# 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 ...
286 views

### 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 ...
34 views

### 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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1 vote
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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 ...
57 views

### 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 ...
1 vote
48 views

### 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 ...
1 vote
145 views

### 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 ...
1 vote
256 views

### 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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1 vote
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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) ...
147 views

### 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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### 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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### 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 ...
1 vote
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 ...