Research article · 2018 · Full text
International Journal of Statistics and Applied MathematicsProlongation and prologational limit sets its topological dynamics
Volume 3(6) · Pages 148–149
Research areasMathematical Dynamics and FractalsAdvanced Differential Equations and Dynamical SystemsAdvanced Topology and Set Theory
Abstract
Author-writtenIn this short paper we try to state and prove some Impovtent theorem which we are given as an exercise to N.P. Bhatia and G.P. Szego and V.V. Stepanove & V.V. Nemytskii. Here we prove that every non-empty compact invariant set MCX contains some minimal sets. Now we also prove that a set MCX is minimal and only if for each x∈M,xR=M from these two rsults. We also prove that the omega limit set Ωx is minimal if for any two poihts y,z∈Ωx and any index a∈A There exists aτ∈R such that π(yτ)∈V_a (z).
Summary
Source-checkedPlain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.
Why it matters
- Prolongation and prolongational limit sets are significant in the study of dynamical systems, specifically within stability and dispersive system theories. View evidence ↗
The question
- The author aims to state and prove several important theorems that were originally presented as exercises in works by Bhatia, Szego, Stepanove, and Nemytskii. View evidence ↗
What they did
- The paper employs mathematical proofs to demonstrate that any non-empty compact invariant set contains minimal sets. View evidence ↗
- The author seeks to prove the necessary and sufficient condition that a set is minimal if and only if the orbit of every point in the set equals the set itself. View evidence ↗
- The study analyzes the conditions under which an omega limit set is minimal based on the existence of a time shift that maps one point into the neighborhood of another. View evidence ↗
What they found
- The omega limit set (Ωx) is always closed and invariant; if it is compact, it is connected (a continuum), but if it is not compact, none of its components are compact. View evidence ↗
- When the closure of the positive orbit γ+(x) is compact, the omega limit set Ωx is guaranteed to be non-empty, compact, connected, and thus a continuum. View evidence ↗
- The first positive prolongation D+(x) is defined as the set of points y reachable by the limit of a flow π(xn, tn) where xn approaches x and tn is in R+. View evidence ↗
- The positive and negative orbits are subsets of their respective prolongations, and the prolongational limit sets contain the omega and alpha limit sets. View evidence ↗
- For any minimal set M, if at least one point is an interior point, then every point within that set must also be an interior point. View evidence ↗
- An omega limit set Ωx is minimal if for any two points y and z in the set, there is a time τ such that the flow of y at that time enters a neighborhood of z. View evidence ↗
What it means
- The author concludes the paper by providing the formal proofs for the theorems originally posed as exercises in classical dynamical systems literature. View evidence ↗
Visual summary
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Hierarchy of Limit and Prolongation Sets
AI-generated conceptual illustration (Abstract mathematical topological spaces represented as nested geometric manifolds). It explains the idea and does not show study data.
View source evidence ↗Introduction Main Results
Introduction Main Results
Limit Set
A point y ∈ X is called positive (omega) limit point of a point x(γ(x)or π x ) if there exists a net {t n } in R + with t n → ∞ and π(x, t n ) → y. The set of all omega limit points of a point x ∈ X is called omega limit set of X and is usually denoted by Ωx or ⋀ + (x).
i.e., Ωx = {y ∈ X : there exists a net {t n } in R + Such that t n → ∞ and π(x, t n ) → y}. The set of all omega limit points of all points x ∈ B c X will be called the omega limit set of B and is denoted by π(B), i. e. , Ω (B) = U{Ωx ∶ X ∈ B}. Similarly, we define negative limit (alpha limit) points of a point x (γ(x) or π x ) and their set is denoted by a x or Λ(x) i.e., a x = {y ∈ X ∶ there exists a net {t n } in R - Such that t n → -∞ and π(x, t n ) → y}. Now we mention some basic properties of limit sets without proof.
Theorem 1: For every For every x ∈ X 1. Ω x is closed and invariant 2.
For the proof we refer to Bhatia [1] .
Theorem 2: If closure of γ + (x) is compact Ω x is non-empty, compact and annected. Hence it is continuum.
