Questions tagged [basins-of-attraction]
The basin of attraction of a given attractor is the set of initial conditions leading to convergence to that attractor.
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Fokker-Planck: uniqueness and attractiveness of stationary distribution (gradient systems)
consider the Langevin equation ($N$-dimensional) with nonlinear drift term but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of ...
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Plotting the bifurcation diagram for Ikeda map
I'm trying to plot the bifurcation diagram for Ikeda map. I wrote a code in Python to get the points of this diagram, but it seems that for $u > 1$ the points diverge and my code doesn't work ...
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the relation between chaos and fractal basin
Does fractal boundary of basin of attraction has something to do with chaos?
I think fractal boundary must lead to chaos, and how about the other way round?
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Does dynamical system has volume preserving property in basin of attraction?
I have a question on the basin of attraction: Does the dynamic flow on every bounded region inside of a basin of attraction has volume preserving property?
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Any idea or direction on how to go about estimating tight region of attraction for this nonlinear system?
I am working with the following discrete nonlinear system:
$\begin{gathered}
{\delta _{s + 1}} &= &{\theta _s}{\delta _s} \hfill \\
{\theta _{s + 1}} &=& {\theta _s} + c\left( {1 - ...
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Understanding where does the second (stochastic) attractor of the system come from.
I am currently reading a paper, studying the population dynamics in 3-dimensional Lotka-Volterra model with the following interaction change descriptive system:
$$
\begin{equation}
\begin{cases}
...
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$p$ being an attracting fixed point of $f$ is equivalent to $p$ being a repelling fixed point of $f^{-1}$
Let $f\colon U\to U$ be a homeomorphism, where $U\subset \mathbb R$ is an open set. By the definition, a fixed point $p\in U$ is attracting if there exists $e > 0$ s.t. if $|x-p| < e$ for some $...
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Number of basins on a N-dimensional surface
I have a real analytic function $f(x_0, x_1, x_2 ... x_N)$, so $\mathbb{R}^N \rightarrow \mathbb{R}$.
I know that all $\frac{\partial}{\partial x_i} f(x_0, x_1, x_2 ... x_N)$ have at most 1 root.
I am ...
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Finding the region of attraction using a Lyapunov function
I'm trying to find an estimate for the region of attraction of an equilibrium point. The notes from Nonlinear Control by Khalil suggest that defining
$$
V(x) = x^TPx,
$$
where $P$ is the solution of
$$...
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Disjoint attractors have a disjoint basin of attraction
Let $\Omega$ be a topological space and $\tau:\Omega\to\Omega$. $A\subseteq\Omega$ is called stable if for every neighborhood $V$ of $A$, there is a neighborhood $U$ of $A$ with $$\tau^n(U)\subseteq V\...
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Show that $(\tau^n(x))$ is eventually in every neighborhood of $A$ iff $\lim_{n\to\infty}d(\tau^n(x),A)=0$
Let $(\Omega,d)$ be a metric space, $\tau:\Omega\to\Omega$, $A\subseteq\Omega$ be compact and $$B:=\{x\in\Omega:(\tau^n(x))_{n\in\mathbb N}\text{ is eventually in every neighborhood of }A\}.$$
How ...
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Show that a set $A$ has a fundamental neighborhood iff its basin of attraction is a neighborhood of $A$
Let $\Omega$ be a topological space, $\tau:\Omega\to\Omega$, $A\subseteq\Omega$, $\mathcal N(A):=\{N\subseteq\Omega:N\text{ is a neighborhood of }A\}$ and $$B(A):=\left\{x\in\Omega\mid\forall N\in\...
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Basin of Attraction of Lorenz Attractor
What exactly is the basin of attraction of the classical Lorenz attractor with standard parameter values (see Wikipedia)?
I often read that "almost all" trajectory starting values do tend to ...
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Continuous function preserves the immediate basin of attraction?
Suppose that $f$ is a differentiable function and $p$ is a fixed point of $f$ such that $|f′(p)|<1$.
Let $K$ be the maximal interval about $p$ in which all points tend asymptotically to $p$ under $...
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Can I say a manifold is partitioned by the basin of attractions?
For smooth continuous dynamical system,
$$\dot{x} = f(x),$$
on manifold $\mathcal{M}$, can I say it is partitioned by countably many basins of attraction?
Motivation
I want to prove something which ...
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What is the basin of attraction of the Dottie Number?
