. Consider the ODE d y² dx(t) = y + x √12² +32 [16- (1² +3²)] d dt ³(t) = ==x+ y √x² +12 [16-(x²+3²)] [16– y² . Convert this system to polar co-ordinates, and by directly analysing the scalar differential equation for r(t), show that there is a stable limit cycle. . Now, use Poincare-Bendixson theorem to show that there is at least one stable limit cycle by constructing an appropriate closed, positively invariant region.
. Consider the ODE d y² dx(t) = y + x √12² +32 [16- (1² +3²)] d dt ³(t) = ==x+ y √x² +12 [16-(x²+3²)] [16– y² . Convert this system to polar co-ordinates, and by directly analysing the scalar differential equation for r(t), show that there is a stable limit cycle. . Now, use Poincare-Bendixson theorem to show that there is at least one stable limit cycle by constructing an appropriate closed, positively invariant region.
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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Question
![. Consider the ODE
d
y²
dx(t) = y +
x
√12² +32 [16- (1² +3²)]
d
dt ³(t) =
==x+
y
√x² +12 [16-(x²+3²)]
[16–
y²
. Convert this system to polar co-ordinates, and by directly analysing the scalar
differential equation for r(t), show that there is a stable limit cycle.
. Now, use Poincare-Bendixson theorem to show that there is at least one stable
limit cycle by constructing an appropriate closed, positively invariant region.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4061eb31-6ba2-4539-9b51-dd7b05481e7e%2F51a1b18f-2aec-4102-a3e0-637065f62d2e%2Fxsj8u0s_processed.png&w=3840&q=75)
Transcribed Image Text:. Consider the ODE
d
y²
dx(t) = y +
x
√12² +32 [16- (1² +3²)]
d
dt ³(t) =
==x+
y
√x² +12 [16-(x²+3²)]
[16–
y²
. Convert this system to polar co-ordinates, and by directly analysing the scalar
differential equation for r(t), show that there is a stable limit cycle.
. Now, use Poincare-Bendixson theorem to show that there is at least one stable
limit cycle by constructing an appropriate closed, positively invariant region.
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