Definition: The basin of attraction is the defined as the set of all initial conditions $x_{0}$ such that $x(t$) tends to an attracting fixed point $x^{\ast}$ as time $t$ tends to $\infty$. Is this basin of attraction necessarily an open set? My text mentioned nothing about the basin of attrac...

The question can be stated in a short way, although I provide motivation after. Given a non-linear vector field (for a first attempt, we can assume polynomial vector field) and an asymptotically stable equilibrium point, is there any result that ensures that the basin of attraction of this equili...

In general, Newton's method for root finding has a "bubbly" boundary between basins of convergence for different roots. This is where fractals are usually created from. But outside these "bubbly" boundaries there are very clear areas where there's no ambiguity about which root Newton's method w...

We've got a function: $ f : \Bbb R \to \Bbb R$ defined by $f(x) = x^3 - 9$. Let $x^* $ be its root, which means $ f(x^*) = 0$. We want to find approximation for $x^*$ using a Newton's method. There are two questions I don't know how to answer: We choose an initial guess: $x_0 = 2,5$. Does it lin...

I want to prove that the immediate basin of attraction of a finite attracting fixed or periodic point is simply connected. We are talking about complex numbers ! According to Remark 2 p. 281 and Exercise 4.2 p. 283 of the text of Devaney [1], If $z_0$ is a finite attracting orbit (i.e., $z_0...

I want to find the basin of attraction of a fixed point. For example, I have $f(x)=\frac 1{x+1}$, whose fixed points are $\frac{-1\pm \sqrt{5}}{2}$. Now, I must create a neighborhood around the $x$ point that would consist of all points around it that would attract to it. How can I figure out the...

For $a,b,c>0$ and $a+b+c=1.$ Prove$:$ $$\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ca+2b^{2}+2b}\geqq \frac{1}{ab+bc+ca}$$ This inequality is easy and there are two nice proof by AM-GM or by C-S also. SOS also help here$:$ $$\text{LHS}-\text{RHS}=\frac{3\Big[\sum\limits_{cyc} (ab+b...

For $a,b,c \geqslant 0.$ Then $$9 ( a+b+c ) ^{2} ( ab+ac+bc ) ^{2}+108a^2b^2c^2-31abc ( a+b+c ) ^{3} \geqslant 0.$$ I use computer and found that the following stronger inequality holds for all reals of $a,b,c.$ $$\sum (a^2 -bc) \Big[9\, \left( a+b+c \right) ^{2} \left( ab+ac+bc \right) ^{2}+108\...