The basin of attraction of a given attractor is the set of initial conditions leading to convergence to that attractor.
Questions tagged [basins-of-attraction]
54 questions
15
votes
1 answer
Bounding the basins of attraction of Newton's method
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…
Jay Lemmon
- 2,228
8
votes
0 answers
Region of attraction of simple ODE with perturbation
There are a few nice discussions about ROA covering a few subtopics:
Region of attraction of : $x'=-y-x^3,y'=x-y^3$ via Lyapunov Function
Region of attraction and stability via liapunov's function (No2)
Basin of attractions with continuity and…
sleeve chen
- 8,576
7
votes
1 answer
How to estimate distance from root to nearest immediate basin boundary for Newton's method in one complex variable?
Context: I want to check that the atom domain size estimate is smaller than the inradius of the Newton immediate basin, for centers of hyperbolic components in the Mandelbrot set, and thus justify using Newton's method to find the center given an…
Claude
- 5,852
7
votes
2 answers
Newton's method — for which initial guesses does it converge?
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…
Anne
- 1,567
6
votes
1 answer
Simple connectedness of basin of attraction
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…
6
votes
1 answer
Does every basin of attraction contain a critical point?
Years and years ago, back when I first became interested in fractals [but didn't know much about anything], I vaguely remember coming across an interesting theorem. The gist of it was that "every basin of attraction contains at least one critical…
MathematicalOrchid
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5
votes
0 answers
Radius of convergence around the origin for infinite composition of $\cos(z)$ in the complex plane
I was reading a textbook on complex dynamics that, in an introductory chapter, demonstrated the complex mapping: $f(z) = \cos(z).$
The text noted that the iteration $ f^{\circ n}(x) $ (infinite composition of $ f $) converges for every real $ x $ to…
user1557030
- 51
5
votes
3 answers
Lyapunov function and an open disk inside the basin of $(0,0)$
a)Find a strict Lyapunov function for the equilibrium point $(0,0)$ of $$x'=-2x-y^2$$ $$y'=-y-x^2$$. b)Find $\delta>0$ as large as possible so that the open disk of radius $\delta$ and center $(0,0)$ is contained in the basin of…
user441848
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4
votes
2 answers
Finding the region of attraction using a Lyapunov function
I'm trying to find an estimate for the region of attraction of an equilibrium point. The notes from Nonlinear Control by Khalil suggest that defining
$$
V(x) = x^TPx,
$$
where $P$ is the solution of
$$
PA+A^TP=-I,
$$
will yield the best results for…
Octavio
- 43
4
votes
3 answers
Largest circle in basin of attraction of the origin.
We're given the following dynamical system:
$$ \begin{aligned} \dot x &= -x + y + x (x^2 + y^2)\\ \dot y &= -y -2x + y (x^2 + y^2) \end{aligned} $$
What's the largest constant $r_0$ such that the circle $x^2 + y^2 < r_0^2$ lies in the origin's basin…
Radost Waszkiewicz
- 1,857
- 8
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4
votes
4 answers
Question about the basin of attraction of the origin
Consider the system $$x'=(\epsilon x+2y)(z+1)$$
$$y'=(\epsilon y-x)(z+1)$$
$$z'=-z^3$$
(a) Show that the origin is not asymptotically stable when $\epsilon=0.$
(b) Show that when $\epsilon <0,$ the basin of attraction of the origin contains the…
user441848
- 1,738
- 1
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- 55
4
votes
3 answers
Show that the equilibrium point $(0,0)$ is asymptotically stable and an estimate of its basin of attraction
Consider the system $$\begin{aligned} \dot{x} &=-y-x^3+x^3y^2\\ \dot{y}&=x-y^3+x^2y^3\end{aligned}$$ Show that the equilibrium point $(0,0)$ is asymptotically stable and an estimate of its attractiveness basin.
Clearly the point $(0,0)$ is a…
user402543
- 1,183
4
votes
3 answers
Is a basin of attraction necessarily an open set?
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…
Mathematicing
- 6,541
3
votes
1 answer
What if the level set of Lyapunov function is disconnected? - when estimating region of attration
Consider $\frac{dx}{dt}=f(x)$, where $x\in\mathbb{R}^n$. Suppose $x=0$ is a stable equilibrium.
It is classical way to estimate region of attraction of $0$ by finding a $C^1$ function $V(x)$ such that when $x\in D\subset \mathbb{R}^n$
(1) V(x) is…
happyle
- 303
3
votes
1 answer
Equilibrium point of a function and its basin of attraction
I'm very lost with the following problem:
Consider $f=(f_1,f_2,f_3)\in\mathcal{C}(\mathbb{R}^3,\mathbb{R}^3)$ such that $f(0,0,0)=(0,0,0)$ and
\begin{equation} x_1f_1+x_2f_2+x_3f_3<0 \tag{*}
\end{equation} for all…
hackerman
- 63