Qualitative Theory of Differential Equations and Dynamics of Quadratic Rational Functions

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1 Nonl. Analysis and Differential Equations, Vol. 2, 2014, no. 1, HIKARI Ltd, Qualitative Theory of Differential Equations and Dynamics of Quadratic Rational Functions Koh Katagata Ichinoseki National College of Technology Takanashi, Hagisho, Ichinoseki, Iwate Japan Copyright c 2013 Koh Katagata. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We study the qualitative theory of first order differential equations consisting of the iteration of complex quadratic rational functions and we focus on the configuration, namely location and stability, of simple equilibrium points which correspond to periodic points of the quadratic rational functions. Our main tools are properties of Julia sets of the quadratic rational functions and the Euler-Jacobi formula. Mathematics Subject Classification: Primary 37C10; Secondary 32A10, 37F10 Keywords: Equilibrium points, The Euler-Jacobi formula, Julia sets 1 Introduction Let f be a holomorphic function. We consider the first order differential equation ż dz dt = f (z), (DE) where t R and z C. A point ζ C 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)

2 46 Koh Katagata in R 2. Equilibrium points are categorized as stable nodes, unstable nodes, centers, stable foci, unstable foci and saddles. For 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 when we classify equilibrium points in general. 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 C be an equilibrium point of the differential equation (DE). Then the equilibrium point ζ is (1) a stable node if and only if f (ζ) < 0, (2) an unstable node if and only if f (ζ) > 0, (3) a center if and only if Im f (ζ) 0 and Re f (ζ) = 0, (4) a stable focus if and only if Im f (ζ) 0 and Re f (ζ) < 0, (5) an unstable focus if and only if Im f (ζ) 0 and Re f (ζ) > 0. Remark 1.2. The differential equation (DE) does not have saddles. The equilibrium point ζ of the differential equation (DE) is a node if and only if the point ζ is a center of the differential equation ż = i f (z). Throughout this paper, we consider the quadratic rational function f c (z) = 1 + c ( z 1 ), 2(1 c) z where c [0, 1/3] is a real parameter, and we deal with the differential equation where f n c ż = fc n (z) fc n (z) z, (DE : c ; n) = f c f c is the n-th iteration of f c. Let EP ( fc n ) be the set of all. Our main result is the following. equilibrium points of f n c Main Theorem. 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 (In fact, they are attracting fixed points of f c ).

3 Qualitative theory of differential equations 47 (b) In the case that c = 1/3, there are no equilibrium points of (DE : c ; n) on C \ R, namely EP ( fc n ) R. (c) All equilibrium points of the differential equation (DE : c ; n) are symmetric with respect to the real axis (Besides, they are symmetric with respect to the origin since f c is an odd function). (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 alternately. are located The main theorem is 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). The main theorem is a counterexample in the case that f is a genuine rational function. 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. (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. 2 Dynamics of the quadratic rational function f c Let f : Ĉ Ĉ be a rational function. The rational function f can be written as f (z) = p(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.

4 48 Koh Katagata Definition 2.1. Let f : Ĉ Ĉ be a non-constant rational function. The Fatou set F ( f ) of f is defined as F ( f ) = { z Ĉ : the family { f n } n=1 is normal in some open neighborhood of z }. 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, The Julia set J( f ) is non-empty. F ( f n ) = F ( f ) and J( f n ) = J( f ). 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. Definition 2.2. Let z 0 = f n (z 0 ) be a periodic point of period n. The multiplier λ = λ(z 0 ) at z 0 is defined as Periodic points are classified as follows. ( f n ) (z 0 ) (z 0 ), λ = 1/ lim( f n ) (z) (z 0 = ). z The periodic point z 0 is superattracting if λ = 0.

