- Email: [email protected]
From isochronous potentials to isochronous systems
Available online at www.sciencedirect.com
ScienceDirect J. Differential Equations 258 (2015) 1791–1800 www.elsevier.com/locate/jde
From isochronous potentials to isochronous systems Andrea Sfecci Università Politecnica delle Marche, Dipartimento di Ingegneria Industriale e Scienze Matematiche, Via Brecce Bianche 12, I-60131 Ancona, Italy Received 25 February 2014 Available online 1 December 2014
Abstract There is a wide literature involving the study of isochronous equations of the type x(t) ¨ + V x(t) = 0, where V is a C 2 -function. In this paper we show how the kinetic energy T (y) = 12 y 2 can be modified still preserving the isochronicity property of the corresponding system. More generally we provide estimates for the periods, and show an application to the Steen’s equation and other systems related to the anharmonic potential V (x) = ax 2 + bx −2 . © 2014 Elsevier Inc. All rights reserved. MSC: 34C25 Keywords: Isochronous potential; Periodic solution; Planar system; Steen’s equation
1. Introduction The study of isochronous centers for differential systems of the type
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jde.2014.11.013 0022-0396/© 2014 Elsevier Inc. All rights reserved.
x˙ = P(x, y) y˙ = Q(x, y),
(1)
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A. Sfecci / J. Differential Equations 258 (2015) 1791–1800
presents a wide literature (see e.g. [8,11] and the references therein). In particular, in this paper, we consider Hamiltonian systems of the type
−y˙ = V (x) x˙ = T (y),
(2)
where T and V are C 2 -functions, with associated Hamiltonian function E(x, y) = T (y) + V (x).
(3)
The simple case T (y) = 12 y 2 is related to the scalar second order differential equation x(t) ¨ + V x(t) = 0,
(4)
where V is a C 2 -function with a strict local minimum point, where V is zero. This equation has been studied widely in literature, see e.g. [4,17,19,21,25,26]. In particular, all the periodic solutions have the period √ τ0 = 2 √ x2
x1
dx , E − V (x)
(5)
where E is the energy of the solution and x1 < x2 are the extremals of the orbit. The study of functions V providing a constant period function τ leads to the following definition. Definition 1.1 (Isochronous potential). A real valued C 2 -function A = A(u), defined on an open interval I , is said to be an isochronous potential if the equation u(t) ¨ + A u(t) = 0
(6)
has a unique equilibrium point and all the other solutions such that u(t) ∈ I for every t ∈ R, are periodic of the same period. In 1961, Urabe [25] proved that the harmonic potential V (x) = 12 k(x − x0 )2 (with k = 0) is the unique isochronous analytic potential, such that V is odd. Later on, it has been proved that it is also the unique isochronous potential among polynomials [2,9]. A different approach, introducing strict involutions, has been introduced in [11]. See also [1,10,14–18,23,26] for other results. There are several well-known isochronous potentials A(u), for instance, besides the harmonic one au2 , we have the anharmonic one au2 + bu−2 , with u > 0, where the constants are assumed to be positive. It has been recently proved in [7], that all the rational isochronous potentials have one of these forms up to constants and shifts. An√example of non-rational isochronous potential has been given by Urabe [25]: A(u) = 1 + u − 1 + 2u for u > −1/2. Other examples can be given where the isochronicity property only holds locally, i.e. in a neighborhood of an equilibrium point. The results established below for the global case could be extended to the local case, as well. For briefness, we will not enter into such details. The following statement, quoting [6] (see also [3,5,11]), gives a geometrical characterization of the graph of an isochronous potential:
A. Sfecci / J. Differential Equations 258 (2015) 1791–1800
1793
