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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Has this chaotic map been studied?

I have recently been playing around with the discrete map $$z_{n+1} = z_n - \frac{1}{z_n}$$ That is, repeatedly mapping each number to the difference between itself and its reciprocal. It shows some interesting behaviour. This map seems so…
Martin Ender
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Is the sequence $(a_n)$ defined by $a_n=\tan{a_{n-1}}$, $a_0=1$, dense in $\Bbb{R}$?

Let $a_0=1,a_n=\tan{a_{n-1}}$. Then is $\{a_n\}_{n=0}^\infty$ dense in $\Bbb{R}$? I've drawn a map of this dynamical system and it seems that the sequence is dense on $\Bbb{R}$.
Dongyu Wu
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If $f(x)=x^2-x-1$ and $f^n(x)=f(f(\cdots f(x)\cdots))$, find all $x$ for which $f^{3n}(x)$ converges.

Let $f:\mathbb{R}\to\mathbb{R}$ be the polynomial defined by $$f(x)=x^2-x-1$$ and let $$g_0(x)=f(x),\quad g_1(x)=f(f(x)),\quad\ldots\quad g_n(x)=f(f(f(\cdots f(x)\cdots)))$$ The positive root of $f(x)$ is the famous golden…
Simon Parker
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Why isn't the 3 body problem solvable?

I'm new to this "integrable system" stuff, but from what I've read, if there are as many linearly independent constants of motion that are compatible with respect to the poisson brackets as degrees of freedom, then the system is solvable in terms of…
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How to go about studying chaos theory/dynamical systems/fluid dynamics in grad school with a physics background?

I'm turning to you as I find myself in need of help as to how to best go about studying something related to chaos theory/dynamical systems/fluid dynamics in postgraduate school. I'm currently in my third year of physics and my first question would…
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Supremum of all y-coordinates of the Mandelbrot set

Let $M\subset \mathbb R^2$ be the Mandelbrot set. What is $\sup\{ y : (x,y) \in M \}$? Is this known? To be more descriptive: What is the supremum of all y-coordinates of all black points in the following picture: Picture File:Mandel zoom 00…
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Identifying this chaotic (?) recurrence relation

Update: I'm working on an interactive bifurcation diagram: http://matt-diamond.com/sineMap.html Here's the image when the starting coordinates are [0.5, 0.5] The bifurcation diagrams differ depending on whether or not the coordinates are equal to…
Matt D
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Why is this family of dynamical systems able to produce spirals and clusters of points?

I have found by trial and error an interesting family of dynamical systems giving some nice strange attractors. They are chaotic complex systems based on the digamma function. It is defined by a complex discrete map as follows: (Disclaimer, tl:dr:…
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Understanding Takens' Embedding theorem

I am having some trouble understanding Takens' embedding theorem, and was hoping that someone with greater knowledge could help out. Formally, the theorem goes as follows: Let $M$ be a compact manifold of dimension $m$. For pairs $(\phi,y)$, where…
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How to know whether an Ordinary Differential Equation is Chaotic?

Assuming we have an ordinary differential equation (ODE) such as Lorenz system: $$ \dot x=\sigma(y-x)\\ \dot y=\gamma x-y-xz\\ \dot z=xy-bz $$ where $$ \sigma = 10\\ \gamma = 28\\ b = \frac{8}{3}\\ x(0)=10\\ y(0)=1\\ z(0)=1 $$ This system is known…
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Is tossing of a coin deterministic experimemt?

This is a question that I practically encountered while I was playing a game: Is tossing a coin a deterministic experiment? It might seem silly to ask but I had some thought over it. By the definition of the term and apparent look, it is not. But I…
user693540
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Why do chaotic systems need dense periodic orbits?

I know that chaotic systems have three properties (by the most cited definition): Sensitive to initial conditions Topological Mixing Dense Periodic Orbits I know that dense periodic orbits means that arbitrarily near any point is a periodic orbit.…
Cort Ammon
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What's the name of this chaotic system? (Cool pics included.)

I found this playing with a 2D-ODE-system plotter I'm writing. Surely, since it's so simple, it's been found and extensively studied by someone. What's it called? I'd like to look it up and learn a little about it. There's a weird pattern were some…
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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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Does an iterated exponential $z^{z^{z^{...}}}$ always have a finite period

Let $z \in \mathbb{C}.$ Let $t = W(-\ln z)$ where $W$ is the Lambert W Function. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = z^{a_n}$ for $n \geq 1$, that is to say $a_n$ is the sequence $z, z^z, z^{z^z} ...$ This is sometimes called…
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