Questions tagged [set-invariance]

A given set is (positively) invariant with respect to a given dynamical system if the following property holds: whenever the initial state is in the set, the state remains in the set thereafter.

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center manifold and bifurcation: 2D Bifurcation system reduction

I have this system to study $$ \left\{ \begin{aligned} \frac{dx}{dt} &= y-x - x^2 \\[5pt] \frac{dy}{dt} &= \mu x - y - y^2 \end{aligned} \right. $$ I have derived the Jacobian around fixed ...
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$V_{\lambda}$ is invariant under $A$

From section 1.5 of Doering & Lopes1: Exercise 10. Let $A \in M(n)$, let $\lambda \in \mathbb{R}$, let $V_{\lambda} := \ker(\lambda I - A)$ and let $x : \mathbb{R} \to \mathbb{R}^{n}$ be a ...
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Invariant curves induce invariant regions in discrete, 2D dynamical systems?

Consider a discrete dynamical system $x_{k+1} = f(x_k)$, where $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$, sufficiently smooth, and let $C \subseteq \mathbb{R}^2$ be an invariant, closed curve in the ...
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Check if region where $\dot{x} > 0$, $\dot{y} > 0$ is positively/negatively invariant

I have the following linear system $\dot{x} = -x$ and $\dot{y} = -2y + 2x^3$. I need to characterise the regions where $\dot{x} > 0, \dot{y} > 0$, $\dot{x} < 0, \dot{y} > 0$ and so on. I ...
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Criterion for finding invariant sets in continuous dynamical system

I'm reading some handouts of a course on dynamical systems, which focuses largely on autonomous systems of ODE's in Euclidean space (i.e. solutions of $\begin{cases} \dot{\mathbf x} = F(\mathbf x) \\ \...
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Is $\pi\left(\bigcap_{n}A_n\right)=\bigcap_{n}\pi\left( A_n\right)$, when $\pi$ is a projection?

For me, $\Bbb N$ includes $0$. I am referencing, yet again, this text, exercise $19$, page $30$. Let $K$ be a compact Hausdorff space, and $\phi:K\to K$ continuous and surjective - i.e. $(K;\phi)$ is ...
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PDE Solution at Large Times and Invariance

I have a few general questions related to PDE solution behavior, specifically as it relates to set invariance. Namely, I've been reading papers that give necessary/sufficient conditions for set ...
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What methods are there for finding Lyapunov functions to prove nonlinear system invariance and stability?

As stated LaSalle's Invariance Principal, a Lyapunov function V can be used to prove certain invariance and stability properties of a dyanimcal system. Are there rules of thumb for finding V? Do ...
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Invariant set for SIRS model

A SIRS model is a set of ODE used to describe the evolution of epidemic/pandemic. It is defined as follows: $$\begin{cases} \dot{s} = -\beta s i + \omega r\\ \dot{i} = \beta s i - \gamma i \\ \dot{r} =...
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Is the set $[0, 1]$ invariant for this ODE (2 strategies replicator equation)?

Consider the replicator equation for $2$ strategies: $$\dot{x} = x(1-x)((a+b)x - b).$$ Prove that for any $x(0) \in [0,1]$, then $x(t) \in [0,1] ~\forall t >0.$ My attempt Suppose that $x(0) \in (...
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Proving a set is positively invariant for a system

I have the following system and set S: $$x' = - y + x - x^3 - xy^2\\y' = y + x - y^3 - x^2y \\\ S = \{(x, y) \in R^2 : x^2 + y^2 \leqslant \sqrt2 \}$$ I need to prove that for this system, the set S ...
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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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Proof question on lemma included in paper on invariant sets of PDEs

I'm not exactly sure how to categorize this question so sorry if it's in the wrong place. I'm trying to understand one part of a lemma given in a paper on invariant sets for PDEs and I'm getting hung ...
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Is the union of stable manifolds of gradient flows an open?

This question is a more general problem but for simplicity let us consider the particular planar dynamical system, which essentially represents a continuous gradient descent of $f$ under constraint $y ...
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How to prove that a set is invariant?

