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Consider the following bifurcation diagram for an autonomous differential equation with parameter r: -1.0 -0.5 0.5 1.0 1.5 2.0 -1 (1) Use the diagram to identify all bifurcation values. Explain your reasoning.

Consider the following bifurcation diagram for an autonomous differential equation with parameter r: -1.0 -0.5 0.5 1.0 1.5 2.0 -1 (1) Use the diagram to identify all bifurcation values. Explain your reasoning.

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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Answer part 1 please

Consider the following bifurcation diagram for an autonomous differential equation with parameter r: -1.0 -0.5 0.5 1.0 1.5 2.0 -1 (1) Use the diagram to identify all bifurcation values. Explain your reasoning. (2) Sketch possible x' vs. x or x vs. t graphs for each of the following parameter values: r = -0.5, r = 0, r = 0.5, and r = 1. (You probably want to draw phase lines for each value of r first.) (3) Set r = 0.5 and imagine a solution function with initial condition x(0) = 2. What is the long-term behavior of this function? Explain what tells you this. (This problem continues on the next page.) Reach: Suppose a long time has passed and your function x(t) has evolved to a value very close to the answer you gave in (3). At this point, will changing your parameter to r = 1.75 impact the long-term behavior of this solution? Briefly explain.
Transcribed Image Text:Consider the following bifurcation diagram for an autonomous differential equation with parameter r: -1.0 -0.5 0.5 1.0 1.5 2.0 -1 (1) Use the diagram to identify all bifurcation values. Explain your reasoning. (2) Sketch possible x' vs. x or x vs. t graphs for each of the following parameter values: r = -0.5, r = 0, r = 0.5, and r = 1. (You probably want to draw phase lines for each value of r first.) (3) Set r = 0.5 and imagine a solution function with initial condition x(0) = 2. What is the long-term behavior of this function? Explain what tells you this. (This problem continues on the next page.) Reach: Suppose a long time has passed and your function x(t) has evolved to a value very close to the answer you gave in (3). At this point, will changing your parameter to r = 1.75 impact the long-term behavior of this solution? Briefly explain.
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