I know that the sole real solution to $\cos x = x$, which I'll call $\textbf{d}$, is a universal attractor of the cosine function for all the real numbers. Meaning that:
$$\forall x\in\mathbb{R} \...
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Region of attraction of simple ODE with perturbation
There are a few nice discussions about ROA covering a few subtopics:
Region of attraction of : $x'=-y-x^3,y'=x-y^3$ via Lyapunov Function
Region of attraction and stability via liapunov&#...
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Region of attraction and stability via Liapunov's function
EXERCISE:
Estimate the region of stability for the stationary point $O(0,0)$ given the differential system:
$$x'=y$$
$$y'=x^7-2\cdot x-y$$
using the liapunov's funcion $V(x,y)=\dfrac{1}{2}\...
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Lyapunov function and an open disk inside the basin of $(0,0)$
a)Find a strict Lyapunov function for the equilibrium point $(0,0)$ of $$x'=-2x-y^2$$ $$y'=-y-x^2$$. b)Find $\delta>0$ as large as possible so that the open disk of radius $\delta$ and center $(0,0)...
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Question about the basin of attraction of the origin
Consider the system $$x'=(\epsilon x+2y)(z+1)$$
$$y'=(\epsilon y-x)(z+1)$$
$$z'=-z^3$$
(a) Show that the origin is not asymptotically stable when $\epsilon=0.$
(b) Show that when $\epsilon <0,$ ...
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Region of attraction of : $x'=-y-x^3,y'=x-y^3$ via Lyapunov Function
PROBLEM:
$1)$ Show that the stationary point $O(0,0)$ is asymptotic stable
$2)$ Find a region of attraction for the system :
$$x'=-y-x^3$$
$$y'=x-y^3$$
given the Lyapunov's function: $$V=x^2+y^2$$
...
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Is the domain of a strict Lyapunov function contained in the basin of attraction of the equilibrium?
I know that if a have a strict Lyapunov $L$ function defined in an open set $O$ that contains the equilibrium point $x_{0}$, then that point must be asymptotically stable. Which means that if my ...
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Show that the equilibrium point $(0,0)$ is asymptotically stable and an estimate of its basin of attraction
Consider the system $$\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 ...
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Basin of attractions with continuity and discontinuity
Suppose $p$ is an attracting fixed point under a continuous map $f$ and that the basin of attraction of $p$ is the interval $(a,b)$. How do I show that $f(a,b)\subset(a,b)$?
So I was able to show ...
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Trajectories in a basin of attraction
For all trajectories in a basin of attraction approaching and ending in a fixed point, are these trajectories one and the same?
Thanks in advance.
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Can the basin of attraction be multiple intervals?
Suppose $f$ is a map from $\mathbb{R} \rightarrow \mathbb{R}$, $f'$ exists everywhere, and $(0,0)$ is an attracting fixed point. Does the basin of attraction of $(0,0)$ have to be one interval? Either ...
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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 ...
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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 ...
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Is every basin of attraction completely invariant?
I can't seem to find a definitive answer in the literature. I believe the answer is yes, but my focus has been on the rational maps on the Riemann sphere. At the very least I'm confident that if the ...
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Fixed point, with basin of attraction
Suppose $p$ is an attracting fixed point under a continuous map $f$ and that the basin of attraction of $p$ is the interval $(a,b)$. How do I show that $f(a,b) \subset (a,b)$?
I said that since $f$ ...
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Show that the basin of attraction contains $(a,b)$
Given $f : [a,b] \to [a,b]$ and $|f(x)-f(y)| < |x-y|$ for all $x,y \in [a,b]$, I have shown so far that there exists a unique fixed point in $[a,b]$ and that fixed point is attracting. How can I ...
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Basin of attraction of a fixed point
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$ ...
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On finding a strict Liapunov function
I need to find a strict Liapunov function for this system at the equilibrium point $(0,0)$
$$x'= -2x-y^{2}$$
$$y'=-y-x^{2}$$
Also need to determine $\delta > 0$ as large as possible so that the ...
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Basin of attraction of the fixed map $f(x) = x-x^3$
Prove that the interval $(-\sqrt 2 ,\sqrt 2 )$ is the basin of attraction of the fixed point $0$ of the map $f(x)=x-x^3$, for $x \in \mathbb{R}$.
How one would prove this? In the examples I've seen ...
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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 ...
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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 ...
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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" ...
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