International Mathematical Forum, Vol. 9, 2014, no. 35, 1725 - 1739
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/imf.2014.410170
Alternate Locations of Equilibrium Points and Poles
in Complex Rational Differential Equations
Koh Katagata
National Institute of Technology, Ichinoseki College
Takanashi, Hagisho, Ichinoseki, Iwate 021-8511 Japan
c 2014 Koh Katagata. This is an open access article distributed under the
Copyright Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
We study configurations of simple equilibrium points of first order
complex differential equations consisting of the iteration of rational functions. Rational functions which we deal with have the unit circle or the
extended real line as Julia sets. Properties of Julia sets and the EulerJacobi formula lead to alternate locations of equilibrium points and
poles of the complex differential equations.
Mathematics Subject Classification: 37C10, 32A10, 37F10
Keywords: Equilibrium points, The Euler-Jacobi formula, Julia sets
1
Introduction
Let D be a domain in C and let f : D → C be a holomorphic function. We
consider the first order differential equation
ż ≡
dz
= f (z),
dt
(DE)
where t ∈ R and z ∈ D. A point ζ ∈ D is an equilibrium point of the
differential equation (DE) if it satisfies that f (ζ) = 0. Let f (x + iy) =
u(x, y) + iv(x, y). We identify the differential equation (DE) with the system of differential equations
ẋ = u(x, y), ẏ = v(x, y)
1726
Koh Katagata
in R2 . Equilibrium points are categorized as stable nodes, unstable nodes,
centers, stable foci, unstable foci and saddles. In general, in order to
classify equilibrium points of a system of the differential equations
ẋ = ϕ(x, y), ẏ = ψ(x, y)
(∗)
in R2 , we have to consider the linearization of the map (x, y) 7→ (ϕ(x, y), ψ(x, y))
and the behavior of solutions of the system of the differential equations (∗) near
equilibrium points. However, the behavior of the solutions of the differential
equation (DE) near equilibrium points is well-known. We can classify equilibrium points complex analytically as follows.
Theorem 1.1 ([1, 2]). Let ζ ∈ D be an equilibrium point of the differential
equation (DE). Then the equilibrium point ζ is
(1) a stable node if and only if f 0 (ζ) < 0,
(2) an unstable node if and only if f 0 (ζ) > 0,
(3) a center if and only if Imf 0 (ζ) 6= 0 and Ref 0 (ζ) = 0,
(4) a stable focus if and only if Imf 0 (ζ) 6= 0 and Ref 0 (ζ) < 0,
(5) an unstable focus if and only if Imf 0 (ζ) 6= 0 and Ref 0 (ζ) > 0.
Besides, the differential equation (DE) does not have saddles.
In [3], the author have studied the complex differential equation
ż = fc∗n (z) ≡ fc◦n (z) − z ,
where
1+c
fc (z) =
2(1 − c)
1
z−
z
(DE : c ; n)
with c ∈ [0, 1/3]
and fc◦n = fc ◦ · · · ◦ fc is the n-th iteration of fc . Configurations of equilibrium
points of the complex differential equation (DE : c ; n) are as follows.
Theorem 1.2 ([3]). For all positive integers n, the following statements
hold.
(a) In the case that 0 ≤ c < 1/3, the number of equilibrium points of
(DE : c ; n) on C \ R is exactly two and the two equilibrium points are
stable nodes.
(b) In the case that c = 1/3, there are no equilibrium points of (DE : c ; n)
on C \ R.
1727
Alternate locations of equilibrium points and poles
(c) All equilibrium points of the differential equation (DE : c ; n) are symmetric with respect to the real axis.
(d) Every equilibrium points of (DE : c ; n) on the real axis are unstable
nodes.
(e) Equilibrium points of (DE : c ; n) on the real axis and poles of fc∗n are
located alternately.
Considering properties of the complex differential equation (DE : c ; n) and the
Julia set of the rational function fc , we can obtain the analogical results to
Theorem 1.2.
