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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Initial conditions for a non-chaotic double pendulum

I know that the initial condition of the full double pendulum being horizontal yields motion that is not chaotic. Is there another set of initial conditions that would yield non-chaotic motion? Or do ...
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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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245 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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Lyapunov exponents of a damped, driven harmonic oscillator

I am supposed to calculate Lyapunov exponent of a damped, driven harmonic oscillator given by $\ddot{x} + 2\beta \dot{x} + \omega_0^2 x = f\cos(\omega t)$ Lyapunov exponent is $\lambda$ in $\delta ...
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Can tearing a piece of paper be chaotic?

When one thinks of chaos, then automatically the thought pops up that a very little difference in the initial conditions enlarges over time and you end up with a totally different end situation. When ...
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If the solar system is a sensitive chaotic system, can gravity waves make orbits unpredictable?

Scott Tremaine says here ..for practical purposes the positions of the planets are unpredictable further than about a hundred million years in the future because of their extreme sensitivity to ...
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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 ...
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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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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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What is a quantum scar?

This notion was proposed by Heller in 1984. But his paper is hard to follow (at least for me). Does anyone has a good understanding? Is it just judged by the naked eye?
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Lyapunov stability of circular orbits

I'm studying Classical mechanics on Arnold's "Mathematical Methods of Classical Mechanics". In a problem i'm asked to find for which $\alpha$ the circular orbits in the central field problem are ...
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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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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 ...
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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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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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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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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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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. ...
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Relationship between entropy and Lyapunov exponent

If $\lambda$ is the largest positive Lyapunov exponent of a piecewise linear dynamical chaotic discrete in time map, then is there a relationship between the entropy and its $\lambda$. I remember ...
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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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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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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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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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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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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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Logistic map and attractors

Does the logistic map have a strange attractor for some "chaotic" values of the parameter?
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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 ...
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T-Symmetry and spatial symmetry of a multivariate conserved quantity

Definition: A reversible system is defined to be any second-order system that is invariant under the map. $t \mapsto -t$ $y \mapsto -y$ Suppose there exists a multivariate function ...
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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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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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110 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 ...