5 Qualitative theory of differential equations 49 The periodic point z 0 is attracting if 0 < λ < 1. The periodic point z 0 is indifferent if λ = 1. The periodic point z 0 is repelling if λ > 1. 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 { } A(z 0 ) = z Ĉ : lim f kn (z) = z 0. 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 quadratic rational function f c (z) = 1 + c ( z 1 ), 2(1 c) z where c [0, 1/3] is a real parameter. Lemma 2.4. The point at infinity belongs to the Julia set J( f c ) for all c [0, 1/3]. Proof. The derivative is that and the multiplier λ at is that f c(z) = λ = 1 + c (1 + 1z ) 2(1 c) 2 1 lim z f c(z) = 2(1 c) 1 + c. In the case that 0 c < 1/3, the point at infinity is a repelling fixed point since 1 < λ 2. In the case that c = 1/3, the point at infinity is a parabolic fixed point since λ = 1. Therefore, the point at infinity belongs to the Julia set J( f c )

6 50 Koh Katagata Lemma 2.5. In the case that 0 c < 1/3, the quadratic rational function f c has two attracting fixed points α ± c, where α ± c = ± i Proof. The multiplier λ ± c at α ± c is that 1 + c 1 3c. λ ± c = f c(α ± c ) = 2c 1 c. Since 0 λ ± c < 1, the two fixed points α ± c are attracting. Proposition 2.6. F ( f c ) = H + H and J( f c ) = R { } for all c [0, 1/3], where H + = {z C : Im(z) > 0} and H = {z C : Im(z) < 0} are the upper half-plane and the lower half-plane respectively. Proof. A simple calculation shows that critical points of f c are ±i for all c [0, 1/3]. Moreover, the graph of the real function f c (x) indicates that the function f c : R { } R { }, f c (x) = 1 + c ( x 1 ) 2(1 c) x is two-to-one and the extended real line R { } is completely invariant under f c, that is to say, f c (R { }) = R { } = f 1 c (R { }). Since the Julia set J( f c ) is the smallest closed completely invariant set containing at least three points, the Julia set J( f c ) is contained the extended real line R { } or J( f c ) R { }. Therefore, we prove that J( f c ) = R { } in the rest of the proof. We assume that there exists a point ζ which is in R { } \ J( f c ), namely is in F ( f c ). In the case that c [0, 1/3), the Fatou set F ( f c ) is the union of two attracting basin A(α ± c ). Hence, the orbit of ζ tends to the attracting fixed point α + c or α c. On the other hand, the orbit of ζ stays on the extended real line R { } since it is invariant under f c. This is a contradiction. Therefore, we obtain that J( f c ) = R { }. Let c = 1/3. For any non-zero real number y, the orbit of y under the function t t + 1/t tends to + or. Since f 1/3 (yi) = yi 1 ( yi = y + 1 ) i y

7 Qualitative theory of differential equations 51 for any purely imaginary number yi, the orbit of yi tends to the point at infinity, especially the orbits of critical points ±i tends to the point at infinity through the half plains H ±. Therefore, the orbit of any point in the Fatou set F ( f 1/3 ) tends to the point at infinity through the half plain H + or H. Hence, the orbit of ζ tends to the point at infinity through the half plain H + or H. However, the orbit of ζ stays on the extended real line R { }. This is a contradiction. Consequently, we obtain that J( f 1/3 ) = R { }. Example 2.7. Let c = 0. The quadratic rational function f 0 (z) = 1 2 ( z 1 ) z is the Newton map of the quadratic polynomial p(z) = z 2 + 1, namely f 0 (z) = z p(z) p (z). Points i and i are superattracting fixed points of f 0. The attracting basin of i is the upper half-plane H + and the attracting basin of i is the lower half-plane H. Example 2.8. Let c = 1/3. The quadratic rational function f 1/3 (z) = z 1 z has the only one fixed point at which is parabolic. The parabolic basin of is H + H. Proposition 2.9. The multiplier of any repelling periodic orbit is greater than one. Proof. An easy computation shows that µ = [ f p c (ζ j ) ] = ζ 1 ζ 2 ζ p ζ 1 p f c(ζ j ) = [ 1 + c 2(1 c) ] p p ζ > 0. 2 j Since ζ j belongs to the Julia set J( f c ), the multiplier satisfies that µ > 1. Therefore, we obtain that µ > 1.

