# Tagged Questions

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### Stability and Asymptotic Stability of Rational Matrix Solutions

If $X(t)$ is a fundamental matrix solution of $\dot{x}=A(t)x$ on $a<t<\infty$ and suppose the entries of $X(t)$ are rational functions of the variable t in the form $x_{ij}=p_{ij}(t)/q_{ij}(t)$. ...
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### Uniform Stability on an Interval

I am trying to prove the following: Supposed $\phi (t)$ is a solution of $\dot{x}=f(t,x)$ defined on $(\alpha , \infty )$ and supposed $\alpha < \beta < \gamma$. They $\phi (t)$ is uniformly ...
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### A Property of Lyapunov Index

Lyapunov Index of a function is defined as $\lambda(f)=\lim\sup_{t\rightarrow\infty}\dfrac{1}{t} \log||{f(t)}||$, where $f:R\rightarrow R^d$ A property of Lyapunov Index is ...
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### Skewed Tent Map

I have the following map: $f(x)=\begin {cases} \mu x &\text x\in[0,a] \\\mu a(1-x)/(1-a) &\text x\in [a,1]\end {cases}$ where $a\in(0,1)$. First of all I would like to check when the map $F$ ...
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### Bernoulli Map properties

I am referring to the function stated here http://en.wikipedia.org/wiki/Dyadic_transformation This map is defined on $[0,1]$ by $f_n(x)=nx [mod 1]$ There are three things I do not quite understand, ...
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### Function from Cantor Set to itself.

I am stuck in getting rational functions (except identity) defined from Cantor set to itself. Please help me to get out these functions.
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### Does “$\exists \delta >0$ S.T $||x(0)-x_e||<\delta\Rightarrow \displaystyle \lim_{t\rightarrow \infty}||x(t)-x_e||=0$” imply stability?

Recall the definition of stable and Asymptotically stable: A fixed point $x_e$ of a vector field is called (Lyapunov) stable if $\forall \varepsilon>0,\exists \delta(\varepsilon)$ such that ...
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### Doubts related to a phase plane diagram.

I want to draw phase plane diagram of the following differential equation $$\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 10 y = 0.$$ Please check if my approach is correct. I have some doubts about it. ...
### For which $\alpha$ do the $\epsilon$-neighborhoods of $\{k\alpha \mod 1 \mid k = 1, \ldots , poly(1/\epsilon) \}$ cover $[0,1]$?
For which $\alpha$ do the $\epsilon$-neighborhoods of $\{k\alpha \mod 1 \mid k = 1, \ldots , poly(1/\epsilon) \}$ cover $[0,1]$? In this paper on quantum computing (last paragraph of page 25), Dorit ...