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A diffeomorphism ƒ : [0, 1] → [0, 1] is called Morse-Smale if f has only hyperbolic periodic points. 1. Prove that a Morse-Smale diffeomorphism has only finitely many periodic points. 2. Prove that a Morse-Smale diffeomorphism of [0, 1] is structurally stable. 3. Prove that the map ƒ(r) = x³ + ³r is a Morse-Smale diffeomorphism on the interval [-1/2,1/2].

A diffeomorphism ƒ : [0, 1] → [0, 1] is called Morse-Smale if f has only hyperbolic periodic points. 1. Prove that a Morse-Smale diffeomorphism has only finitely many periodic points. 2. Prove that a Morse-Smale diffeomorphism of [0, 1] is structurally stable. 3. Prove that the map ƒ(r) = x³ + ³r is a Morse-Smale diffeomorphism on the interval [-1/2,1/2].

Advanced Engineering Mathematics
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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A diffeomorphism f : [0, 1] → [0, 1] is called Morse-Smale if f has only hyperbolic periodic points. 1. Prove that a Morse-Smale diffeomorphism has only finitely many periodic points. 2. Prove that a Morse-Smale diffeomorphism of [0, 1] is structurally stable. 3. Prove that the map f(x) = x³ + ³r is a Morse-Smale diffeomorphism on the interval [-1/2,1/2].
Transcribed Image Text:A diffeomorphism f : [0, 1] → [0, 1] is called Morse-Smale if f has only hyperbolic periodic points. 1. Prove that a Morse-Smale diffeomorphism has only finitely many periodic points. 2. Prove that a Morse-Smale diffeomorphism of [0, 1] is structurally stable. 3. Prove that the map f(x) = x³ + ³r is a Morse-Smale diffeomorphism on the interval [-1/2,1/2].
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