8 52 Koh Katagata 3 The Euler-Jacobi formula The main ingredient of the proof of the main theorem 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 w 1, w 2,..., w d of f are simple, then d g(w j ) f (w j ) = 0 for any polynomial g satisfying that deg(g) < deg( f ) = 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 Γ g(z) d ( ) g f (z) dz = 2πi Res f, w j = 2πi On the other hand, we obtain that g(z) f (z) dz Γ 2πr max z =r g(z) f (z) d. g(w j ) f (w j ). 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. (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. Proof. (a) Since w 1, w 2,..., w d 1 are nodes, let f (w j ) = c j with c j R \ {0} for j = 1, 2,..., d 1. Applying the Euler-Jacobi formula with g(z) 1, we obtain that d 1 d 1 f (w j ) = 1 c j + 1 f (w d ) = 0.

9 Qualitative theory of differential equations 53 Hence, the equality / d 1 f 1 (w d ) = 1 R \ {0} c j holds and the equilibrium point w d is a node. (b) Using Remark1.2, we obtain the result. (c) Let f (w j ) = a j + ib j for j = 1, 2,..., d. Applying the Euler-Jacobi formula with g(z) 1, we obtain that d 1 f (w j ) = d a j ib j a 2 j + b2 j = 0. Taking the real part of the equality, we obtain that d a j a 2 j + b2 j = 0. Since not all equilibrium points are centers, not all non-zero a j have the same sign. Therefore, there exist at least two of equilibrium points of (DE) 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. (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. Proof. Without loss of generality, we can assume that the straight line L is the real axis, namely all d 2k equilibrium points a 1, a 2,..., a d 2k on L are real. Then

10 54 Koh Katagata the other 2k equilibrium points are represented by w j and its conjugate w j for j = 1, 2,..., k. Therefore, the differential equation (DE) can be written as the form d 2k ż = f (z) = (α + iβ) (z a j ) k (z w j )(z w j ), where a j < a j+1 for all j = 1, 2,..., d 2k 1 and all the w j are different and Im(w j ) 0 for all j = 1, 2,..., k. Applying the Euler-Jacobi formula with the function d 2k k g(z) = (z a j ) (z w j )(z w j ), we obtain that d j=3 g(z j ) f (z j ) = g(a 1) f (a 1 ) + g(a 2) f (a 2 ) = 0. Since g(a 1 ) and g(a 2 ) are real and they have the same sign, then (1) (i) f (a 1 ) is real if and only if f (a 2 ) is real, namely a 1 is a node if and only if a 2 is a node. (ii) In the above case, a 1 and a 2 have the opposite stability because the inequality f (a 1 ) f (a 2 ) < 0 holds. (2) f (a 1 ) is purely imaginary if and only if f (a 2 ) is purely imaginary, namely a 1 is a center if and only if a 2 is a center. (3) (i) a 1 is a focus if and only if a 2 is a focus. (ii) In the above case, a 1 and a 2 have the opposite stability because the inequality Re f (a 1 ) Re f (a 2 ) < 0 holds. Replacing a 1 and a 2 by a j and a j+1 respectively, we obtain the following statements. (1 ) (i) f (a j ) is real if and only if f (a j+1 ) is real, namely a j is a node if and only if a j+1 is a node. (ii) In the above case, a j and a j+1 have the opposite stability because the inequality f (a j ) f (a j+1 ) < 0 holds. (2 ) f (a j ) is purely imaginary if and only if f (a j+1 ) is purely imaginary, namely a j is a center if and only if a j+1 is a center. (3 ) (i) a j is a focus if and only if a j+1 is a focus. (ii) In the above case, a j and a j+1 have the opposite stability because the inequality Re f (a j ) Re f (a j+1 ) < 0 holds.

