Consider the discrete-time dynamical systems (R, No, S¿) given by the iteration of the map f:R →R below. Determine and prove the attractivity and stability properties of I, using the definition: • attractivity (attracts points in a neighbourhood W of I , attracts points in R, attracts all compact sets in R, attracts all sets in R) • and stability (unstable, stable, asymptotically stable, globally asymptotically stable) (a) f(x)= 0. Determine the (only) compact and invariant set I. (b) f(x) = –x. Show that I = {0} is a compact and invariant set, and determine its attractivity and stability. Is {0} the only compact and invariant set? Prove your answer.
Consider the discrete-time dynamical systems (R, No, S¿) given by the iteration of the map f:R →R below. Determine and prove the attractivity and stability properties of I, using the definition: • attractivity (attracts points in a neighbourhood W of I , attracts points in R, attracts all compact sets in R, attracts all sets in R) • and stability (unstable, stable, asymptotically stable, globally asymptotically stable) (a) f(x)= 0. Determine the (only) compact and invariant set I. (b) f(x) = –x. Show that I = {0} is a compact and invariant set, and determine its attractivity and stability. Is {0} the only compact and invariant set? Prove your answer.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Consider the discrete-time dynamical systems (R, No, St) given by the iteration of the
map f:R → R below. Determine and prove the attractivity and stability properties of I,
using the definition:
• attractivity (attracts points in a neighbourhood W of I, attracts points in R, attracts
all compact sets in R, attracts all sets in R)
• and stability (unstable, stable, asymptotically stable, globally asymptotically stable)
(a) f(x) = 0.
Determine the (only) compact and invariant set I.
(b) f(г) — —г.
= -:
Show that I
{0} is a compact and invariant set, and determine its attractivity
and stability.
Is {0} the only compact and invariant set? Prove your answer.
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