“a C 2 -potential is isochronous if and only if its graph arises by horizontally shearing the graph of a parabola, by a shear which preserves monotonicity of the two sides” Let us start with a simple observation: the elastic potential V (x) = 12 kx 2 has the same form of the kinetic energy function T (y) = 12 my 2 , so that both are isochronous potentials in the sense of Definition 1.1. The natural question is the following: can the isochronicity of the potentials T and V be a necessary and sufficient condition for the isochronicity of the planar system (2)? This question has already been faced in [10,27], where a partial answer was given assuming the functions T and V to be analytic. In this paper, just assuming these functions to be of class C 2 , we will prove that, if one of the two potentials is isochronous, then system (2) is isochronous if and only if the other potential is isochronous as well. The paper is organized as follows: in Section 2 we will evaluate the periods of all the periodic orbits of system (2). In particular, when T is assumed to be an isochronous potential, we will show that there exists a constant CT > 0, determined by T , for which all the periodic solutions to system (2) have the period √ τ = CT 2 √ x2
x1
dx , E − V (x)
where E is the constant energy of the orbit, and x1 < x2 are the extremals of the orbit. In the classical case T (y) = 12 y 2 we will consistently obtain CT = 1. As a consequence we will obtain, in Theorem 2.5, that system (2) is isochronous when T and V are isochronous potentials. In Section 3 we provide an example of application: from the estimate of the period of the isochronous system related to the Steen’s equation x¨ −
c + μx = 0, x3
where x > 0,
we obtain that the system (all the constants are assumed to be positive) ⎧ c ⎪ ⎪ −y˙ = − x 3 + μx ⎪ ⎨ d x˙ = − 3 + νy ⎪ ⎪ y ⎪ ⎩ x > 0, y > 0, which we denominate Steen’s planar system, is an isochronous system. 2. Main results In this section we assume that system (2) has a center at a certain point (x0 , y0 ). We can assume without loss of generality that V (x0 ) = T (x0 ) = 0. Let us introduce some notations and results referring to [6]. Definition 2.1 (Normal potential). A real valued C 2 -function A = A(u), defined in (a, b) with −∞ ≤ a < 0 < b ≤ +∞, is said to be a normal potential if it satisfies the following properties
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Fig. 1. An example of the orbit of energy level E = E0 .
• A(0) = A (0) = 0, A (0) = 2, • A (u) > 0 when u > 0 and A (u) < 0 when u < 0, • limu→a A(u) = limu→b A(u) = ∈ (0, +∞]. Given an isochronous potential, it is possible to obtain, by shifts of coordinates and scaling time, a normal potential. Let us consider a normal potential A. It is possible to find two local −1 −1 inverse functions A−1 + : [0, ) → [0, b) and A− : [0, ) → (a, 0] such that A(A± (ξ )) = ξ for every ξ ∈ [0, ). Defining −1 A (ξ ) = A−1 + (ξ ) − A− (ξ ),
(7)
we have the following proposition, cf. [3,5,6,10,11]. Proposition 2.2. A normal potential is isochronous if and only if
A (ξ ) = 2 ξ . This result describes the property that an isochronous potential is obtained from a parabola by shearing, as quoted in the introduction. There are normal isochronous potentials such that the interval (a, b) is bounded and < ∞: for example Urabe’s normal potential A(u) = 2[1 + u − √ 1 + 2u], defined in (−1/2, 3/2), with = 1. See [6] for other examples. We are now ready to state our main result for system (2). Theorem 2.3. Suppose that T is a normal isochronous potential. Then every non-trivial closed orbit of (2) corresponds to a periodic solution with the period x2 τ= x1
dx , √ E0 − V (x)
where E0 is the energy of the solutions and x1 < x2 are its horizontal extremals, as in Fig. 1.
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Proof. Consider a non-trivial closed orbit of energy E0 as in the assumption of the theorem, with horizontal extremals x1 < x2 . So, the orbit is the image of a solution (x(t), y(t)) of system (2) which reaches the point (x1 , 0) such that V (x1 ) = E0 at a certain time t0 . Denote with t1 > t0 the smallest time in which the solution reaches the point (x2 , 0), and by t2 > t1 the smallest time in which the solution reaches again (x1 , 0). See Fig. 1. Hence, τ = t2 − t0 . By the second equation in (2) one has, using the previously introduced notation, and setting φ(x(t)) = E0 − V (x(t)), −1 T (T+ (φ(x(t)))) if y(t) > 0 x(t) ˙ = T y(t) = T (T−−1 (φ(x(t)))) if y(t) < 0 [(T+−1 ) (φ(x(t)))]−1 if y(t) > 0 = [(T−−1 ) (φ(x(t)))]−1 if y(t) < 0.