Throughout this paper, we consider two rational functions A and B. The
rational function B is a Blaschke product
B(z) = e2πiθ z
z − aν
z − a1 z − a2
···
,
1 − a1 z 1 − a2 z
1 − aν z
where aj ∈ D = {z ∈ C : |z| < 1} for 1 ≤ j ≤ ν and θ ∈ [0, 1). The rational
function A is a conjugate of B, namely
A(z) = φ ◦ B ◦ φ−1 (z), φ(z) = i
z+1
.
z−1
Moreover, we deal with the differential equation
ż = A∗n (z) ≡ A◦n (z) − z,
(DE : a ; θ ; n)
where a = (a1 , . . . , aν ) ∈ Dν and A◦n = A ◦ · · · ◦ A is the n-th iteration of A.
Let EP (A∗n ) be the set of all equilibrium points of A∗n . Our main result is
the following.
Main Theorem . For any a = (a1 , . . . , aν ) ∈ Dν , θ ∈ [0, 1) and n ≥ 1, the
following statements hold.
(a) Equilibrium points of (DE : a ; θ ; n) on C \ R are ±i and the two equilibrium points are stable nodes or stable foci.
(b) All equilibrium points of the differential equation (DE : a ; θ ; n) are
symmetric with respect to the real axis.
(c) Every equilibrium points of (DE : a ; θ ; n) on the real axis are unstable
nodes.
(d) Equilibrium points of (DE : a ; θ ; n) on the real axis and poles of A∗n
are located alternately.
1728
2
Koh Katagata
Dynamics of rational functions A and B
Let f : Ĉ → Ĉ be a rational function. The rational function f can be written
as
p(z)
,
f (z) =
q(z)
where p and q are polynomials with no common roots. The rational function
f is continuous with respect to the spherical metric. The degree deg(f ) of f
is the maximum of the degrees of p and q. The degree deg(f ◦n ) of f ◦n is equal
to (deg(f ))n .
Definition 2.1. Let f : Ĉ → Ĉ be a non-constant rational function. The
Fatou set F(f ) of f is defined as
n
o
F(f ) = z ∈ Ĉ : the family {f ◦n }∞
is
normal
in
some
open
neighborhood
of
z
.
n=1
The Julia set J (f ) of f is the complement J (f ) = Ĉ \ F(f ). The Fatou set
F(f ) is open and the Julia set J (f ) is closed.
We suppose that d = deg(f ) ≥ 2. Here are some basic properties of the
Fatou set and the Julia set.
• The Fatou set and the Julia set are completely invariant, namely
f F(f ) = F(f ) = f −1 F(f ) and f J (f ) = J (f ) = f −1 J (f ) .
• For a positive integer n,
F(f ◦n ) = F(f ) and J (f ◦n ) = J (f ).
• The Julia set J (f ) is non-empty.
• The Julia set J (f ) has no isolated point.
• The Julia set J (f ) is the smallest closed completely invariant set containing at least three points.
• If the Julia set J (f ) has an interior point, then J (f ) = Ĉ.
• If the Julia set J (f ) is disconnected, it has uncountably many components.
• The Fatou set F(f ) has either zero, one, two or countably many components.
Alternate locations of equilibrium points and poles
1729
Let z0 be a point in Ĉ. The point z0 is a periodic point of f if there
exists a positive integer n such that f ◦n (z0 ) = z0 . Such the smallest n is called
the period of z0 . The point z0 is a fixed point of f if the period of z0 is one.
Definition 2.2. Let z0 = f ◦n (z0 ) be a periodic point of period n. The
multiplier λ = λ(z0 ) at z0 is defined as
(
(f ◦n )0 (z0 )
(z0 6= ∞),
λ=
◦n 0
1/ lim (f ) (z) (z0 = ∞).
z→∞
Periodic points are classified as follows.
• The periodic point z0 is superattracting if λ = 0.
• The periodic point z0 is attracting if 0 < |λ| < 1.
• The periodic point z0 is indifferent if |λ| = 1.
• The periodic point z0 is repelling if |λ| > 1.
Indifferent periodic points are classified into the following two cases.
• The periodic point z0 is parabolic if λ is a root of unity.