I am a Master 1 student studying dynamical systems. I am new to it. There's a problem, I have with invariant sets. Excuse me, I didn't know how to start the following question. I have the following ...
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Intersection between invariant curves for a map that coming from an autonomus vector field.

Consider the system of differential equation $$\dot x = f(x) $$ with $x\in{M}$ where $f$ is a function of the only $x$, so the system is autonomus. Now let $\phi^t(x)$ be the flow of the system, that ...
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Show without solving the ODE that an equality is an invariant after initial condition

I am studying invariants for systems of ODEs. For example, I have proved that if $x'(t)=f(x(t))$ such that $x\, (t_0) > c$, and if $f(k)>0$ for all $k \geq c$, then the derivative will keep ...
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Positively invariant set with ODE undefined on part of boundary

Consider the system $$\begin{cases} \dot x = x\left(5-\frac{5x}{12}-\frac{y}{1+x}\right)\\ \dot y = y\left(1-\frac{y}{5x}\right) . \end{cases}$$ I want to show that there is a positively invariant set ...
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Looking for a proof of LaSalle's invariance principle for a dynamical system on a manifold.

I found the following version of LaSalle's theorem and it appears to be stayed differently from the original. Consider the smooth dynamical system on an $n-$manifold $M$ given by $\dot{x} = X(x)$ and ...
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The level sets of integral are invariant sets (Wiggins' textbook)

I am reading the following book: Introduction to applied nonlinear dynamical systems and chaos, Stephen Wiggins On p. 77, for a general vector field $$\dot{x} =...
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When does the trajectory lie in the positive orthant?

I have the following differential equation $$M \dot y(t) = y(t) + K p(t)$$ where $M$ is a nilpotent matrix of degree $m$ and $K$ is some matrix of suitable order. Functions $p(t)$ are piecewise ...
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Closed positively invariant subset is a subset of a basin in an autonomous ODE

I have a doubt about a theorem in Hirsch and Smale's Differential Equations, Dynamical Systems, and Linear Algebra textbook. In the chapter of Stability of equilibria they state Theorem 2 Let $\bar{...
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For which values of $\alpha$ is the disk $B = \{(x, y) \mid x^2+y^2 \leq 1\}$ positively invariant?

Given the following dynamical system $$\begin{aligned} \dot x &= f(x,y) = -x + \alpha y \\ \dot y &= g(x,y) = -y\end{aligned}$$ for which values of $\alpha$ is the disk $B = \{(x, y)\mid x^2+y^...
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Show that the annulus $1 \leq x^2 + y^2 \leq 3$ is positively invariant

Given the planar dynamical system $$\dot x = x - y - x^3, \\ \dot y = x + y - y^3$$ show this is positively invariant in the annulus $1 \leq x^2 + y^2 \leq 3$. Hints: $$x^4 + y^4 = (x^2 + y^2)^2 − 2x^...
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Limit set of a non-autonomous system is not invariant

I am trying to understand why the (positive or omega) limit set of a non-autonomous dynamical system \begin{equation} \dot{x}=f(t,x) \label{ftx} \tag{1} \end{equation} is not necessarily (positively) ...
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Find the positively invariant annulus

Given the following dynamical system $$\begin{aligned} \dot r &= r(1-r^2) \\ \dot \theta &= r^2 \left( (1-r^2)^2 + \sin(\theta)^2 \right)\end{aligned}$$ find the annulus $A$ defined as follows ...
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Proving a set is positive invariant for a dynamical system

I have the following dynamical system: $$ \begin{align} \dot{x}&=-x-2y^2, \\ \dot{y}&=-x^2y-y^3. \end{align} $$ My task is to show that, for the dynamical system, the set $$S=\left\{ (x,y) \...
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Geometric intuition of an invariant set, positively invariant and negatively invariant

Definition: Invariant set A set $S \subseteq \mathbb{M}$ is invariant if $\Psi_{t}\left ( S \right )\subseteq S,\forall t$ -if $\forall x \in S$, $\Psi_{t}\left ( x \right )\subseteq S,\forall t$ -if $...
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Compactness and positive invariance of set under flow of ODEs

