By writing the ODEs in their normal form near the bifurcation point show that (a) ˙x = 1 + rx x cos(x) + sin(r) undergoes a saddle-node bifurcation at (r∗ , x∗ ) = (0, 0), (b) ˙x = (r2 2 r + 1)(x + x3) ) ln(1 + x) undergoes a transcritical bifurcation at (r∗ , x∗ ) = (1, 0). Please specify the order of the approximation as an error in ε > 0 and the corresponding size of r, x, e.g. |x - x∗| < ε.
By writing the ODEs in their normal form near the bifurcation point show that (a) ˙x = 1 + rx x cos(x) + sin(r) undergoes a saddle-node bifurcation at (r∗ , x∗ ) = (0, 0), (b) ˙x = (r2 2 r + 1)(x + x3) ) ln(1 + x) undergoes a transcritical bifurcation at (r∗ , x∗ ) = (1, 0). Please specify the order of the approximation as an error in ε > 0 and the corresponding size of r, x, e.g. |x - x∗| < ε.
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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By writing the ODEs in their normal form near the bifurcation point show that
(a) ˙x = 1 + rx x cos(x) + sin(r)
undergoes a saddle-node bifurcation at (r∗ , x∗ ) = (0, 0),
(b) ˙x = (r2 2 r + 1)(x + x3) ) ln(1 + x)
undergoes a transcritical bifurcation at (r∗ , x∗ ) = (1, 0).
Please specify the order of the approximation as an error in ε > 0 and the corresponding
size of r, x, e.g. |x - x∗| < ε.
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