Hence, we have the estimate of the period t1 τ=
−1 φ x(t) dt + x(t) ˙ T+
t0
=
t2
−1 φ x(t) dt x(t) ˙ T−
t1
x2
T+−1
φ(x) − T−−1 φ(x) dx
x1
=
x2
T+−1 − T−−1
φ(x) dx.
x1
Using the notation in (7), we have by Proposition 2.2,
T+−1 (ξ ) − T−−1 (ξ ) = T (ξ ) = 2 ξ , with derivative (when ξ > 0)
1 T+−1 − T−−1 (ξ ) = √ , ξ
thus giving us x2 τ= x1
dx = √ φ(x)
x2 x1
dx . √ E0 − V (x)
2
Let us consider now a system of the type (2) where the function T is an isochronous potential which is not normal. Let y0 be the unique non-degenerate minimum of T with T (y0 ) = 0. With the change of variable w = y − y0 , scaling the time variable with t˜ = kt and indicating with † the derivative with respect t˜, we obtain
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(x) −w † = V † x = T (w),
(8)
where
(x) = 1 V (x) and T (w) = 1 T (y0 + w), V k k with 1 k = T (y0 ). 2 With such a procedure we have a new potential T which is normal isochronous, thus permitting us to apply Theorem 2.3, obtaining the estimate of the period τ of a periodic orbit of system (8), where the associated energy function is
w) = T (w) + V
(x) = 1 E(x, y0 + w). E(x, k Hence, the period τ of periodic orbits of system (2) is 1 1 τ= τ= k k
x2 x1
1 =√ k
x2 x1
dx
(x)
E0 − V
dx . √ E0 − V (x)
We have just proved the following corollary of Theorem 2.3. Corollary 2.4. Suppose that T is an isochronous potential with minimum in y0 . Then every non-trivial closed orbit of system (2) corresponds to a periodic solution with the period √ τ = CT 2 √ x2
x1
dx , E0 − V (x)
where E0 is the energy of the solution, x1 < x2 are its horizontal extremals (see Fig. 2), and −1/2 CT = T (y0 ) . In the next result we focus our attention on systems where both the functions T and V are isochronous potentials. We thus obtain a generalization of [27, Theorem 1.3], where the potentials are required to be analytic.
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Fig. 2. An example of the orbit of energy level E = E0 , when the potentials T and V are not normal potentials.
Theorem 2.5. Suppose that T is an isochronous potential. Then system (2) is isochronous if and only if V is an isochronous potential. In such a case, the period is τ = 2πCT CV , with −1/2 CT = T (y0 )
and
−1/2 CV = V (x0 ) ,
where y0 and x0 are the minimum points of T and V , respectively. Proof. Applying Corollary 2.4, we obtain τ = CT τ0 , with τ0 as in (5). On the other hand, it is well-known that the period of all the periodic solutions of Eq. (4) is constant if and only if V is an isochronous potential, having −1/2 τ0 = 2π V (x0 ) = 2π CV . Therefore, τ = CT τ0 = 2π CT CV .
2
The previous theorem leaves open the problem of characterizing isochronous systems of the type (2), when neither T nor V are isochronous potentials. Let us finally state the following immediate consequence of Theorem 2.5. Corollary 2.6. If T and V are C 2 -potentials obtained by horizontally shearing the graph of a parabola, by a shear which preserves monotonicity of the two sides, then system (2) is isochronous.
