1. Show that both (0,0) and (1,0) are both equilibria of the ODE d d *x(t)=x+xy−(x+y)(x² + y²)¹/2 y(t)=y-x²+(x − y)(x² + y²)1/2 2. Linearise the system at both of these equilibria points. What can you conclude regarding stability of each equilibrium point? 3. Convert the system to polar co-ordinates (r(t), 0(t)). 4. Find any equilibria * of the ODE for the radius r(t) only. What do these equilibria indicate about the dynamics of the planar system (x(t), y(t))? 5. Now, let r(t) = r* where r* is the non-zero equilibrium of the ODE for r(t). Solve the ODE for e(t). Use the explicit solution for 0 (t) to determine if any limit cycles exist. Hint: The following identities might be useful 1) 1- cos(0) (1+cos(0) 1-cos² (0) (1+ cos(0) sin²(0) csc²(0)+ cos(0) sin²(0) d 2)cot(t) = csc² (t) == dt 3) cot(x) + csc(x) = cot(x/2).
1. Show that both (0,0) and (1,0) are both equilibria of the ODE d d *x(t)=x+xy−(x+y)(x² + y²)¹/2 y(t)=y-x²+(x − y)(x² + y²)1/2 2. Linearise the system at both of these equilibria points. What can you conclude regarding stability of each equilibrium point? 3. Convert the system to polar co-ordinates (r(t), 0(t)). 4. Find any equilibria * of the ODE for the radius r(t) only. What do these equilibria indicate about the dynamics of the planar system (x(t), y(t))? 5. Now, let r(t) = r* where r* is the non-zero equilibrium of the ODE for r(t). Solve the ODE for e(t). Use the explicit solution for 0 (t) to determine if any limit cycles exist. Hint: The following identities might be useful 1) 1- cos(0) (1+cos(0) 1-cos² (0) (1+ cos(0) sin²(0) csc²(0)+ cos(0) sin²(0) d 2)cot(t) = csc² (t) == dt 3) cot(x) + csc(x) = cot(x/2).
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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Just 4 and 5 please
Transcribed Image Text:1. Show that both (0,0) and (1,0) are both equilibria of the ODE
d
d
*x(t)=x+xy−(x+y)(x² + y²)¹/2
y(t)=y-x²+(x − y)(x² + y²)1/2
2. Linearise the system at both of these equilibria points. What can you conclude
regarding stability of each equilibrium point?
3. Convert the system to polar co-ordinates (r(t), 0(t)).
4. Find any equilibria * of the ODE for the radius r(t) only. What do these equilibria
indicate about the dynamics of the planar system (x(t), y(t))?
5. Now, let r(t) = r* where r* is the non-zero equilibrium of the ODE for r(t). Solve
the ODE for e(t). Use the explicit solution for 0 (t) to determine if any limit cycles
exist.
Hint: The following identities might be useful
1)
1- cos(0)
(1+cos(0)
1-cos² (0)
(1+ cos(0)
sin²(0)
csc²(0)+
cos(0)
sin²(0)
d
2)cot(t) = csc² (t)
==
dt
3) cot(x) + csc(x) = cot(x/2).
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