1
Topic 7, HW 7:
4.1# 1:
Solve the following equations in order to determine whether each of the steady states is stable or
unstable.
a)
b)
????
????
????
????
= ???? ? ???? 2 , ??, ?? > 0 , ?? > 0
= ????? + ???? 2 , ??, ?? > 0 , ?? > 0
4.1 # 2:
Use phase space techniques to deduce the stability of the steady states in
a)
b)
????
????
????
????
= ???? ? ???? 2 , ??, ?? > 0 , ?? > 0
= ????? + ???? 2 , ??, ?? > 0 , ?? > 0
4.3 #1:
Find the form of the following systems when each of their steady states is translated to the origin.
a)
????
????
= ?? (?? ? 2)(?? ? 2) ,
????
????
= (?? + 2)(?? ? 1)2
2
4.3 #2:
Find the trajectories, integral curves and stability of the critical point at the origin for the
following systems
a)
????
= ???,
????
????
= 2?? ? ??
????
b)
????
= ?3?? + ??,
????
????
= ?? ? 3??
????
4.4 #2:
For the following exercises find the critical points at which the system is almost linear and
discuss their stability.
????
????
= (?? ? 1)(?? ? 2)2 (?? ? 1) ,
????
????
= (?? ? 1)2 (?? ? 2)
4.4 # 3:
For the following exercises find the critical points at which the system is almost linear and
discuss their stability.
????
????
= ??(1 + cos ??),
????
????
= ??(1 + sin ??)
3
4.5 #4:
For the following systems classify and draw the phase portrait for the steady state at the origin.
????
????
= ?? + ?? + 2?? 2 ? ?? 2 ,
????
????
= ?? ? 2?? + ?? 3
4.5 #5:
For the following systems classify and draw the phase portrait for the steady state at the origin.
????
????
= 2?? ? ?? 3 ,
????
????
= 2?? + ?? 2 (?????????????? ??????? ????? ?????????? ????????????????)
4
4.7 # 3:
Determine the periodic solutions and their stability for the system, Equation
????
????
= ??? + ???(??) ,
????
????
= ?? + ???(??)?? 2 = ??2 + ?? 2 if
a) ?(??) = ??(?? ? 1)2 (?? ? 3)(?? ? 4)
b) ?(??) = ?? 2 ? 4
5.2 #5:
Draw the bifurcation diagrams for the following systems
a) ??? = (???? ? ?? 3 )(?? + 2?? ? 1)
b) ??? = (???? ? ?? 3 )(?? ? ?? 2 )

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