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3. An example of application In [6], Bolotin and MacKay present a method to create different isochronous potentials starting from suitable C 2 -functions. In this section we focus our attention to systems where one or both the potentials are anharmonic. Consider the Steen’s equation x¨ −
c + μx = 0, x3
where x > 0,
(9)
with μ > 0 and c > 0, and the associated planar system of the form (2), with V (x) =
√ 1 −2 cx + μx 2 − μc 2
1 and T (y) = y 2 . 2
It is well-known that V is an isochronous potential (see e.g. [22], and [12,13,20,21,24] for related problems), having minimum at x0 = (c/μ)1/4 with V (x0 ) = 4μ. Hence, the period of all the √ periodic solutions is τ = π/ μ, independently of the choice of c. A natural generalization of Steen’s equation (9), is what we call Steen’s planar system ⎧ c ⎪ −y˙ = − 3 + μx ⎪ ⎪ x ⎨ d (10) x˙ = − 3 + νy ⎪ ⎪ y ⎪ ⎩ x > 0, y > 0, where again all the constants are assumed to be positive. The associated potentials V (x) =
√ 1 −2 cx + μx 2 − μc 2
and T (y) =
√ 1 −2 dy + νy 2 − νd 2
are isochronous. Hence, applying Theorem 2.5, we obtain the next result. Proposition 3.1. The Steen’s planar system (10) is an isochronous system of the period π τ= √ , 2 μν independently of the constants c and d. In what follows we want to briefly study the relations between the Steen’s equation (9), the Steen’s planar system (10), assuming for simplicity ν = 1, and the second order differential equation x¨ +√μx = 0. These are three examples of isochronous systems with periods, respectively, √ √ τ2 = π/ 4μ, τ1 = π/ μ and τ0 = 2π/ μ. Let us consider the respective energy functions √ 2 = 1 dy −2 + y 2 + 1 cx −2 + μx 2 − d − √μc, E 2 2 1 1 1 = y 2 + cx −2 + μx 2 − √μc, E 2 2 1 μ 0 = y 2 + x 2 , E 2 2
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2 = 2, with c = 1/2, μ = 1/3, and d ∈ {4, 1, 0.25, 0.01}. When d tends to zero the Fig. 3. Level curves of value E 1 = 2, with c = 1/2, μ = 1/3. branches of level curves go to glue together forming the level curve E
2 = 2 with μ = 1/3, and c = d ∈ {4, 1, 0.25, 0.01}. When both c and d tend to zero the Fig. 4. Level curves of value E 0 = 2, with μ = 1/3. branches of level curves go to glue together forming the level curve E
2 tend to the as defined on their full domain. Fig. 3 shows how the branches of level curves of E x-axis when the constant d goes to zero, and seem to glue together forming the branches of level 1 . Similarly, in Fig. 4, the branches of the level curves of E 2 tend to the axes when curves of E 0 . both the constants c and d go to zero, and seem to glue together forming the level curves of E It has been proved (cf. [13,24]) that the speed of a solution to (9), or resp. to (10), increase to infinity when it approaches the y-axis, or resp. one of the axes. Hence, the largest part of the period of a solution is spent when it is far from the y-axis, or resp. the axes. Consistently, we have τ0 = 4 τ2 and τ1 = 2 τ2 . Acknowledgments I express my gratitude to Alessandro Fonda for useful discussions. I thank also Jean Mawhin and Rafael Ortega for their suggestions.
