1. Show that both (0,0) and (1,0) are both equilibria of the ODE d dt d x(t)=x+xy−(x+y)(x² + y²) 1/2 d³(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 r* 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) = * 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+cos(0) 1-cos² (0) = (1+cos(0) sin² (0) = = csc² (0) + cos(0) sin²(0) 1 1) 1 - cos(0) d =- dt 2) cot(t) = csc²(t) 3) cot(x)+csc(x) = cot(x/2).
1. Show that both (0,0) and (1,0) are both equilibria of the ODE d dt d x(t)=x+xy−(x+y)(x² + y²) 1/2 d³(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 r* 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) = * 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+cos(0) 1-cos² (0) = (1+cos(0) sin² (0) = = csc² (0) + cos(0) sin²(0) 1 1) 1 - cos(0) d =- dt 2) cot(t) = csc²(t) 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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can you please show the working out for part 5 please
Transcribed Image Text:1. Show that both (0,0) and (1,0) are both equilibria of the ODE
d
dt
d
x(t)=x+xy−(x+y)(x² + y²) 1/2
d³(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 r* 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) = * 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+cos(0)
1-cos² (0)
=
(1+cos(0)
sin² (0)
=
= csc² (0) +
cos(0)
sin²(0)
1
1)
1 - cos(0)
d
=-
dt
2) cot(t) = csc²(t)
3) cot(x)+csc(x) = cot(x/2).
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