• The periodic point z0 is irrationally indifferent if |λ| = 1 and λ is not
a root of unity.
In the case that the periodic point z0 of period n is superattracting or attracting,
the attracting basin A(z0 ) of z0 is defined as
n
o
◦kn
A(z0 ) = z ∈ Ĉ : lim f (z) = z0 .
k→∞
Every superattracting and attracting periodic point belongs to the Fatou
set, and every parabolic and repelling periodic point belongs to the Julia set.
Attracting basins of periodic points are subsets of the Fatou set. It is difficult
to distinguish whether an irrationally indifferent periodic point belongs to the
Fatou set or the Julia set. The Julia set is characterized by repelling periodic
points.
Theorem 2.3. The Julia set J (f ) is equal to the closure of the set of all
repelling periodic points of f .
We investigate dynamics of the Blaschke product
B(z) = e2πiθ z
z − aν
z − a1
···
= e2πiθ z b1 (z) · · · bν (z)
1 − a1 z
1 − aν z
1730
Koh Katagata
and its conjugate A = φ ◦ B ◦ φ−1 , where θ ∈ [0, 1) is a real parameter, aj ∈ D
is a complex parameter (1 ≤ j ≤ ν) and
bj (z) =
z+1
z − aj
, φ(z) = i
.
1 − aj z
z−1
It is easy to check that the Möbius transformation bj maps the unit disk onto
itself and the Möbius transformation φ maps the unit disk onto the lower
half-plane.
Lemma 2.4. The Blaschke product B has attracting fixed points at the
origin and the point at infinity.
Proof. Fixed points of B are the solutions of the equation B(z) = z or
z e2πiθ b1 (z) · · · bν (z) − 1 = 0.
Since the Möbius transformation bj maps the unit disk onto itself, the inequality
2πiθ
e
b1 (z) · · · bν (z) < 1
holds for any z ∈ D. Therefore, there are no fixed points of B in the unit disk
except for the origin. The derivative of B is that
B 0 (z) = e2πiθ b1 (z) · · · bν (z) + e2πiθ z
ν
X
b1 (z) · · · b0j (z) · · · bν (z)
j=1
and the multiplier at the origin is that
B 0 (0) = e2πiθ b1 (0) · · · bν (0) = (−1)ν e2πiθ a1 · · · aν .
Since |B 0 (0)| < 1, the origin is an attracting fixed point. The Blaschke product
B is conjugate to itself via ι : z 7→ 1/z̄. This indicates that there are no fixed
points of B in Ĉ \ D̄ except for the point at infinity. The multiplier λ at ∞ is
that
1
= (−1)ν e−2πiθ a1 · · · aν .
λ=
0
lim B (z)
z→∞
Since |λ| < 1, the point at infinity is also an attracting fixed point.
Lemma 2.5. The rational function A = φ ◦ B ◦ φ−1 has attracting fixed
points at ±i.
Proof. Since φ(∞) = i and φ(0) = −i, the result follows by the above lemma.
Lemma 2.6. The Julia set J (B) is the unit circle S 1 .
Alternate locations of equilibrium points and poles
1731
Proof. The Möbius transformation bj maps the unit disk onto itself. Moreover,
bj maps S 1 and Ĉ \ D̄ onto themselves respectively. This indicates that the
unit circle S 1 is completely invariant under B, namely
B S 1 = S 1 = B −1 S 1 .
Since the Julia set J (B) is the smallest closed completely invariant set containing at least three points, the Julia set J (B) is contained in the unit circle
S 1 or
J (B) ⊂ S 1 .
We prove that J (B) = S 1 in the rest of the proof. We assume that there
exists a point ζ which is in S 1 \ J (B), namely ζ is in F(B). In this case, the
Fatou set F(B) is the union of two attracting basins A(0) and A(∞). Hence,
the orbit of ζ tends to the attracting fixed point 0 or ∞. On the other hand,
the orbit of ζ stays on the unit circle S 1 since it is invariant under B. This is
a contradiction. Therefore, we obtain that J (B) = S 1 .