Proof: Now we first prove that if γ + (x) is compact then Ω x is non-empty and compact. For, this we write Ω x =∩ γ + (π(x, t)).
t > 0
But γ + (π(x, τ))cγ + (x) for τ > 0. This implies that {γ + (π(x, τ)): τ ∈ R + } is a decreasing family of closed. Subsets of γ + (x). Weting that it has finite intersection property and γ + (x). Noting that it has intersection property and γ + (x) is compact. As given from the characterization of compactness by finite intersection property, we have γ + (π(x, r)) ≠ ∅. Therefore Ω x is non-empty and Ω x cγ + (x) is compact, closed subset of a compact space. Furthermore the compactness of Ω x implies its connectedness [Theorem 1]. Consequently the omega limit set is a continuum, whenever it is compact.
Prolongation and prolongational limit sets
Let X be a topological space. For any point x ∈ X the sets D + (x) → (D(x)) and J + (x)(J -(x)) defined as below are respectively called the first prolongation and prolongational limit sets of x. The '+' sign denotes the positive prolongation and the '-' sign stands for the negative protongation. These sets plays an important role in the theory of dynamical system, Particularly, in the stability theory and the theory of dispersive system. Their formal definitions run as follows: i) D + (x) = {y ∈ X: there exists a net {x n } is X and there is a net {t n } in R + with x n → x such that π(x n , t n ) → y}. ii) D -(x) = {y ∈ X: there exists a net {x n } is X and a net {t n } in R -with x n → x such that π(x n , t n ) → y}. iii) J + (x) = {y ∈ X: there exists a net {x n } is X and a net {t n } in R + with x n → x and t n → ∞ such that π(x n , t n ) → y}. iv) J -(x) = {y ∈ X: there exists a net {x n } is X and a net {t n } in R -with x n → x and t n → ∞ such that π(x n , t n ) → y}.
From above it is obivious that D + and D -maps for X to 2 x . It is clear from the above definitions that for any x ∈ X, γ + (x) ⊂ D + (x) and γ -(x) ⊂ D -(x). For this we take x n = X and t n = t for all n in the above definitions. Also +(x) ⊃ Ωx and J -(x) ⊃ α x Minimal Set: A set M ⊂ X is called minimal if it is nonempty closed and invariant and does not have any proper subset with these three properties that 𝑥𝑅 ̅̅̅̅ ⊂ 𝑁 ⊂ M, as N being closed and invariant this implies 𝑥𝑅 ≠ 𝑀, is a contradiction to our assumption. Therefore, M is minimal.
Theorem 3: If a minimal set M ⊂ X has an interior point then all its. points interior points.
Proof: Let x ∈ M be in interior point of M, then there exists an index a such V 1 (x) ⊂ M. Since M is minimal, for each t ∈ R, 𝜋(V a (x), 𝑡) ⊂ M this with the homomorphic property of 𝜋 (V a (x), t) is a knighthood of 𝜋(x, t). Thus every point of the trajectory is an interior point.
Theorem 4: the omega limit set Ω x is minimal if for any two points y and z Ω x and index a ϵ𝐴 there exists a τ ∈ R such that π(y, τ)ϵV a (z).
Proof: Let us suppose on the contrary, i.e., Ω x is not minimal. This implies there is a minimal set M ⊂ Ω x Thus M and Ω x M are two disjoint, empty, invariant sets.
for ∈ M, π(y, t) ∉ Ω x -M for all t ∈ R.
Compact set M and any point z ∈ Ω x-M of a Housedroff space X can be by two disjoint neighbourhoods Hence there exists a and b in A that V a (z) ∩ V b (M) = ∅. This implies π(y, t) ∉ V a (z)forall. t ∈ R.
References
3 in the original paper- Bhaitha NP, SZEGO GP. Dynamical System: Stability Theory and. Application L.N.M.-35 springer Varlag New York, 1967.
- Birkhoff GD. Dynamical system Amer, Math Soc, Collq. Pub. New York 1927, 9.
- Nemytskii VVI, Sepanove VV. Qualitative theory of differential equations Priceton University Press New Jersy, 1960.