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Questions tagged [chaos-theory]

For questions in chaos theory.

Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, engineering, economics, biology, and philosophy.

Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the butterfly effect.

Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general.

The theory was pioneered by Lorenz and Devaney, who states the 'three laws of chaos' as:

  • it must be sensitive to initial conditions
  • it must be topologically mixing
  • it must have dense periodic orbits
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Where does Feigenbaum's Constant (4.6692...) originate?

Feigenbaum discovered a ratio between bifurcations that were found in all known chaotic-dynamic systems, from dripping water faucets to abstract equations on population fluctuations (as elucidated in James Gleick's book "Chaos"). How should one…
Marcos
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Is there a known mathematical foundation to the concept of emergence?

I'm researching many topics including emergence and chaos theory, and I cannot for the life of me find strictly mathematical treatments of the idea of emergence. Is there any form or field of mathematics that can predict the emergence of one…
Garrett Miller
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Are there chaotic systems with explicit solutions?

Are there such continuous chaotic systems, for which an explicit solution exists, which would allow to practically compute state at any given position in time just knowing the initial conditions? Or are they all the explicit solutions limited by the…
Ruslan
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Is 'level one chaos', as Yuval Noah Harari describes, a real concept in chaos theory?

In Sapiens by Yuval Noah Harari, it is mentioned that Chaotic systems come in two shapes. Level one chaos is chaos that does not react to predictions about it. The weather, for example, is a level one chaotic system... Level two chaos is chaos that…
Nico Damascus
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Why do we need folding and a finite domain for chaos?

One of the generic ways to obtain chaos in phase space is when the system causes trajectories to stretch and fold. I understand that the stretching will cause neighboring initial conditions to diverge, which is one of the conditions of chaos, but…
ddc
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Stock analysis and forecast using chaos theory

Yesterday, I was going through an article in which the user had mentioned that he has used chaos theory to predict stock prices and ended up with 30% + profit.(I am not intersted in the profits :P) After that I read a bit about chaos theory and…
Egalitarian
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How do mathematicians deal with some predictability in otherwise-chaotic systems

I've been looking at systems which are predictable in some senses and chaotic in others. For example, consider a double pendulum that considers the movements within the pendulum body itself. The outward double pendulum is a known chaotic system,…
Cort Ammon
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feigenbaum constant

how can i predict a chaos with feigenbaum constant in the logistic map?
yosibaryosef
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How can one prove chaos in logistic map?

Many introductions to chaos start with logistic map $$ x_{n+1}=\lambda x_n(1-x_n) $$ and claim it is chaotic at some values of $\lambda$. Unfortunately, all proofs of chaos I saw were numerical and not rigorous. How does one prove that such a map is…
Pavlo. B.
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Does the set of points where the logistic map is chaotic have positive measure?

Let $r$ be a real number in $(0,4)$, let $x_0$ be any real number in $(0,1)$, and define a sequence: $$ x_{n+1} = rx_n(1-x_n). $$ This is the logistic map. For some choices for the value of $r$, the resulting sequence will converge to periodic…
Lorioch
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What does the phrase "invariant under f" mean?

Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $f(x,y)=(\frac{x}{3}+11y^2,-2y)$. Let $A=\{(3y^2,y)|y\in \mathbb{R}\}$. Show that $A$ is invariant under $f$. What is it asking?
Chuck Franklin
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Is there a proof that the double pendulum path reaches every point in a closed region eventually?

The simplest double pendulum are just two rods attached to each other. The main rod may or may not flip depending on the initial conditions. If it flips, the path that the end of pendulum passes covers the space between two concentric circles. Is it…
user318107
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Devaney's definition of sensitive dependence on initial conditions

Devaney defines that a funtion $f:J \to J$ has sensitive dependence on initial conditions if there exists $\delta > 0$ such that, for every $x \in J$ and any neighborhood $N$ of $x$, there exists $y \in N$ and $n \geq 0$ such that…
ThePunisher
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Are there continuous chaotic systems where nearby paths diverge at rates other then exponential?

I know that in general, nearby paths in a chaotic system tend to diverge exponentially, but are there continuous systems where paths diverge at other rates? For example, is there a system where nearby paths diverge say double exponentially or at…
Colonizor48
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Henon maps and strange attractors

What I understand of chaos is that, it cannot occur in 2D space. A strange attractor, as seen in a Lorenze system, is a feature of chaos. Now, I am reading about the Henon map which is a 2d map with a strange attractor. My question, then, is: Does a…
dimwit_recluse
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