Lemma 2.7. The Julia set J (A) is the extended real line R ∪ {∞}.
Proof. Since A is conjugate to B via φ or A = φ ◦ B ◦ φ−1 , we obtain that
J (A) = φ J (B) = φ S 1 = R ∪ {∞}.
Lemma 2.8. The equation
bj ◦ φ−1 (z) =
z + i a∗j
1 − aj
·
1 − aj
z − i a∗j
holds, where
a∗j =
1 + aj
.
1 − aj
Proof.
bj ◦ φ−1 (z) = bj
z+i
z−i
z + i − aj z − i
(1 − aj ) z + i (1 + aj )
=
= (1 − aj ) z − i (1 + aj )
z − i − aj z + i
1 + aj
1 − aj
1 − aj
=
·
1 + aj
1 − aj
z−i
1 − aj
z+i
=
z + i a∗j
1 − aj
·
1 − aj
z − i a∗j
1732
Koh Katagata
We calculate exact form of the rational function A. By the above lemma,
−1
2πiθ
B ◦ φ (z) = e
−1
φ (z)
ν
Y
−1
bj φ (z) = e
j=1
2πiθ
ν
z + i a∗j
z + i Y 1 − aj
·
.
z − i j=1 1 − aj
z − i a∗j
Therefore, we obtain that
A(z) = φ ◦ B ◦ φ−1 (z)
ν
z+i Y
e2πiθ
z − i j=1
= i
ν
Y
2πiθ z + i
e
z − i j=1
z + i a∗j
1 − aj
+1
·
1 − aj
z − i a∗j
z + i a∗j
1 − aj
−1
·
1 − aj
z − i a∗j
ν
ν
Y
Y
eπiθ z + i
(1 − aj ) z + i a∗j + e−πiθ z − i
(1 − aj ) z − i a∗j
= i
j=1
j=1
ν
ν
Y
Y
πiθ
∗
−πiθ
e
z+i
(1 − aj ) z + i aj − e
z−i
(1 − aj ) z − i a∗j
j=1
= i
j=1
A1 (z) + A2 (z)
,
A1 (z) − A2 (z)
where
πiθ
A1 (z) = e
z+i
ν
Y
(1 − aj ) z +
i a∗j
, A2 (z) = e
−πiθ
ν
Y
z−i
(1 − aj ) z − i a∗j .
j=1
j=1
Proposition 2.9. A1 (x) = A2 (x) and A01 (x) = A02 (x) hold for all x ∈ R.
Moreover, the following inequality holds for all x ∈ R :
A0 (x) =
Im [ A1 (x) A02 (x) ]
> 0.
h
i2
Im A1 (x)
Proof. The first two equations are obvious. The derivative of A is that
A0 (z) = i
2 [ A1 (z) A02 (z) − A01 (z) A2 (z) ]
.
h
i2
A1 (z) − A2 (z)
We transform the numerator and the denominator of the above equation as
A1 (x) A02 (x) − A01 (x) A2 (x) = A1 (x) A02 (x) − A02 (x) A1 (x) = 2i Im [ A1 (x) A02 (x) ]
and
A1 (x) − A2 (x) = A1 (x) − A1 (x) = 2i Im A1 (x)
1733
Alternate locations of equilibrium points and poles
for all x ∈ R. Therefore, we obtain that
A0 (x) = i
Im [ A1 (x) A02 (x) ]
2 · 2i Im [ A1 (x) A02 (x) ]
=
h
i2
h
i2
2i Im A1 (x)
Im A1 (x)
for all x ∈ R. We calculate the numerator of the last equation.
A1 (x) A02 (x)
ν
Y
πiθ
x+i
= e
(1 − aj ) x + i a∗j
j=1
"
× e−πiθ
ν
Y
∗
(1 − aj ) x − i aj
ν
ν Y
Y
X
+ x−i
(1 − aj ) ·
x − i a∗k
j=1
=
j=1
j=1 k6=j
ν
ν
Y
2 2 Y
x+i
|1 − aj |2 x + i a∗j + x + i |1 − aj |2
j=1
j=1
#
!
