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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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15answers
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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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1answer
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Are there necessary and sufficient conditions for ergodicity?

What are the necessary and sufficient conditions (if any) for ergodicity (or non-ergodicity)? I see for instance that some integrable systems are not ergodic. For instance a linear chain of harmonic ...
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Question on the stability of the solar system

One of the pertinent questions about many body systems that causes me much wonder is why the solar system is so stable for billions of years. I came across the idea of "resonance" and albeit an useful ...
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2answers
872 views

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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2answers
727 views

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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1answer
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Poincaré maps and interpretation

What are Poincaré maps and how to understand them? Wikipedia says: In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the ...
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7answers
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Chaos theory and determinism

My professor in class went a little over chaos theory, and basically said that Newtonian determinism no longer applies, since as time goes to infinity, no matter how close together two initial points ...
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2answers
218 views

Conditions for periodic motions in classical mechanics

What are the conditions for classical motion to be periodic? In one dimension, if the motion is bounded, then it is also periodic. However, I don't think this generalizes to higher dimensions. I am ...
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2answers
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How and why can random matrices answer physical problems?

Random matrix theory pops up regularly in the context of dynamical systems. I was, however, so far not able to grasp the basic idea of this formalism. Could someone please provide an instructive ...
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2answers
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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, or ...
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4answers
2k views

Randomness, Chaos, Quantum mechanical probability functions

Can someone explain these 3 concepts into a unified framework. Randomness : Randomness as seen in a coin toss, where the system follows known and deterministic (at the length and scale and precision ...
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2answers
2k views

What is the highest energy position for a double pendulum? And for which energy positions is it chaotic?

Math/physics teachers love to break out the double pendulum as an example of chaotic motion that is very sensitive to initial conditions. I have some questions about specific properties: For a ...
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2answers
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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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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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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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1answer
366 views

Are bifurcations in dynamical systems related to phase transitions? [closed]

Bifurcation is a qualitative measure for a dynamical system changing the system parameter. Does the statistical behavior in the system shows phase transition-like characteristics?
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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 http://...
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1answer
325 views

Max Lyapunov Exponent of a Double Pendulum [closed]

Using Euler's method I got this graph. I used separation between angles $10^{-10}$, $\Delta t$ of integration 0.0001s and max time 100s. The initial angles are the same ($\theta_1=\theta_2$). I ...
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4answers
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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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1answer
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Chladni figures: avoided crossings of nodal lines

As khown, Chladni figures display nodal lines of eigenfunctions satisfying the equation $\Delta^2\psi=k^4\psi$ with appropriate boundary conditions. One can note these lines don't like to cross each ...
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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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1answer
877 views

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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3answers
2k 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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Physics-oriented books on fractals

I'm looking for some good books on fractals, with a spin to applications in physics. Specifically, applications of fractal geometry to differential equations and dynamical systems, but with emphasis ...
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1answer
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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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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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3answers
495 views

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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2answers
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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 ...
4
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1answer
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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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4answers
853 views

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
78 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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Does the “Andromeda Paradox” (Rietdijk–Putnam-Penrose) imply a completely deterministic universe?

Wikipedia article: http://en.wikipedia.org/wiki/Rietdijk–Putnam_argument Abstract of 1966 Rietdijk paper: A proof is given that there does not exist an event, that is not already in the past for ...
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2answers
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Is there a connection between Bertrands theorem and Chaos theory?

Bertrand's theorem states Among central force potentials with bound orbits, there are only two types of central force potentials with the property that all bound orbits are also closed orbits, ...
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4answers
817 views

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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4answers
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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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2answers
551 views

Motion of a bouncing sphere with a spring attached inside

Imagine a sphere with inside a spring attached (between opposite sides). You let it fall from a certain height, after which it bounces from a flat surface. The sphere is rigid. Will the following ...
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0answers
52 views

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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1answer
186 views

Can forces be predicted?

I seem to remember an exhibit in the Science Museum years ago that showed a rotating set of arms (under power), of say 4 arms. Each arm then had another rotating set of arms that rotated due to the ...
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1answer
117 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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2answers
159 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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2answers
1k views

Is chaos theory essential in practical applications yet?

Do you know cases where chaos theory is actually applied to successfully predict essential results? Maybe some live identification of chaotic regimes, which causes new treatment of situations. I'd ...
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1answer
270 views

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 question ...
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2answers
563 views

How to visualize an electron existing in two different places at the same time?

Let's consider a hypothetical situation where there are two electrons. The first electron is in superposition, simultaneously existing in two different locations. Let the locations be ...
3
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1answer
354 views

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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1answer
374 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 ...
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1answer
204 views

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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1answer
237 views

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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2answers
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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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1answer
186 views

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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1answer
248 views

Do the Lyapunov exponents depend on integrals of motion

When calculating the Lyapunov exponents it is usual to average over initial conditions. In a Hamiltonian system is it correct to average of energy as well, or do we pick an ensembles of trajectories ...