Phase Portraits of Uniform Isochronous Centers with Homogeneous Nonlinearities

We classify the phase portraits in the Poincaré disc of the differential equations of the form x′=−y+xf(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x^{\prime } = -y + x f(x,y)$\end{document}, ẏ=x+yf(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\dot y =x + y f(x,y)$\end{document} where f(x,y) is a homogeneous polynomial of degree n − 1 when n = 2,3,4,5, and f has only simple zeroes. We also provide some general results on these uniform isochronous centers for all n ≥ 2. All our results have been revised by the program P4; see Chaps. 9 and 10 of Dumortier et al. (UniversiText, Springer-Verlag, New York, 2006).


Introduction and Statement of the Main Results
The firstinvestigation in isochronicity goes back to Huygens in [7] with the studyof the cycloidal pendulum in the seventeenth century. Nowadays, isochronicity appears in many physical problems and it is closely related to the existence and uniqueness of solutions for certain bifurcation problems or boundary value problems (see for instance [9] and the references therein). In the last decade, the study of the isochronicity has been growing specially in the case of polynomial differential systems due to the appearance of powerful methods of computational analysis; see for instance [1,3,5,14] to cite just few of them. A polynomial differential system of degree n in R 2 is a differential systeṁ x = P (x, y),ẏ = Q(x, y), (1) with P and Q polynomials such that the maximum of their degrees is n. We denote by χ = (P , Q) the polynomial vector field associated to system (1). An equilibrium point p ∈ R 2 of χ is a center if there exists a punctured neighborhood of p filled by periodic orbits of the flow of χ . A center p is called isochronous if these periodic orbits have all the same period. We say that p is a uniform isochronous center of χ if the system associated to χ can be written in some polar coordinates in the forṁ r = G(θ, r),θ = κ, κ ∈ R \ {0}.
In other words, the angular velocity of the orbits of a uniform isochronous center does not depend on the radius. We recall that a period annulus of a center p is the maximum neighborhood U of the point p in R 2 such that U \ {p} is filled up only with periodic orbits. We say that p is a global center if its period annulus is R 2 \ {p}.
The polynomial differential systems of R 2 can be extended analytically to infinity in the so-called Poincaré compactification. Roughly speaking, the Poincaré compactification identifies R 2 with the interior of the unit disc centered at the origin and its boundary, the circle S 1 , with the infinity of R 2 . Note that in the plane R 2 we can go to infinity as many directions as points has the circle S 1 . Then, the polynomial differential system defined in the interior of the unit disc can be extended to an analytic differential system in the closed unit disc and in particular to the infinity S 1 . This closed disc is called the Poincaré disc. The equilibrium points of the extended differential system in S 1 are called the infinite equilibria of the initial polynomial differential system, and the ones which are in the interior of the unit disc are called finite equilibria. For more details on this Poincaré compactification, see Chap. 5 of [6] or the Appendix 1.
Two polynomial vector fields are topologically equivalent if there exists a homeomorphism which preserves the infinity carrying orbits of the flow induced by the first compactified vector field to orbits of the flow induced by the second compactified vector field.
A limit cycle is a periodic orbit of system (1) isolated in the set of all periodic orbits of system (1).
A non locally constant C 1 function H : U → R 2 (where U is an open and dense subset of R 2 ) is a first integral of system (1) in U if H is constant on the solution curves of system (1), or equivalently in U . Of course, H x denotes the partial derivative of H with respect to x. System (1) has a Liouvillian first integral if it has an integrating factor given by a Darboux function (see Section 2 for the definition for integrating factor and Darboux function). For a more detailed information about these three notions see [6,Chapter 8], the paper of Singer [15], and the paper of Christhopher [4]. Roughly speaking, the Liouvillian functions are those functions which can be obtained "by quadratures" of elementary functions.
In the present paper, we classify the phase portraits in the Poincaré disc of uniform isochronous centers of the polynomial differential systeṁ with homogeneous polynomial nonlinearities xf (x, y) and yf (x, y), being f (x, y) a homogenous polynomial of degree n − 1. We do the complete classification under the generic assumption that the singular points at infinity are simple in the sense that in the factorization of the homogeneous polynomial f (x, y) as product of linear and quadratic real homogeneous polynomials every linear factor appears with multiplicity one. All the points of the line at infinity of the polynomial differential systems (1) are equilibria, as it will be proved in statement (f) of Theorem 1. When we remove this line of equilibria, there can remain some additional equilibria, that we call special equilibria. The notion of special equilibria will become clear in the proofs of statements (g) and (h) of Theorem 1.
In the following are our results.
Theorem 1 Consider the polynomial differential system (2) of degree n ≥ 2 in R 2 such that f is a homogeneous polynomial of degree n − 1 and we shall write it as The following statements hold for system (2).
then it is a uniform isochronous center, otherwise it is a focus. (h.1) If n > 2 is even, then all the special equilibrium points at infinity are cusps and their local phase portrait might be one of the two shown in Fig. 1. Moreover between two cusps with local phase portrait as in Fig. 1a, there must be a cusp with local phase portrait as in Fig. 1b. (h.2) If n is odd, then k ≥ 2 is even and there are k/2 pairs of special equilibria at infinity which are saddles and k/2 pairs of special equilibria at infinity which are centers or foci. Moreover, between two saddles there must be a center or a focus.
The proof of statements (d) and (e) are in Theorem 3.1 of [5] but the proof there is more complicated than the one done in this paper and so we decided to include it here. Theorem 1 is proved in Section 3.
Theorem 2 Consider the polynomial differential system (2) of degree n ≥ 2 such that f is a homogeneous polynomial of degree n − 1 satisfying (3) when n is odd, and all the special equilibria come from simple roots. The phase portraits in the Poincaré disc for n = 2, 3, 4  Fig. 2 is achieved with the polynomial differential systeṁ Figure 3 is achieved with the polynomial differential systeṁ and the local phase portraits A, B, C, D, and E of Fig. 4 are achieved with the polynomial differential systemṡ Phase portrait of the polynomial differential system (2) with n = 3, and also with n = 5 with a ∈ (9/16, 5/8), respectively. The proof of Theorem 2 for the case n = 2 is given in [12]. We recall that there is no infinite special equilibrium in this case.
The proof of Theorem 2 for the case n = 3 is given in [2]. We recall that there are two special equilibria that are a saddle and a center or focus.
The proof of Theorem 2 for the case n = 4 is given in [11]. We recall that in the phase portraits A, B, D, and E of Fig. 4 there are three pairs of special equilibria that are cusps and in Fig. 4c there is only one pair of special equilibria (again a cusp). Figure 4e can also be realized without any special equilibrium in the local chart U 1 and one pair of special equilibria in the charts U 2 ∪ V 2 , for example the systeṁ Figure 4c and d are missing in [11]. The phase portraits of these two figures are obtained taking into account all the possibilities provided by the result of statement (h) of Theorem 1 that we realize with the differential systems provided in the statement of Theorem 2.
The following is the second main theorem of the paper.
Theorem 3 Consider the polynomial differential system (2) of degree n = 5 such that f is a homogenous polynomial of degree 4 satisfying (3) and with all its real zeros being simple. The different topologically equivalent phase portraits in the Poincaré disc are given in Fig. 3 and in Fig. 5a and b. Figure 3 is achieved with the polynomial differential systeṁ . Figure 5a and b are achieved with the polynomial differential systemṡ respectively.
The proof of Theorem 3 is given in Section 4.

