Consider the system dx dt = J = (11 + x) sin(y), a) Determine all critical points of the given system of equations. The system has two types of critical points. Critical points of type I: ,2km) J = Critical points of type II: ( where k = 0, ±1, 2, ... b) Find the corresponding linear system near each critical point. NOTE: Find the Jacobian matrix J, and evaluate it at each critical point. At the critical points of type I: ? ? ? ? dy dt = 11- x - cos(y) At the critical points of type II: ? ? ? ,(2k + 1)π)
Consider the system dx dt = J = (11 + x) sin(y), a) Determine all critical points of the given system of equations. The system has two types of critical points. Critical points of type I: ,2km) J = Critical points of type II: ( where k = 0, ±1, 2, ... b) Find the corresponding linear system near each critical point. NOTE: Find the Jacobian matrix J, and evaluate it at each critical point. At the critical points of type I: ? ? ? ? dy dt = 11- x - cos(y) At the critical points of type II: ? ? ? ,(2k + 1)π)
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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Transcribed Image Text:c) Find the eigenvalues r, and r2 of of each linear system. What
conclusions can you then draw about the nonlinear system?
At critical points of type I: r₁=
1^2 =
At critical points of type II:r₁
=
r2 =
DOU
d) Using an appropriate graphing utility, draw a phase portrait of the
nonlinear system to confirm your conclusions, or to extend them if
the linear system does not provide definite information about the
nonlinear system.
The critical points of type I are/resemble
Choose one
saddle points
The critical points of type II are/resemble nodal points
spiral points
centers
Transcribed Image Text:Consider the system
dx
dt
San
(11 + x) sin(y),
Critical points of type II:
J =
a) Determine all critical points of the given system of equations.
The system has two types of critical points.
Critical points of type I:
,2kπ)
J =
dy
dt
where k = 0, 1, 2, ...
b) Find the corresponding linear system near each critical point.
NOTE: Find the Jacobian matrix J, and evaluate it at each critical point.
At the critical points of type I:
? ?
? ?
= 11- x - cos(y)
At the critical points of type II:
(??)
,(2k + 1)π)
What
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