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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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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Question about a Attractors in Non-linear Systems

I've recently been reading up on non-linear dynamics and came across the concept of attractors. I'd like to ask if the concept of attractors can be used for pedestrian egress from a room? Since ...
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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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Homoclinic orbit and a particle in a double well

The physical set-up is a classical particle in a parabolic double well: Physically, a particle with reasonable amount of potential energy would be able to roll down the slope of the well, roll past ...
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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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How to compute the Liapunov exponents in henon map?

I am trying to compute the Liapunov exponents in Henón map, but i don't know the theory that i need, in logistic map is easy but in 2-dimensions? how is in general case? i need program it in Fortran ...
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Symbolic dynamics of Lorenz system [migrated]

Based on paper : Symbolic dynamics and periodic orbits of the Lorenz attractor download link and a previous post Symbolic dynamics of a multidimensional system From FIgure 1 in the paper says a ...
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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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Help in understanding a coding technique based on inverse mapping of a dynamical system [migrated]

Based on paper titled : Simultaneous Arithmetic Coding and Encryption Using Chaotic Maps by Kwok-Wo Wong et. al The Authors use a non-linear dynamical system for generating keys to be used in ...
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Help in understanding an application chaos theory in time series modeling

I read the following papers: (1) Iacomi et al., Mammographic images segmentation based on chaotic map clustering algorithm, BMC Medical Imaging 2014, 14 :12*. Download link (2) L. Angelini et. al, ...
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Symbolic dynamics of a multidimensional system

Let $x_t = F(x_{t-1})$ be a discrete-time dynamical system in the chaotic regime. Starting from an initial condition $x_0$, we can generate a time series $(x_t)$ where $t =1,2,...,T$ indicates the ...
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Mapping between numbers and symbolic representations

I am not a physicist but applying symbolic dynamics for information coding in signal processing. Is there any mapping between symbols and numbers?
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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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Would two identical universes evolve identically? [duplicate]

What if there were 2 universes (completely disconnected - not part of the same multiverse) which were identical and a given point in time (say when they first began). Would these 2 universes evolve in ...
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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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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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When is an attractor meaningful?

I’m originally a computer scientist; so I hope my question is not trivial. I’m working with time series and want to reconstruct the phase space from the time series based on time-lagged versions of ...
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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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How can noise be used in physics simulations? [closed]

I have been studying chaotic dynamical systems and noise. What is the difference between chaos and noise? I have looked over the internet for a good definition of what noise is but I haven't managed ...
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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 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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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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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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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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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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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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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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Chaos is predictable?

I'm reading a book of computational physics [1] where the driven nonlinear pendulum is studied in depth. This is the equation derived in the book: $$ \frac{d^2\theta}{dt^2} = -\frac{g}{l}\sin\theta - ...
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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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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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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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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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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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Infinitesimal input, macroscopic output

I must admit that I never got well how physicists handle infinitesimal quantities, mainly because of my education as a mathematician. So the following lines (taken from the preface of Berezin and ...
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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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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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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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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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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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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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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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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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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 ...