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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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 ...
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In the definition for sensitivity to initial conditions what exactly does the distance between trajectories mean?
I've seen this definition for sensitive dependance in Modeling Life (Garfinkel et al, 2010):
$$d(M_{t} - N_{t}) = e^{\lambda * t} * d(M_0 - N_0)$$
or alternatively from Wikipedia:
$$ {|\delta \mathbf {...
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Fractal basin of attraction in dynamical systems
My goal is to proof the following:
Theorem: Let $(M,\mathbb{R},F)$ be a DS on a smooth (compact) manifold $M$, with dynamical law $\dot{x}=F(x)$. Assuming it has two attractors $A_1,A_2$ with basin of ...
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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{...
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$\textbf{Reference Request:}$ Analytically Characterizing Basins of Attraction Boundaries and Measures
I understand that doing the above is not possible in general. However, when it is possible, what are common methods people use to analytically characterize basin of attraction boundaries (i.e. via a ...
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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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a question regarding subsets of basin of attraction
I am reading the review paper named review on computational methods for lyapunov functions (Doi: 10.3934/dcdsb.2015.20.2291) which can be seen here
My question regarding the lower part of page 4 in ...
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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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What's the difference between a total basin of attraction and an immediate basin of attraction?
I understand that for an attracting fixed point $\hat{p}$ of a holomorphic self-map defined on some Riemann surface $S$ we define the total basin of attraction as $\mathcal{A}=\text{Bas}(\hat{p})=\{p\...
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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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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 $$B(A):=\left\{x\in\Omega\mid\forall N\in\...
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How to estimate distance from root to nearest immediate basin boundary for Newton's method in one complex variable?
Context: I want to check that the atom domain size estimate is smaller than the inradius of the Newton immediate basin, for centers of hyperbolic components in the Mandelbrot set, and thus justify ...
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Basin of attraction of the Lorenz attractor
What exactly is the basin of attraction of the classical Lorenz attractor with standard parameter values?
I often read that "almost all" trajectory starting values do tend to the Lorenz ...
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Continuous function preserves the immediate basin of attraction?
Suppose that $f$ is a differentiable function and $p$ is a fixed point of $f$ such that $|f′(p)|<1$.
Let $K$ be the maximal interval about $p$ in which all points tend asymptotically to $p$ under $...
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Largest circle in basin of attraction of the origin.
We're given the following dynamical system:
$$ \begin{aligned} \dot x &= -x + y + x (x^2 + y^2)\\ \dot y &= -y -2x + y (x^2 + y^2) \end{aligned} $$
What's the largest constant $r_0$ such that ...
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Can I say a manifold is partitioned by the basin of attractions?
For smooth continuous dynamical system,
$$\dot{x} = f(x),$$
on manifold $\mathcal{M}$, can I say it is partitioned by countably many basins of attraction?
Motivation
I want to prove something which ...
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Basin of attraction of simple nonlinear coupled ODE
Consider ($\epsilon = 0.1$)
\begin{equation}\label{eq:general eq}
\begin{aligned}
\dot{x}_1(t) &= x_1(x_1-0.5)(x_1+0.5)+\epsilon x_2\\
\dot{x}_2(t) &= x_2(x_2-0.5)(x_2+0.5)+\epsilon x_1
\end{...
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What is the basin of attraction of the Dottie Number?
I know that the sole real solution to $\cos x = x$, which I'll call $\textbf{d}$, is a universal attractor of the cosine function for all the real numbers. Meaning that:
$$\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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Lyapunov stability
i have a question regarding Lyapunov stability and basin of attraction.
Let
$${x}'=-x-y$$
$${y}'=2x-y+y^3$$
Use $$V(x,y)=x^{2}+\frac{1}{2}y^{2}$$ to determine the stability of (0,0) and a basin of ...
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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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Determining the size of a basin of attraction
$$\dot x = -2x-y^2$$
$$\dot y = -y-x^2$$
$(0,0)$ is an obvious attractive fixed point, and I'll only look at this one.
I need to get the maximal radius $r > 0$ for a ball centered on the origin so ...
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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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Basin of attraction
Let $$g(x)=\frac{2}{5}x^3-\frac{7}{5}x$$. The fixed points are 0 and $$\sqrt6$$. There is a period-2 orbit of 1 and -1. The critical points are $$\sqrt\frac{7}{6}$$
a. calculate the Schwarzian ...
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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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