Fig. 5
Phase portraits of the polynomial differential system (2) with n = 5

Preliminary Results
The following result was proved in [5, Theorem 2.1].
Theorem 4 Let f be a homogeneous polynomial of degree n − 1 of degree n ≥ 2. Then system (2) has a uniform isochronous center at the origin if either n is even or n is odd and condition (3) holds.
The following proposition deals with the existence of limit cycles. Its proof can be found in [8,Theorem 9]. We recall that an inverse integrating factor of a C 1 vector field χ = (P , Q) is a non-locally constant C 1 function V satisfying the partial differential equation The domain of definition of the function V depends on the functions P and Q. Of course, the integrating factor is the inverse of the inverse integrating factor.
Theorem 5 Let χ = (P , Q) be a C 1 vector field defined in a subset U ⊂ R 2 . Let V be an inverse integrating factor of the vector field. If γ is a limit cycle of χ then γ is contained An invariant algebraic curve of system (1) is the zero set of a non-zero polynomial g which satisfies for a polynomial K called the cofactor of g. Note that an invariant algebraic curve is invariant by the dynamics of system (1) in the sense that if a trajectory starts on the curve it does not leave it. An exponential factor of system (1) is a function F = exp(g/ h) with coprime polynomials g and h such that for a polynomial L, whose degree is the degree of the system minus one, is called the cofactor of F . For additional properties of invariant algebraic curves and exponential factors see [6,Chap. 8].
We recall that in view of [6, Theorem 8.7 (iv)] we have that Proposition 6 Assume that a polynomial differential system χ admits p invariant algebraic curves g i = 0 with cofactors K i for i = 1, . . . , p and q exponential factors F i with cofactors L j for j = 1, . . . , q. Then there exists λ j , μ j ∈ C not all zero such that if and only if the function is an integrating factor of χ . Here, div stands for the divergence of the system.