ν
X
x + i a∗j
j=1
Y
|x + i a∗k |2 .
k6=j
Since
x + i a∗j
1 − αj2 + βj2 + i · 2βj
1 + aj
= x+i
= x+i
1 − aj
(1 − αj )2 + βj2
and
(aj = αj + i βj ∈ D)
2
2
+
β
1
−
α
j
j
Im x + i a∗j =
> 0,
(1 − αj )2 + βj2
we obtain that
Im [ A1 (x) A02 (x) ]
=
ν
Y
2
|1 − aj |2 x + i a∗j j=1
! ν
ν
2
2
X
Y
2 Y
1
−
α
+
β
j
j
+x + i |x + i a∗k |2 > 0.
|1 − aj |2
2
2
(1 − αj ) + βj k6=j
j=1
j=1
Therefore, the inequality
Im [ A1 (x) A02 (x) ]
A (x) =
>0
h
i2
Im A1 (x)
0
holds for all x ∈ R.
Corollary 2.10. The multiplier of any repelling periodic orbit of A
ζ1 7→ ζ2 7→ · · · 7→ ζp 7→ ζ1
is greater than one.
1734
Koh Katagata
Proof. We may assume that p = 1. Let ζ ∈ J (A) = R ∪ {∞} be a repelling
fixed point. Its multiplier µ satisfies that |µ| > 1 and
(
A0 (ζ) > 0
(ζ 6= ∞),
µ=
0
1/ lim A (x) ≥ 0 (ζ = ∞).
x→∞
Therefore, we obtain that µ > 1.
3
Configurations of equilibrium points
The proof of the main theorem relies on Lemma 3.4 which determines configurations of equilibrium points and poles. The main ingredient of the proof of
Lemma 3.4 is the Euler-Jacobi formula.
Theorem 3.1 (The Euler-Jacobi Formula). Let f : C → C be a polynomial
of degree d. If all zeros w1 , w2 , . . . , wd of f are simple, then
d
X
g(wj )
=0
f 0 (wj )
j=1
for any polynomial g satisfying that deg(g) < deg(f 0 ) = d − 1.
Proof. We consider the rational function g(z)/f (z). Let Γ be a circle with a
large radius r surrounding all zeros of f . Applying the residues theorem, we
obtain that
Z
Γ
d
d
X
X
g(z)
g
g(wj )
dz = 2πi
Res
, wj = 2πi
.
f (z)
f
f 0 (wj )
j=1
j=1
On the other hand, we obtain that
Z
g(z) g(z)
.
dz ≤ 2πr max |z|=r
f (z) Γ f (z)
Since deg(g) + 1 < deg(f ) = d, the right hand side of the inequality tends to
zero as the radius r tends to infinity.
There are some applications of the Euler-Jacobi formula in [1].
Proposition 3.2 ([1, Proposition 2.6]). Let f be a polynomial of degree d ≥
2. We consider the differential equation (DE) and assume that all equilibrium
points w1 , w2 , . . . , wd of (DE) are simple, namely all zeros w1 , w2 , . . . , wd of f
are simple. Then the following statements hold.
Alternate locations of equilibrium points and poles
1735
(a) If w1 , w2 , . . . , wd−1 are nodes, then wd is also a node.
(b) If w1 , w2 , . . . , wd−1 are centers, then wd is also a center.
(c) If not all equilibrium points are centers, then there exist at least two of
them that have different stability.
Proposition 3.3 ([1, Proposition 2.7]). Let f be a polynomial of degree
d. We consider the differential equation (DE) and assume that all equilibrium
points of (DE) are simple. Moreover, we assume that d−2k equilibrium points
z1 , . . . , zd−2k are located on a straight line L for some k ≥ 0 and the other 2k
equilibrium points zd−2k+1 , . . . , zd are symmetric with respect to the line L .
Then the following statements hold.
(a) All the points on L are of the same type and if they are not centers,
then they have alternated stability.
(b) If all the points on L are of center type, then each pair of symmetric
points with respect to L is formed by two points of the same type and
if they are not centers, then they have opposite stability.
