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]
55 questions
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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
- 5,375
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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
- 410
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
- 8,576
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
- 9,986
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votes
1 answer
Find the invariant manifolds of the equilibrium
Consider the system
$$\begin{cases}\dot{x}=x+y\cos(y)\\
\dot{y}=-y
\end{cases}$$
which has the unique equilibrium point $(0,0)$. I want to find the invariant manifolds for this system. The Jacobian at the origin…
Anon
- 939
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
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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
- 2,016
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
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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 that
$$\Omega = \left\{ (S,I) : I\geq 0, S \geq 0,…
JOEF
- 269
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votes
1 answer
Is there a translational invariant probability measure over $\mathbb Z$?
I hope to find a non-trivial probability measure $\mu:\mathcal B\subset\mathcal P(\mathbb Z)\to[0, 1]$ such that $\mu(A) = \mu(c+A),~\forall A\in\mathcal B,~\forall c\in\mathbb Z$.
Of course, no finite subset of $\mathbb Z$ could be $\mu-$measurable…
Alma Arjuna
- 6,521
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votes
1 answer
Proving a set is positive invariant for a dynamical system
I have the following dynamical system:
$$
\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\}$$ is positive…
George Wilson
- 592
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votes
1 answer
When are attracting sets invariant?
Consider a control system of the form $\dot{x}(t) = f(x(t), u(t))$ where $u(t)$ is the control input, $x \in \mathbb{R}^{n}$, $u \in \mathbb{R}^{m}$. Assume $f$ is Lipschitz continuous so that the existence and uniqueness of solutions…
Truong
- 673
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votes
0 answers
Intersection of Bouligand tangent cones is the tangent cone of the intersection.
I am reading the book "Set-Valued analysis" by Jean-Pierre Aubin and Hèlène Frankowska. On page 152, table 4.4, statement 5b), we read:
If $K_1$ and $K_2$ are closed derivable subsets contained in $X, x \in K_1 \cap K_2$ satisfies…
Olayo
- 119
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votes
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Using Poincare-Bendixson to show there is a periodic orbit
I am having difficulty using Poincare-Bendixson to show that a particular system of differential equations has a periodic orbit. The system is
$$
\left\{\begin{array}{rcl}
\dot{x} & = & -3y+x(1-x^2-y^2+y)
\\
\dot{y} & = &…
Anon
- 939
2
votes
1 answer
A doubt on the positively invariant sets for a system of ODEs
Let $f:\mathbb{R}^2 \to (-\infty,0]$ be such that $f(0,0)=0$. Consider the following system of ODEs
$$ \begin{aligned} x'(t) &= f\big( x(t), y(t) \big)\\ y'(t) &= y^2(t) \end{aligned} $$
for $t > 0$. A set $S \subseteq \mathbb{R}^2$ is said to be…
MathRookie2204
- 779