11 Qualitative theory of differential equations 55 Therefore, we obtain the statement (a). Next, we prove the statements (b) and (c). It is easy to check that f (w j ) = α + iβ α iβ f (w j ). If all the points on L are of center type, we obtain that α = 0 and f (w j ) = f (w j ). Let f (w j ) = α j + iβ j, then f (w j ) = α j + iβ j. Therefore, w j and w j are the same type. Moreover, if they are not centers (α j 0), then they have opposite stability. Similarly, if all the points on L are of node type, we obtain that β = 0 and f (w j ) = f (w j ). Therefore, w j and w j are the same type. Moreover, if they are not centers (α j 0), then they have the same stability. 4 Configurations of equilibrium points Let F(z) = P(z)/Q(z) be a rational function, where P and Q are polynomials with real coefficients and with no common factors. We suppose that deg(p) = n and deg(q) = m. Moreover, we suppose that the two polynomials P and Q have only simple roots. Let a 1, a 2,..., a n 2k be the real zeros of P with the order a 1 < a 2 < < a n 2k, and let w 1, w 1,..., w k, w k be the other zeros of P with Im(w j ) 0. Similarly, let b 1, b 2,..., b m 2l be the real zeros of Q with the order b 1 < b 2 < < b m 2l, and let v 1, v 1,..., v l, v l be the other zeros of Q with Im(v j ) 0. The derivative of the function F is that and we obtain that for any zero z j of P. Let F (z) = P (z)q(z) P(z)Q (z) Q(z) 2 F (z j ) = P (z j ) Q(z j ) = P (z) Q(z) P(z)Q (z) Q(z) 2 n 2k R 12 (z) = (z a j ) j=3 k (z w j )(z w j ) and we rewrite P as P(z) = A n n 2k (z z j ) = A (z a j ) k (z w j )(z w j ), where A is a real constant. Applying the Euler-Jacobi formula to the rational function R 12 (z)/p(z), we obtain that n R 12 (z j ) P(z j ) = R 12(a 1 ) P (a 1 ) + R 12(a 2 ) P (a 2 ) = 0

12 56 Koh Katagata or We transform the last equation as and we obtain that P (a 2 ) = R 12(a 2 ) R 12 (a 1 ) P (a 1 ). P (a 2 ) Q(a 2 ) = Q(a 1) Q(a 2 ) R12(a 2 ) R 12 (a 1 ) P (a 1 ) Q(a 1 ) F (a 2 ) = Q(a 1) Q(a 2 ) R12(a 2 ) R 12 (a 1 ) F (a 1 ). ( ) Since R 12 (a 2 )/R 12 (a 1 ) is positive and the polynomial Q have the expression m 2l Q(z) = B (z b j ) l (z v j )(z v j ) for some real number B, the signs of values Q(a 1 ) and Q(a 2 ) depend on configurations a 1, a 2 and b 1,..., b m 2l. Conversely, the following lemma holds. Lemma 4.1. The equation ( ) indicates that the signs of values F (a 1 ) and F (a 2 ) control the signs of values Q(a 1 ) and Q(a 2 ). More precisely, the configuration of equilibrium points a 1, a 2 and b 1,..., b m 2l depends on the signs of values F (a 1 ) and F (a 2 ). For example, the inequality F (a 1 ) F (a 2 ) < 0 implies that the number of zeros of Q between a 1 and a 2 is even since the signs of values Q(a 1 ) and Q(a 2 ) are the same. In general, the following lemma holds. Lemma 4.2. The equation holds, where s < t and F (a t ) = Q(a s) Q(a t ) Rst(a t ) R st (a s ) F (a s ) (#) R st (z) = (z a j ) j s,t k (z w j )(z w j ). Proof of the Main Theorem. Equilibrium points of the differential equation (DE : c ; n) correspond to periodic points of period k (k n) of f c or fixed points of fc n. Let { } a n 2 n 2 EP ( f n ) c