A function of the form in Eq. 4 is called a Darboux function.
The following theorem was proved in [15].

Theorem 7 The polynomial differential system (1) has a Liouvillian first integral if and only if it has an integrating factor which is a Darboux function.
The next result is proved in [6, Theorem 3.5]. (c) If m is odd, a < 0 and m < 2n + 1, then the origin is a center or a focus.

Proof of Theorem 1
Suppose that (x, y) is an equilibrium point of system (2); note that if x = 0 then y = 0, and, conversely, if y = 0 then x = 0. Then the origin is the only singular point of system (2) because if x = 0 and y = 0 we have y = xf (x, y) and x + xf 2 (x, y) = x(1 + f 2 (x, y)) = 0, which is never zero. In short, statement (a) is proved.
To prove statement (b), note that g(x, y) = x 2 + y 2 = 0 is an invariant algebraic curve whose cofactor is 2f (x, y). The divergence of system (2) is 2f +xf x +yf y (as usual f x denotes the partial derivative of f with respect to the variable x, similarly by f y ). Since f is a homogeneous polynomial of degree n−1, by Euler's Theorem for homogeneous functions we obtain that the divergence of the system is (n + 1)f . Now by Proposition 6 we have that (x 2 + y 2 ) −(n+1)/2 is an integrating factor of system Eq. 2. Consequently, in view of Theorem 7, we conclude that system (2) has a Liouvillian first integral. This completes the proof of statement (b). Statement (c) is an immediate consequence of Theorem 4. Statement (d) is an immediate consequence of Theorem 5. Indeed, if system (2) has a limit cycle, since we have shown that it has an inverse integrating factor (x 2 + y 2 ) (n+1)/2 , in view of Theorem 5 the limit cycle must be contained in x 2 + y 2 = 0, which is not possible. Hence, statement (d) is proved.
For statement (e), we compute the infinite equilibrium points in the local chart U 1 . Then, in the local chart U 1 system (2) becomeṡ see the Appendix 1. Therefore, all points (u, 0) for all u ∈ R are infinite equilibrium points in U 1 . Note that the Jacobian matrix at (u, 0) is (1, u) .
Hence, the equilibrium points that are not special, that is, the points that after reparameterizing by the time ds = v dt are not equilibrium points of the system are normally hyperbolic. So in view of the results on the normally hyperbolic points described in Appendix 2, on these equilibrium points it starts or ends a unique orbit, implying that there are no global centers in system (2).This concludes the proof of statements (e) and (f).
In a similar way to the obtaining of system (5) for the special equilibria in the local chart U 1 , we can obtain from Appendix 1 the system for the special equilibria in the local chart U 2 . We are only interested if the origin of U 2 is a special equilibrium, and this happens only if x divides the polynomial f . So if x divides f statement (g) holds. Assume now that x does not divide f , so the monomial y n−1 must appear in f . If n is even then f (1, u) has an odd degree and so it has at least one real zero, implying that there is at least one special equilibrium in the local chart U 1 . If n is odd and f (1, u) has real zeros, statement (g) follows. Finally assume that n is odd, the monomial y n−1 appears in f , and that f (1, u) has no real zeros. Then, the polynomial f (x, y) is written in the form f (x, y) = (n−1)/2 j =1 (α j x 2 +β j xy +γ j y 2 ) with β 2 j −4α j γ j < 0 for j = 1, . . . , (n − 1)/2. Then clearly 2π 0 f (x, y)| x=cos θ,y=sin θ dθ = 0, in contradiction with condition (3) because the origin is a uniform isochronous center. So, statement (g) is proved. Now, we start the proof of statement (h). Under our assumption, all the singular points of system (5) are of the form (ū, 0), whereū is a simple zero of f (1, u), i.e., f (1,ū) = 0 and f (1,ū) = 0. Since the linear part of system (5) at the equilibrium points is the equilibrium point (u, 0) is nilpotent. Since at most one additional infinite equilibrium point can appear, which is the origin of the local chart U 2 , without loss of generality we can assume that all special infinite equilibrium points of system (5) are on the local charts U 1 ∪ V 1 , otherwise doing a rotation in the coordinates (x, y) this would be the case. So in what follows, we do not need to study whether the origin of the local chart U 2 is a special infinite equilibrium point.
To study the special infinite equilibrium points of system (5) (and so of system (2)), we write where 0 is a positive integer, 1 is a non-negative integer and r 1 < r 2 < · · · < r 0 , and β 2 k − 4γ k < 0. Now, we study the equilibrium point (r p , 0) with 1 ≤ p ≤ 0 . We perform a translation to the origin of the point (r p , 0) taking the new variables In these new variables, system (5) becomes Note that in order to write this system under the normal form for applying Theorem 8 we make a scaling. Note that where (r 2 p + 2r p β k + γ k ) = 0, and A 1 , A 2 , . . . , A 0 belong to the higher order terms in U for V . Now we introduce the scaling dτ = A 0 ds and system (8) becomeṡ whereÃ i = A i /A 0 for i = 2, . . . , 0 and the dot means derivative with respect to the new time τ . Note that system (9) is in the normal form for applying Theorem 8 with (x, y) = (V , U ). Solving U = f (V ) fromV = 0 we get that Moreover and If n is even applying Theorem 8, we conclude that (r p , 0) is a cusp. Therefore, modulo a translation to the origin and undoing the rescaling of time the local phase portrait for each equilibrium point (r p , 0) might be one of the two shown in Fig. 1. Moreover, recall that if A 0 is positive for r p then it is negative for r p+1 and r p−1 because r p−1 < r p < r p+1 and these zeroes are simple. Therefore between two cusps with local phase portrait as in Fig. 1a there must be a cusp with local phase portrait as in Fig. 1b. This concludes the proof of statement (h.1).
If n is odd applying Theorem 8, we conclude that (r p , 0) is a saddle if (1 + r 2 p )/A 0 < 0, and it is a focus or a center if (1 + r 2 p )/A 0 > 0. Moreover, recall that if A 0 is positive for r p then it is negative for r p+1 and r p−1 because r p−1 < r p < r p+1 . So, if (r p , 0) is a saddle, (r p+1 , 0) and (r p−1 , 0) must be foci or centers, or vice versa. Moreover, if n is odd, n − 1 is even and then 0 is even. So there are 0 /2 pairs of special equilibrium points which are saddles and 0 /2 pairs of special equilibrium points which are foci or centers at the infinity in the local chart U 1 . Moreover, we have shown that between two saddles there must be a center or a focus. This concludes the proof of statement (h.2) and concludes the proof of the theorem.

