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Describe the bifurcation that occurs when A= -1. 1 hifunantion diagram nes

Describe the bifurcation that occurs when A= -1. 1 hifunantion diagram nes

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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Exercises 6-14 deal with the logistic family of functions given by F(x) = dz(1 – 1). 6. For which values of A does F, have an attracting fixed point at z = 0? %3D 7. For which values of A does F, have a nonzero attracting fixed point? 8. Describe the bifurcation that occurs when A = 1. 9. Sketch the phase portrait and bifurcation diagram neard = 1. 10. Describe the bifurcation that occurs when A = 3. %3D 11. Sketch the phase portrait and bifurcation diagram near A = 3.
Transcribed Image Text:Exercises 6-14 deal with the logistic family of functions given by F(x) = dz(1 – 1). 6. For which values of A does F, have an attracting fixed point at z = 0? %3D 7. For which values of A does F, have a nonzero attracting fixed point? 8. Describe the bifurcation that occurs when A = 1. 9. Sketch the phase portrait and bifurcation diagram neard = 1. 10. Describe the bifurcation that occurs when A = 3. %3D 11. Sketch the phase portrait and bifurcation diagram near A = 3.
68 DYNAMICAL SYSTEMS 12. Describe the bifurcation that occurs when =-1. 13. Sketch the phase portrait and bifurcation diagram near =-1 14. Compute an explicit formula for the periodic points of period 2 f 15. Consider F(x) = Ar - a. Show that the 2-cycle given by ±Vti: repelling whend> -1. 16. Consider the family of functions F(x) = x° - Ax³. Discuss the k: furcation of 2-cycles that occurs when = 2. Hint: Note that this is odd function for each A, so points of period 2 may be found by solving F(2) = %3D = -r.
Transcribed Image Text:68 DYNAMICAL SYSTEMS 12. Describe the bifurcation that occurs when =-1. 13. Sketch the phase portrait and bifurcation diagram near =-1 14. Compute an explicit formula for the periodic points of period 2 f 15. Consider F(x) = Ar - a. Show that the 2-cycle given by ±Vti: repelling whend> -1. 16. Consider the family of functions F(x) = x° - Ax³. Discuss the k: furcation of 2-cycles that occurs when = 2. Hint: Note that this is odd function for each A, so points of period 2 may be found by solving F(2) = %3D = -r.
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