(c) If all the points on L are of node type, then each pair of symmetric
points with respect to L is formed by two points of the same type and
if they are not centers, then they have the same stability.
Theorem 1.2 and the main theorem are motivated by Proposition 3.3. If
all simple equilibrium points of the “polynomial” differential equation (DE)
are located on a straight line and they are not centers, then they have alternated stability (Proposition 3.3.a). Theorem 1.2 and the main theorem are
counterexamples in the case that f is a genuine rational function.
Lemma 3.4. Let F (z) = P (z)/Q(z) be a rational function, where P and
Q are polynomials with no common factors and of degrees deg(P ) = n and
deg(Q) = m respectively. We suppose that the two polynomials P and Q have
only simple roots. Let s1 , s2 , . . . , sn−2k be the real zeros of P with the order
sn−2k < · · · < s2 < s1 , and let w1 , w1 , . . . , wk , wk be the other zeros of P with
Im(wj ) 6= 0. Then the equation
F 0 (sδ ) = −
Q(sγ )
Rγδ (sδ )
·
· F 0 (sγ )
Q(sδ )
Rγδ (sγ )
holds, where γ < δ and
Rγδ (z) =
Y
j6=γ,δ
(z − sj )
k
Y
j=1
(z − wj ) (z − wj ) .
(#)
1736
Koh Katagata
If δ − γ is odd, then Rγδ (sγ ) and Rγδ (sδ ) have the same sign or
Rγδ (sγ ) · Rγδ (sδ ) > 0.
Proof. The derivative of the function F is that
F 0 (z) =
P 0 (z)Q(z) − P (z)Q0 (z)
P (z)Q0 (z)
P 0 (z)
−
=
Q(z)2
Q(z)
Q(z)2
and we obtain that
F 0 (zj ) =
P 0 (zj )
Q(zj )
for any zero zj of P . The polynomial P has the form
P (z) = C
n
Y
(z − zj ) = C
n−2k
Y
j=1
j=1
(z − sj )
k
Y
(z − wj ) (z − wj ) ,
j=1
where C is a constant. Applying the Euler-Jacobi formula to the rational
function Rγδ (z)/P (z), we obtain that
n
X
Rγδ (sγ )
Rγδ (sδ )
Rγδ (zj )
=
+
=0
0
0 (s )
P
(z
)
P
(s
)
P
j
γ
δ
j=1
or
P 0 (sδ ) = −
Rγδ (sδ )
· P 0 (sγ ).
Rγδ (sγ )
We transform the last equation as
Q(sγ )
Rγδ (sδ )
P 0 (sγ )
P 0 (sδ )
=−
·
·
Q(sδ )
Q(sδ )
Rγδ (sγ )
Q(sγ )
and we obtain that
F 0 (sδ ) = −
Q(sγ )
Rγδ (sδ )
·
· F 0 (sγ ).
Q(sδ )
Rγδ (sγ )
In the case that δ − γ is odd, the statement is obvious.
Proof of the Main Theorem. Equilibrium points of the differential equation
(DE :a ; θ ; n) correspond to periodic points of period k (k | n) of A or fixed
points of A◦n . Let
n
o(ν+1)n −ε
hni
sj
⊂ EP (A∗n )
j=1
be the set of all fixed points of A◦n in C except attracting fixed points ±i,
where ε = 2 if ∞ is a fixed point of A◦n or ε = 1 if ∞ is not a fixed point of
A◦n .
1737
Alternate locations of equilibrium points and poles
(a) By Lemma 2.7, there are no attracting or parabolic periodic points of A
hni
except attracting fixed points ±i. Hence, each sj is a repelling fixed point of
A◦n or
n
o(ν+1)n −ε
hni
sj
⊂ J (A).
j=1
Let λ± be the multiplier at ±i. Since |λ± | < 1 and
Re (A∗n )0 (±i) = Re (λ± )n − 1 < 0,
equilibrium points ±i are stable nodes or stable foci.