13 Qualitative theory of differential equations 57 be the set of all fixed points of f n c except attracting fixed points α ± c. (a) In the case that 0 c < 1/3, there are no attracting or parabolic periodic points of period k (k n) except attracting fixed points α ± c (in Lemma 2.5) because of Proposition 2.6. Therefore, we obtain that each a n is a repelling fixed point of fc n, namely { } a n 2 n 2 J( f c). Since equilibrium points α ± c are stable nodes. ( f n) c (α ± c ) = ( λ ± ) n c 1 < 0, (b) In the case that c = 1/3, there are no attracting or parabolic periodic points of period k (k n). Therefore, we obtain that EP ( fc n ) { } = a n 2 n 2 R. (c) Since f c is an odd function and EP ( fc n ) R i R, all equilibrium points of the differential equation (DE : c ; n) are symmetric with respect to the real axis. (d) By Proposition 2.9, let µ be the multiplier of a n, then ( f n) ( c a n n ) = µ k 1 > 0. Therefore, the equilibrium point a n is an unstable node. (e) We assume that The rational function f n c a n c, 2 n 2 < < a n +1 < a n < < a n c, 1. can be written as f n c (z) = P n c (z) Q n c (z), where P n c and Q n c are polynomials with no common roots. Then equilibrium points of the differential equation (DE : c ; n) correspond to zeros of P n c and poles of fc n correspond to zeros of Q n c. By the proof of Proposition 2.6, we obtain that ( f n) 1 c ( ) \ { } R because the extended real line R { } is completely invariant under f c. Let { } b n 2 n 1 R

14 58 Koh Katagata be the set of all poles of fc n, where b n c, 2 n 1 < < b n +1 < b n < < b n c, 1. By symmetricity (c), we obtain the following two inequalities holds: a n c, 1 < a n c, 2 < < a n c, 2 n 1 1 < 0 < a n c, 2 n 1 1 < < a n c, 2 < a n c, 1, b n c, 1 < b n c, 2 < < b n c, 2 n 1 1 < b n c, 2 n 1 < b n c, 2 n 1 1 < < b n c, 2 < b n c, 1. = We apply Lemma 4.2 to F = fc n for s = j and t = j+1. In the case that 0 c < 1/3, let R st (z) = (z a n c, k ) (z α+ c )(z α c ). In the case that c = 1/3, let Since then ( f n c k s,t ) ( a n 0 R st (z) = (z a n k s,t c, k ). ) ( > 0 and f n) ( c a n +1) > 0, ( a n +1) < 0. ( ) Q n c a n Q n c The last inequality indicates that there is an odd number of poles between equilibrium points a n and a n +1 for all j 1. Since the number of equilibrium points of the differential equation (DE : c ; n) is 2 n 2 and the number of poles of fc n is 2 n 1, there is just one pole between equilibrium points a n and a n +1 for all j 1, namely the following inequality holds: b n c, 1 < a n c, 1 < b n c, 2 < a n c, 2 < < b n c, 2 n 1 1 < a n c, 2 n 1 1 < b n c, 2 n 1 < a n c, 2 n 1 1 < b n c, 2 n 1 1 < < a n c, 2 < b n c, 2 < a n c, 1 < b n c, 1. = 0 Therefore, equilibrium points of the differential equation (DE : c ; n) on the real axis and poles of f n c are located alternately. Acknowledgements. I would like to thank professor Masakazu Onitsuka for many helpful discussions.

15 Qualitative theory of differential equations 59 References [1] M. J. Álvarez, A. Gasull and R. Prohens, Configurations of critical points in complex polynomial differential equations, Nonlinear Anal. 71 (2009), [2], Topological classification of polynomial complex differential equations with all the critical points of center type, J. Difference Equ. Appl. 16 (2010), [3] J. Milnor, Dynamics in One Complex Variable, Vieweg, 2nd edition, Received: August 28, 2013