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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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11
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361 views

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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1answer
219 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 ...
4
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1answer
104 views

Is there a prediction when our solar system would fall apart?

Within a few billion years our Sun would become a red giant and destroying Mercury and Venus and perhaps even the Earth. But nevertheless our solar system will still have some planets like Jupiter and ...
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91 views

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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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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94 views

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 ...
3
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115 views

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 ...
2
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28 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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131 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 ...
2
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0answers
300 views

Local Hamiltonian

When I read the paper "Chaos and complexity by design", I found the following sentence in the Introduction part confusing: $$W(t)=\sum_{j=0}^{\infty}\frac{(it)^j}{j!}\underbrace{[H,\dots[H}_{j},W\...
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31 views

Fermi-Pasta-Ulam for the beam equation

The Fermi-Pasta-Ulam numerical experiment is based upon the discrete wave equation, with a small non-linearity added to the forcing term. Does anybody know of similar research performed on the beam ...
2
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1answer
75 views

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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174 views

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+ \frac{q^2}{8\pi\epsilon_0\sqrt{r^2+z^...
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1answer
121 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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31 views

Are my voltmeter readings expected for a correct chua's circuit?

I am following this article Robust OP Amp Realization of Chua's Circuit by Kennedy to implement a Chua's circuit. I use exactly the same design as the article; the only difference is that I use a 15mH ...
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34 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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42 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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2answers
38 views

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 ...
1
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1answer
310 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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39 views

Example of an adiabatically perturbed integrable 2 d.o.f. Hamiltonian?

Consider the following (classical) Hamiltonian system: $H(u,v,p,q, \tau)$, where $(u,v)$ and $(p,q)$ are conjugated variables and $\tau = \epsilon t$ is a slowly varying parameter, $0 < \epsilon &...
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134 views

What is neuroavalanche and how does it relate to the idea of supersymmetry?

In a wiki on stochastic differential equations, one passage regarding supersymmetry states the following: The spontaneous breakdown of... supersymmetry... explains the associated long-range ...
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1answer
59 views

Predicting the behaviour of a spring-mass system under the influence of regular periodic impulses

Here is the question: I have a mass 'm' connected by a spring to a wall. The setup is horizontal. The time period of the mass is 'T0' determined by the spring constant 'k'. The mass obeys a position ...
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1answer
76 views

Why do the KAM tori divide the phase space into disjoint parts in $d\leq 2$ systems?

Here $d$ is the degree of freedom. It is not the case when $d \geq 3$? Can anyone give an intuitive explanation? When $d =2$, the phase space is 4 dimensional. The tori are 2 dimensional. In this ...
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83 views

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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30 views

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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0answers
273 views

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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0answers
97 views

spectral eigenvalue staircase and quantum system

in a d-dimensional system of Quantum physics , does the Eigenvalue staircase $ N(E)= \sum_{E_{n}\le E} 1 $ determine ALL the properties of Quantum System ?? for example, let us assume that the ...
1
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1answer
69 views

Discontinuities in a Poincare map for a double pendulum

I'm generating poincare sections of a double pendulum, and they mostly look okay, but some of them have weird discontinuities that seem wrong. The condition for these sections is the standard $\...
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24 views

Can the onset of atrial fibrillation be compared to the onset of chaos in a dripping tap?

Atrial fibrillation, roughly speaking, starts when the end of a complete cycle of a heartbeat overlaps with the beginning of the next heartbeat, which makes the heart behave in a chaotic way (...
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16 views

Why is the angular momentum of a 3D kicked rotor non-negative?

We know that for a 2D kicked rotor the angular momentum quantum number can be any integer from minus infinity to infinity. However, for a 3D kicked rotor this is not the case: it can only be positive ...
0
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1answer
64 views

Double pendulum Poincaré section issues

I'm generating high-resolution Poincaré sections of a double pendulum, and I'm running into some issues in creating the initial conditions for a given section. In general, I describe my pendulum with ...
0
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1answer
106 views

Master equation for the mechanical modes

Consider the standard model of optomechanical systems with a single optical cavity mode coupled to a mechanical oscillator, which is canonically modeled as a FP cavity with one fixed mirror and one ...
0
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1answer
51 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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16 views

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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22 views

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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45 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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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
52 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
53 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
101 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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99 views

Existence of time-dependent invariants for a classical Hamiltonian system

Suppose the Hamiltonian is time-dependent, $H(t)$. The condition on a time-dependent invariant is $$ 0 = \frac{d I}{d t } = \frac{\partial I }{\partial t} +\{ I, H\} . $$ We can rewrite it as $$ ...
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93 views

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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2answers
49 views

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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1answer
223 views

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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1answer
46 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 ...