Proof of Theorem 3
Note that system (2) is invariant under the change (x, y) → (−x, −y). Therefore, this system is symmetric with respect to the origin; and thus, it is enough to study the phase portrait in the half plane x ≥ 0. Note that in this system there are either two pairs of special infinite equilibrium points or four pairs of special equilibrium points. If there are two special equilibrium points, then they are a saddle and a center or a focus. In this case, the phase portrait is the same as the one in Fig. 3.
When there are four pairs of special equilibrium points then, by statement (h) of Theorem 1, two pairs are saddles and the other two are centers or foci, and the saddles alternate with the centers or foci. So taking into account the symmetry, the existence of the uniform isochronous center at the origin, that in all the infinite equilibrium points which are not special equilibria arrive or exit a unique orbit, and that the boundary of the period annulus of the center must be formed by the separatrices of the saddles at infinity (otherwise we have a contradiction between the period annulus and the orbits which arrive or exit at the infinite equilibria), the unique possible phase portraits are the ones described in Fig. 5. Since we can realize these two phase portraits with the systems provided in the statement of Theorem 3, these complete the proof of that theorem. study them on the local chart U 1 , and to check if the origin of the local chart U 2 is or not a equilibria.

Appendix 2: Normally Hyperbolicity
Let ϕ t a smooth flow on a manifold M and let C be a submanifold of M formed only for equilibria of the flow ϕ t . C is normally hyperboic if the tangent bundle to M over C splits into three subbundles T C, E s and E u invariant under dϕ t and satisfying (a) dϕ t contracts E s exponentially, (b) dϕ t expands E u exponentially, (c) T C = tangent bundle of C.
For normally hyperbolic submanifolds, one has the usual existence of smooth stable and unstable manifolds together with the persistence of these invariant manifolds under small perturbations. More precisely, there is the following result proved in [10].
Theorem 9 Let C be a normally hyperbolic submanifold of equilibrium points for ϕ t . Then, there exist smooth stable and unstable manifolds tangent along C to E s ⊕T C and E u ⊕T C respectively. Moreover, both C and the stable and unstable manifolds are permanent under small perturbations of the flow.