idea behind them. The Wigner function arises when ...
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Non-integrability of the 2D double pendulum

Context: For a system with $n$ degrees of freedom (DOF), one has to deal with $2n$ independent coordinates ($2n$ dimensional phase space), of position $q$ and $\dot{q}$ in Lagrangian formulation, ...
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How far ahead can we predict solar and lunar eclipses?

The solar system is non-integrable and has chaos. The sun-earth-moon three-body system might be chaotic. So, how far into the future can we predict solar eclipses and/or lunar eclipses? How about ...
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The natural metric of a phase space and the Lyapunov exponent

For me, it seems that there is no apparent metric on a phase space of a dynamical system. Of course one can naively define an Euclidean metric on it, but it seems that this metric has not much to do ...
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What is the temperature evolution of the conductance of a quantum chaotic system?

I read this really nice article by Abanin and Levitov: http://arxiv.org/pdf/0704.3608.pdf They argued that the mixing of the quantum edge channel at the vicinity of a PN junction is described by the ...
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Is quantum indeterministic? [duplicate]

The question might look clear from a viewpoint of a non-physics guy but let me be more specific. Can we say quantum leaps or waves or maybe the universe itself are completely indeterministic or do ...
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Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant in essential ...
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Diffusion in the standard map

Consider the standard map (also known as Chirikov map): $$ p_{n+1} = p_n + K \sin(\theta_n) \\ \theta_{n+1} = \theta_n + p_{n+1} $$ I know that the diffusion coefficient according to the ...
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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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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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Is it really impossible to calculate in advance the result of throwing dice?

Is it really impossible to calculate in advance the result of throwing dice? After all, the physics of dice throwing is in the world of classical mechanics, rather than quantum mechanics.
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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 ...