Math Problem Statement
I'm solving problems dealing with bifurcations in 1D systems. The initial function given is x_dot = r + (1/2)x - (x/(1+x)). In detailed steps: determine the bifurcation point(s), sketch the bifurcation diagram of the fixed point(s) versus parameter r, and sketch all the qualitatively different phase portraits as r is varied. Double check the math before answering.
Solution
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Math Problem Analysis
Mathematical Concepts
Bifurcation Theory
Dynamical Systems
Fixed Points
Saddle-Node Bifurcation
Formulas
ẋ = r + (1/2)x - (x / (1 + x))
Quadratic formula: x = [-b ± √(b² - 4ac)] / 2a
Discriminant Δ = (r - 1/2)² - 2r
Theorems
Saddle-Node Bifurcation
Suitable Grade Level
Undergraduate (Math, Physics, or Engineering)
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