Alternate Locations of Equilibrium Points and Poles in Complex Rational Differential Equations

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


Introduction
Let D be a domain in C and let f : D → C be a holomorphic function.We consider the first order differential equation 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) in R 2 .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 R 2 , we have to consider the linearization of the map (x, y) → (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.Besides, the differential equation (DE) does not have saddles.
In [3], the author have studied the complex differential equation where Configurations of equilibrium points of the complex differential equation (DE : c ; n) are as follows.
(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. Throughout this paper, we consider two rational functions A and B. The rational function B is a Blaschke product where a j ∈ D = {z ∈ C : |z| < 1} for 1 ≤ j ≤ ν and θ ∈ [0, 1).The rational function A is a conjugate of B, namely Moreover, we deal with the differential equation where a = (a 1 , . . ., a ν ) ∈ D ν and A Let EP (A * n ) be the set of all equilibrium points of A * n .Our main result is the following.
(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.

Dynamics of rational functions A and B
Let f : Ĉ → Ĉ be a rational function.The rational function f can be written as 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 Here are some basic properties of the Fatou set and the Julia set.
• The Fatou set and the Julia set are completely invariant, namely • For a positive integer n, • 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.
Let z 0 be a point in Ĉ.The point z 0 is a periodic point of f if there exists a positive integer n such that f •n (z 0 ) = z 0 .Such the smallest n is called the period of z 0 .The point z 0 is a fixed point of f if the period of z 0 is one.
Periodic points are classified as follows.
Indifferent periodic points are classified into the following two cases.
• The periodic point z 0 is parabolic if λ is a root of unity.
• The periodic point z 0 is irrationally indifferent if |λ| = 1 and λ is not a root of unity.
In the case that the periodic point z 0 of period n is superattracting or attracting, the attracting basin A(z 0 ) of z 0 is defined as 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 It is easy to check that the Möbius transformation b j maps the unit disk onto itself and the Möbius transformation φ maps the unit disk onto the lower half-plane.
and the multiplier at the origin is that Since |B (0)| < 1, the origin is an attracting fixed point.The Blaschke product B is conjugate to itself via ι : z → 1/z.This indicates that there are no fixed points of B in Ĉ \ D except for the point at infinity.The multiplier Since |λ| < 1, the point at infinity is also an attracting fixed point.
Lemma 2.6.The Julia set J (B) is the unit circle S 1 .
Proof.The Möbius transformation b j maps the unit disk onto itself.Moreover, b j maps S 1 and Ĉ \ D onto themselves respectively.This indicates that the unit circle S 1 is completely invariant under B, namely 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 ∪ {∞}.
We calculate exact form of the rational function A. By the above lemma, Therefore, we obtain that where Proposition 2.9.A 1 (x) = A 2 (x) and A 1 (x) = A 2 (x) hold for all x ∈ R.Moreover, the following inequality holds for all x ∈ R : Proof.The first two equations are obvious.The derivative of A is that We transform the numerator and the denominator of the above equation as for all x ∈ R. Therefore, we obtain that 2 for all x ∈ R. We calculate the numerator of the last equation.
we obtain that Therefore, the inequality holds for all x ∈ R.
Corollary 2.10.The multiplier of any repelling periodic orbit of A is greater than one.

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.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 g(w j ) f (w j ) .
On the other hand, we obtain that 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 w 1 , w 2 , . . ., w d of (DE) are simple, namely all zeros w 1 , w 2 , . . ., w d of f are simple.Then the following statements hold.(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 s 1 , s 2 , . . ., s n−2k be the real zeros of P with the order s n−2k < • • • < s 2 < s 1 , and let w 1 , w 1 , . . ., w k , w k be the other zeros of P with Im(w j ) = 0. Then the equation holds, where γ < δ and (z − w j ) (z − w j ) .

( 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 f * n c are located alternately.Considering properties of the complex differential equation (DE : c ; n) and the Julia set of the rational function f c , we can obtain the analogical results to Theorem 1.2.

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 equationB(z) = z or z e 2πiθ b 1 (z) • • • b ν (z) − 1 = 0.Since the Möbius transformation b j maps the unit disk onto itself, the inequalitye 2πiθ b 1 (z) • • • b ν (z) < 1holds 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

( a )
If w 1 , w 2 , . . ., w d−1 are nodes, then w d is also a node.(b) If w 1 , w 2 , . . ., w d−1 are centers, then w d 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 z 1 , . . ., z d−2k are located on a straight line L for some k ≥ 0 and the other 2k equilibrium points z d−2k+1 , . . ., z d 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.