(b) Since
n
hni
sj
o(ν+1)n −ε
j=1
∗n
⊂ J (A) = R ∪ {∞} and EP (A ) =
n
hni
sj
o(ν+1)n −ε
∪ { ±i },
j=1
all equilibrium points of the differential equation (DE : a ; θ ; n) are symmetric
with respect to the real axis.
hni
(c) Let µ be the multiplier of sj . By Corollary 2.10,
n
hni
= µ k − 1 > 0.
(A∗n )0 sj
hni
Therefore, the equilibrium point sj
is an unstable node.
(d) We assume that
hni
hni
hni
s(ν+1)n −ε < · · · < sj+1 < sj
hni
< · · · < s1 .
The rational function A∗n can be written as
A∗n (z) =
P hni (z)
,
Qhni (z)
where P hni and Qhni are polynomials with no common roots. Then equilibrium
points of the differential equation (DE : a ; θ ; n) correspond to zeros of P hni and
poles of A∗n correspond to zeros of Qhni . Since the Julia set J (A) = R ∪ {∞}
is completely invariant under A, all poles of A∗n in C belong to R. Let
n
hni
tj
o(ν+1)n +1−ε
j=1
= (A∗n )−1 (∞) \ {∞} ⊂ R
be the set of all poles of A∗n in C, where
hni
hni
hni
t(ν+1)n +1−ε < · · · < tj+1 < tj
hni
< · · · < t1 .
1738
Koh Katagata
Then we can assume that
P hni (z) = Cn
(ν+1)n −ε Y
hni
z − sj
× z+i z−i
j=1
and
hni
Q
(z) =
(ν+1)n +1−ε Y
hni
z − tj
,
j=1
where Cn is a constant. We apply Lemma 3.4 to F = A∗n for γ = j and
δ = j + 1. Let
Y hni
Rj (j+1) (z) =
z − sk × z + i z − i .
k6=j, j+1
hni
Since sj
hni
and sj+1 are unstable nodes,
hni
hni
∗n 0
∗n 0
> 0 and (A ) sj+1 > 0.
(A ) sj
By Lemma 3.4,
hni
hni
Rj (j+1) sj
· Rj (j+1) sj+1 > 0.
Therefore, we obtain that
hni
Q
hni
sj
hni
·Q
hni
sj+1
< 0.
The last inequality indicates that the number of poles between equilibrium
hni
hni
points sj and sj+1 is odd for all j ≥ 1. Since the number of equilibrium
points of the differential equation (DE : a ; θ ; n) on the real axis is (ν + 1)n − ε
and the number of poles of A∗n is (ν +1)n +1−ε, there is just one pole between
hni
hni
equilibrium points sj and sj+1 for all j ≥ 1, namely the following inequality
holds:
hni
hni
hni
hni
t(ν+1)n +1−ε < s(ν+1)n −ε < t(ν+1)n −ε < · · · < sj
hni
< tj
hni
hni
hni
< · · · < t2 < s1 < t1 .
Therefore, equilibrium points of the differential equation (DE : a ; θ ; n) on the
real axis and poles of A∗n are located alternately.
References
[1] M. J. Álvarez, A. Gasull and R. Prohens, Configurations of critical points
in complex polynomial differential equations, Nonlinear Anal. 71 (2009),
923–934. http://dx.doi.org/10.1016/j.na.2008.11.018
Alternate locations of equilibrium points and poles
1739
[2] M. J. Álvarez, A. Gasull and R. Prohens, Topological classification of
polynomial complex differential equations with all the critical points of
center type, J. Difference Equ. Appl. 16 (2010), 411–423.
http://dx.doi.org/10.1080/10236190903232654
[3] K. Katagata, Qualitative Theory of Differential Equations and Dynamics
of Quadratic Rational Functions, Nonl. Analysis and Differential Equations 2 (2014), 45–59. http://dx.doi.org/10.12988/nade.2014.3819
[4] J. Milnor, Dynamics in One Complex Variable, Vieweg, 2nd edition, 2000.
http://dx.doi.org/10.1007/978-3-663-08092-3
Received: October 11, 2014; Published: December 3, 2014