# Questions tagged [chaos-theory]

For questions in chaos theory.

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### Prove that the orbit of a semi-conjugacy is dense in a metric space

A semi-conjugacy $h$ is a surjective, continuous map for which the conjugacy relation $h\circ f=g\circ h$ holds. Assume that $X$ and $Y$ are metric space, and let \begin{split} ...
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1 vote
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### If a diffeomorphism has a dense orbit, are almost all of its orbits dense?

Let $M$ be a closed manifold. Suppose that a diffeomorphism $f:M\to M$ has a dense orbit. Is it true that almost every ofbit of $f$ is dense in $M$? Or, maybe, if the orbit of $x_0$ is dense, then all ...
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### A chaotic function related to the $3x+1$ problem? (Li-Yorke and the Collatz problem)

Let $x$ be an infinite binary string. Define the function $f(x)$ mapping $x$ to the Cantor set of $I = [0,1]$ as: $$f(x) = \sum_{n=0}^{\infty} \frac{2 x_n}{3^{n+1}}$$ where $x_n$ are the ...
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### Chaotic one-dimensional system

Why can the solution of a one-dimensional equation of the form $$m\ddot{x}=F(x)$$ not be chaotic if $F$ is not explicitly time-dependent? Multiplying by $\dot{x}$ and integrating with respect to time, ...
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### Nonlinear Dynamics and Chaos - Convergence of a Map to the Logistic Map

The cosine–map is defined as:xn+1 = r/4((a+1)cos[k(xn - 1/2)]-a), with k = 2arccos(a/a+1), a > 0. Show that in the limit a → ∞, the cosine–map is the logistic map. I am really struggling in where ...
1 vote
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### Can probe trajectories to compute Lyapunov exponents get "stuck in more regular orbits" after rescaling?

I am computing Lyapunov exponents, and there is something that I do not understand about the data. The model has a regime for $\delta \approx 1$ (in some units) where it is fully chaotic, and the ...
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### Finding critical value of bifurcation

I have equations such as $\frac{dx}{dt} = y \\ \frac{dy}{dt} = \mu y + x - x^{2} + xy$ This system is known to be homoclinic bifurcation at the origin. To find the critical value of $\mu_{c}$, one ...
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### Duffing oscillator - Understanding the phase diagrams

I am trying to simulate a duffing oscillator. I am not sure how to read the phase diagrams I am obtaining. In the image, we can see at the top the phase diagram before the bifurcations; it has an ...
150 views

### Why is the number 3 come up so often in Chaos theory and Undecidability as a boundry? Is it just a coincedence?

What I mean is that the number 3 comes up a lot in these fields as sort of a boundary between decidability and undecidability, or chaos and order. Examples: Quadratic Diophantine equations are always ...
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### Density of points under dyadic transformation

My question concerns the Bernoulli map (a.k.a. the shift map and as dyadic transformation). I understand that most numbers $x \in [0,1)$ do not have a periodic orbit under this transformation, and ...
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### Showing that the horseshoe set is locally minimal

I'm trying to prove the Smale's horseshoe set is locally minimal. More specifically, let $H$ be the horseshoe set described in Section 1.8 in the book "Introduction to Dynamical Systems" by ...
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### Calculate Lyapunov Exponent for Double Pendulum

I have the x and y coordinates for two trajectories separated by a small difference in initial conditions (for a double pendulum). Since the pendulum is physically constrained, there can only be local ...
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### Is this a viable way to calculate the Lyapunov Exponent?

Is this a viable way to compute the Lyapunov Exponent for a double pendulum? Here is the code for the double pendulum (You don't have to look at this part, it just returns the angles and angular ...
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1 vote
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### How is Chaos defined in Math?

When I look at the English definition of the word "Chaos", I get the following definition : "Behavior so unpredictable as to appear random, owing to great sensitivity to small changes ...
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### Why does this distance estimator method render The Mandelbrot set incorrectly (non-divergent regions as divergent)?

I am using the following algorithm to render the Mandelbrot set and the exterior: • for each test point, calculate $c$ • initialise $z_{0}=(0+0i)$ • also initialise the gradient $dz_{0}=(0+0i)$ • ...
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