Maths 761: Dynamical Systems Assignment 3

Q1。考虑微分方程组
8
<

_ x = y x z
_ y = x y3
_ z = z 2 x y 2 x4
+ x2

（a）证明原点是非双曲平衡。
（b）推导原点中心歧管的幂级数近似值。
（c）使用您的方程式计算中心歧管，以得出近似的动力学

（d）在中央歧管上绘制动力学局部相位图和相位图

1 + x2
2。
[30分]

Q2。再次考虑一维微分方程组
_ x = x + 4 x3
（1 x2

（a）回想一下，您为此系统确定了三个分支。对于每个分叉

（b）注意，对于相同的参数值= 1，发生两个分叉。

[10分]
Q3。两参数族
_ x = x + 4 x3
（1 x2

（a）首先，x = 0：1仅略大于0。

（b）接下来，x = 1并使用XPPAUT以相同的方式计算分叉图

（c）用手证明=的任何平衡

=
。证明在（;）=（
; </ s> </ s> </ s> </ s> </ s>

; </ s> </ s> </ s> </ s> </ s>
）。
（d）使用XPPAUT来计算两个参数同时变化的两参数分叉集。

Q1. Kǎolǜ wéifēn fāngchéng zǔ

Q1. Consider the system of di erential equations
8
<
:
_ x = y x z
_ y = x y3
_ z = z 2 x y 2 x4
+ x2
de ned for (x; y; z) 2 R3
.
(a) Show that that the origin is a non-hyperbolic equilibrium.
(b) Derive a power series approximation for the centre manifold of the origin.
(c) Use your equation for the centre manifold to nd an approximation to the dynamics on
the centre manifold.
(d) Draw a local phase portrait of the dynamics on the centre manifold and a phase portait
of the three-dimensional phase space locally near the origin.
Note: Stability analysis of the equilibrium solution on the centre manifold is perhaps best
done using Theorem 2.11 from Glendinning (page 38) with a Lyapunov function
of the form V (x1; x2) = x2
1 + x2
2.
[30 marks]

Q2. Consider again the one-parameter family of di erential equations
_ x =  x + 4 x3
(1 x2
)
from Q4. of Assignment 2.
(a) Recall that you identi ed three bifurcations for this system. For each of the bifurcation
points, verify that the appropriate genericity conditions are satis ed.
(b) Note that two of the bifurcations occur for the same parameter value  = 1. Explain why
you would or would not expect this to be a coincidence.
[10 marks]
Q3. The two-parameter family
_ x =   x + 4 x3
(1 x2
)
with ;  2 R contains the family from the above Q2., obtained when  = 0.
(a) First, x  = 0:1 only slightly larger than 0.
 Show that x = 1 are equilibria of this system if  = 0:1.
 Use XPPAUT to compute the bifurcation diagram and hand in this bifurcation diagram
with your assignment. Choose an appropriate range for  so that your diagram shows all
the bifurcations that occur for this system. Colour and/or label the curves (by hand) to
indicate clearly the stability of all equilibria and the locations of the bifurcation points.
(b) Next, x  = 1 and use XPPAUT to compute the bifurcation diagram in the same way
as for part (a).
(c) Show (by hand) that any equilibrium for  = 
and  = 
also exists for  = 
and
 = 
. Show that any bifurcation for such equilibria also occur for both (; ) = (
; 
)
and (; ) = (
; 
).
(d) Use XPPAUT to compute a two-parameter bifurcation set with both  and  varying.
 Identify the bifurcations you found for  = 0,  = 0:1, and  = 1 in the two-parameter
bifurcation set.
 The two-parameter bifurcation curves intersect in several points. Explain what happens
at such intersection points.

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