In dynamical systems, the motion of a particle in some geometric space, governed by some time dependent rules, is studied. The process can be discrete (where the particle jumps from point to point) or continuous (where the particle follows a trajectory). Dynamical systems is used in mathematical ...

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115 views

Are the values generated by non-linear equations truly random?

I was recently studying some literature on chaos theory and non-linear equations . where in various ciphers were created using non- linear equations like Lorenz equation . Are the data generated from ...
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0answers
193 views

How to find a function which satisfies such functional equation?

How to find a function which satisfies: $$a^x=\lim_{h\to\infty} \left( f_a \left(f_a^{[-1]}(x)+\frac xh\right)\right)^{[h]}$$ where $f^{[n]}(x)$ is the number if iterations of a function (if n=-1 ...
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1answer
144 views

Minimizing the cost of a path in a dynamic system

So suppose I want a path from 0 to $c>0$ on the real line, and I am going to use the function $S(t)$ to get there in (discrete) time $T$. That is, my position at time 0 is 0, my position at time $T$ ...
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2answers
364 views

Schwarzian Derivative and One-Dimensional Dynamics - how are they connected?

During the summer, I did an REU where we focused primarily on one-dimensional dynamics and more specifically kneading theory. One thing that I was always confused about is why the Schwarzian ...
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2answers
706 views

Fekete's conjecture on repeated applications of the tangent function

A high-school student named Erna Fekete made a conjecture to me via email three years ago, which I could not answer. I've since lost touch with her. I repeat her interesting conjecture here, in case ...
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1answer
447 views

Large Deviation Properties of a function of a geometric random variable

Suppose I have a variable $s$ that has a geometric distribution with success parameter x. So the probability of success on trial $s$ is $p_s = (1 - x)^{s - 1} x$, Consider the following function of ...
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1answer
351 views

FitzHugh- Nagumo caricature of the Hodgkin-Huxley equations

I've been trying to solve this problem from Elements of Applied Bifurcation Theory, but even with the hints given I haven't found out how to proceed yet. I would appreciate any further hints or ...
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1answer
308 views

Polygonal billiards and uniform distribution

According to this article in Wikipedia: A billiard is a dynamical system in which a particle alternates between motion in a straight line and specular reflections from a boundary. When the particle ...
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1answer
865 views

Determine a conserved quantity in a dynamical system Lotka-Volterra

I have a two state dynamical system. The two state variables are $P$ and $Z$ and $a,b,c,d$ are parameters. The system equations are: $\frac{dP}{dt}=a\cdot P-b\cdot PZ=P\left(a-bZ\right)$ ...
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5answers
821 views

Period of 3 implies chaos

Let $f(x)$ be a continuous function from $\mathbb{R}\rightarrow\mathbb{R}$. Let's denote $k$-times repeated application of the function, $f(f(f(...f(x)...)))$ as $f^{(k)}(x)$. Let's call any $x$ a ...
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11answers
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Mandelbrot-like sets for functions other than $f(z)=z^2+c$?

Are there any well-studied analogs to the Mandelbrot set using functions other than $f(z)= z^2+c$ in $\mathbb{C}$?