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.

# Questions tagged [set-invariance]

42 questions

**5**

votes

**2**answers

### 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 phase space.
By Jordan's theorem, $C$ gives rise…

temo

- 4,989
- 2
- 25
- 56

**5**

votes

**1**answer

### 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 invariance of parabolic PDE systems, and have noticed…

Leif Ericson

- 340
- 1
- 13

**5**

votes

**1**answer

### 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 $$\dot{x} = f(x), \ \ \ x\in \mathbb{R}^n.$$
A scalar valued function $I(x)$ is said to be an…

sleeve chen

- 7,641
- 7
- 40
- 87

**5**

votes

**1**answer

### 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 that
$$A^TQ+QA=-I$$
and where $A=Df(0,0)$, to find a…

sequence

- 9,002
- 5
- 36
- 102

**4**

votes

**3**answers

### 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 defined by $or(x_{0})=\left\{ {x\in X: x=\phi^{t}x_{0},…

pcepkin

- 347
- 2
- 11

**4**

votes

**2**answers

### 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 a surjective topological dynamic system.
Let…

FShrike

- 8,766
- 3
- 7
- 25

**3**

votes

**1**answer

### $V_{\lambda}$ 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 solution of $\dot x = Ax$ such that $x(t_0) \in…

jokadeka055

- 107
- 4

**3**

votes

**1**answer

### 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 $\begin{cases} \dot{\mathbf x} = F(\mathbf x) \\ \mathbf x(0) = \mathbf x_0\end{cases}$ where…

cip999

- 1,976
- 9
- 21

**3**

votes

**1**answer

### 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 $$S=\left\{ (x,y) \in \mathbb{R}^2:x \leq0 \right\}$$ is positive…

George Wilson

- 592
- 3
- 11

**3**

votes

**1**answer

### 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 existence and uniqueness of solutions…

Truong

- 613
- 5
- 15

**3**

votes

**1**answer

### 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 that
$$\Omega = \left\{ (S,I) : I\geq 0, S \geq 0,…

JOEF

- 259
- 2
- 9

**2**

votes

**0**answers

### 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 point
$$
\phi = (0, 0).
$$
I have found…

WeirdoChicken

- 21
- 1

**2**

votes

**1**answer

### 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:
$$\begin{cases}
\dot{s} = -\beta s i + \omega r\\
\dot{i} = \beta s i - \gamma i \\
\dot{r} = \gamma i - \omega r
\end{cases},$$
where $s$,…

the_candyman

- 12,956
- 4
- 31
- 60

**2**

votes

**0**answers

### 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 continuous.
If $y(0) \ge 0$ lies in the positive…

miosaki

- 4,605
- 4
- 26
- 69

**2**

votes

**2**answers

### 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 $B = \{(x, y)\mid x^2+y^2 \leq 1\}$ positively invariant?
Now what I have…

Gragbow

- 641
- 3
- 17