# Tagged Questions

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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### Starting digits of $2^n$.

Prove that for any finite sequence of decimal digits, there exists an $n$ such that the decimal expansion of $2^n$ begins with these digits.
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### Formal proof of Lyapunov stability

I was trying to solve the question of AeT. on the (local) Lyapunov stability of the origin (non-hyperbolic equilibrium) for the dynamical system $$\dot{x}=-4y+x^2\\\dot{y}=4x+y^2$$ The streamplot ...
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### Does $\lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(…\cos x))}}$ exist? [duplicate]

Possible Duplicate: Explaining $\cos^\infty$ Does the following limit exist? $$\lim_{n\to\infty}\underset{n}{\underbrace{\cos(\cos(...\cos x))}}$$ If yes, find the limit. If no, please ...
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### Is Collatz' conjecture the only stable solution of its type?

The Collatz Conjecture is well known with the sequence $$f(n) = \begin{cases} n/2 &;\text{if } n \equiv 0 \pmod{2}\\ k\,n+1 &; \text{if } n\equiv 1 \pmod{2} \end{cases}$$ and $k=3$; the ...
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### Discuss the convergence of $\left \{ a_n \right\}$ where $a_{n+1}=\frac{a_0}{2}+\frac{a_n^2}{2},n\geq 1$

Let $$a_{n+1} = \dfrac{a_0}{2} + \dfrac{a_n^2}{2}$$ where $a_1 = \dfrac{a_0}{2}$ and $n\geq 1$ Discuss the convergence of $\left\{a_n\right\}$
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### Iterates of $f_b(x) = x - \log_b(x)$ - for $\log(b) \approx 0.399$: convergence to accumulation points or chaos?

In a previous question I arrived at the family of functions (depending on a real parameter b for the base of exponentiation/logarithm): $$f(x) = x - \log_b(x) \qquad \qquad b \gt 0$$ with the ...
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### Why is the Koch curve homeomorphic to $[0,1]$?

Henning Makholm has provided a nice proof that the limiting curve is a continuous function from $[0,1]$ to the plane. I was curios if the function is homeomorphism. A quick search gave me many sources ...
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### convergence of the iterated cosine

it can be demonstrated by elementary means that the curves $y=\cos x$ and $y=x$ meet exactly once, at a value $x=\alpha$ satisfying: $$\cos \alpha = \alpha$$ it is also evident (empirically) that ...