Given a system of ODEs, $$x'=y$$ $$y'=x-x^3-y$$ $$x(0)=x_0$$ $$y(0)=y_0,$$ also given a set $S=\{(x,y):V(x,y)\le k, x>0\}$, $V(x,y)=-\frac{x^2}{2}+\frac{x^4}{4}+\frac{y^2}{2}$, where $-\frac{1}{4}&...
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Positive invariance of a set under a system of ODEs

Given the system of ODEs, $$x'=x(1-x-y)$$ $$y'=y(x-1),$$ $Q=\{(x,y):x\ge 0, y\ge 0\}$, and $S=(x,y)\in Q:x+y\le k$, $k>1$, I need to show that $S$ is invariant under this system of ODEs. Attempted ...
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5 votes
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Positively invariant neightbourhood using Lyapunov function

Given the following system of nonlinear ODEs, $$x_1'=-x_1-x_2$$ $$x_2'=2x_1-x_2^3$$ I need to use the quadratic Lyapunov function $$V(x) = x^TQx$$ where $Q$ is a positive definite matrix such ...
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Show a positively invariant set open

Suppose $f := R^n \to R^n$ is continuously differentiable. Given $\phi(t,y)$ is the solution to the IVP: $\dot{x} = f(x), \ x(0) = y$. Assume $x_0$ is an asymptotically stable equilibrium of the ...
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Positively invariant $(S,I)$-triangle for SIS dynamical system

Consider the following differential equations $${dS \over dt} = \lambda-\beta SI-\mu S+\theta I$$ $${dI \over dt} =\beta SI-(\mu +d)I-\theta I$$ In all papers that I have read it is only mentioned ...
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2 votes
1 answer
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Determine whether a set is Invariant, Positively invariant or negatively invariant

I have just started a dynamical systems course and I am a bit confused as to how to determine if something is positively or negatively invariant, or just invariant. I know the defintions for ...
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2 answers
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Nexus between connectedness of invariant set and stability of dynamical system

Let an autonomous dynamical system is characterized by the state equation $$ \dot x(t) = f(x(t)),\quad x(0)=x_0 $$ with state $x(t)\in \mathbb R^n$. The definition of invariant set, as I came across, ...
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ODE system and show infinite number of positively invariant ellipsoids

The system of ODEs is: $$ \dot{x} = -2x+yz \\ \dot{y} = x-xz \\ \dot{z} = xy $$ I found two lines of equilibria etc. but I now need to find the parameters for this "energy" or Lyapunov function, so ...
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4 votes
3 answers
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Dynamical systems and invariant sets

I have basic questions to understand the invariant sets of dynamical systems. Let me define a dynamical system $\left\{ {T, X, \phi^{t}}\right\}$. Here an orbit with a starting value $x_{0}$ is ...
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Showing that a nonlinear system is positively invariant on a subset of $\mathbb{R}^2$

I have the following nonlinear system: \begin{align} ...
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2 votes
1 answer
4k views

Omega limit set is invariant

In the ODE where $y'=f(y(t))$ and $y(0)=y_0$. The omega limit set $\omega(y_0)$ is positively invariant and also negatively invariant. I want to prove first that its positively invariant and then ...
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Existence of invariant set in dynamical system generated by ODE

Is there any nonempty, compact and invariant set in dynamical system generated by this system of equations? $x'=x+\sin{(xy+2)}-7$ $y'=-y+\arctan{(x^2+y^3-6)}$ My idea is to use this fact: Not empty ...
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Maximal Positive Invariant Set -- Some fine print

I would like to share something I noticed on the definition of Maximal Positively Invariant Sets. Definition 1. For a discrete-time system of the form $x_{k+1}=f(x_{k})$ (and $x_{k}\in \mathbb{X}\...
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3 votes
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When are attracting sets invariant?

Consider a control system of the form $\dot{x}(t) = f(x(t), u(t))$ where $u(t)$ is the control input, $x \in \mathbb{R}^{n}$, $u \in \mathbb{R}^{m}$. Assume $f$ is Lipschitz continuous so that the ...
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