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References [1] A. Algaba, E. Freire, E. Gamero, Isochronicity via normal form, Qual. Theory Dyn. Syst. 1 (2000) 133–156. [2] V.V. Amelkin, Isochronism of a center for two-dimensional analytic differential systems, Differ. Equ. 13 (1977) 667–674. [3] C. Antón, J.L. Brun, Isochronous oscillations: potentials derived from a parabola by shearing, Am. J. Phys. 76 (2008) 537–540. [4] P. Appell, Traité de mécanique rationelle, vol. 1, Gauthiers-Villars, Paris, 1902. [5] D. Bonheure, C. Fabry, D. Smets, Periodic solutions of forced isochronous oscillators at resonance, Discrete Contin. Dyn. Syst. 8 (2002) 907–930. [6] S. Bolotin, R.S. MacKay, Isochronous potentials, in: Proceedings of the Third Conference: Localization and Energy Transfer in Nonlinear Systems, World Scientific, London, 2003, pp. 217–224. [7] O.A. Chalykh, A.P. Veselov, A remark on rational isochronous potentials, J. Nonlinear Math. Phys. 12 (2005) 179–183. [8] J. Chavarriga, M. Sabatini, A survey of isochronous centers, Qual. Theory Dyn. Syst. 1 (1999) 1–70. [9] C. Chicone, M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989) 433–486. [10] A. Cima, A. Gasull, F. Mañosas, Period function for a class of Hamiltonian systems, J. Differential Equations 168 (2000) 180–199. [11] A. Cima, F. Mañosas, J. Villadelprat, Isochronicity for several classes of Hamiltonian systems, J. Differential Equations 157 (1999) 373–413. [12] A. Fonda, M. Garrione, A Landesman–Lazer type condition for asymptotically linear second order equations with a singularity, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012) 1263–1277. [13] A. Fonda, R. Toader, Periodic orbits of radially symmetric systems with a singularity: the repulsive case, Adv. Nonlinear Stud. 11 (2011) 853–874. [14] E. Freire, A. Gasull, A. Guillamon, First derivative of the period function with application, J. Differential Equations 204 (2004) 139–162. [15] E. Freire, A. Gasull, A. Guillamon, A characterization of isochronous centres in terms of symmetries, Rev. Mat. Iberoam. 20 (2004) 205–222. [16] G. Gorni, G. Zampieri, Global isochronous potentials, Qual. Theory Dyn. Syst. 12 (2013) 407–416. [17] J.J. Levin, S.S. Shatz, Nonlinear oscillations of fixed period, J. Math. Anal. Appl. 7 (1963) 284–288. [18] F. Mañonas, P.J. Torres, Isochronicity of a class of piecewise continuous oscillators, Proc. Amer. Math. Soc. 133 (2005) 3027–3035. [19] Z. Opial, Sur le périodes des solutions de l’équation différentielle x + g(x) = 0, Ann. Polon. Math. 10 (1961) 49–72. [20] R. Ortega, Periodic perturbations of an isochronous center, Qual. Theory Dyn. Syst. 3 (2002) 83–91. [21] E. Pinney, The nonlinear differential equation y + p(x)y + cy −3 = 0, Proc. Amer. Math. Soc. 1 (1950) 681. [22] R. Redheffer, Steen’s equation and its generalizations, Aequationes Math. 58 (1999) 60–72. [23] R. Schaaf, A class of Hamiltonian systems with increasing periods, J. Reine Angew. Math. 135 (1985) 129–138. [24] A. Sfecci, Positive periodic solutions for planar differential systems with repulsive singularities on the axes, J. Math. Anal. Appl. 415 (2014) 110–120. [25] M. Urabe, Potential forces which yield periodic motions of a fixed period, J. Math. Mech. 10 (1961) 569–578. [26] M. Urabe, The potential force yielding a periodic motion whose period is an arbitrary continuously differentiable function of the amplitude, J. Sci. Hiroshima Univ. 26 (1962) 93–109. [27] L. Yang, Isochronous centers and isochronous functions, Acta Math. Appl. Sin. Engl. Ser. 18 (2002) 315–324.
ScienceDirect J. Differential Equations 258 (2015) 1791–1800 www.elsevier.com/locate/jde
From isochronous potentials to isochronous systems Andrea Sfecci Università Politecnica delle Marche, Dipartimento di Ingegneria Industriale e Scienze Matematiche, Via Brecce Bianche 12, I-60131 Ancona, Italy Received 25 February 2014 Available online 1 December 2014
Abstract There is a wide literature involving the study of isochronous equations of the type x(t) ¨ + V x(t) = 0, where V is a C 2 -function. In this paper we show how the kinetic energy T (y) = 12 y 2 can be modified still preserving the isochronicity property of the corresponding system. More generally we provide estimates for the periods, and show an application to the Steen’s equation and other systems related to the anharmonic potential V (x) = ax 2 + bx −2 . © 2014 Elsevier Inc. All rights reserved. MSC: 34C25 Keywords: Isochronous potential; Periodic solution; Planar system; Steen’s equation
1. Introduction The study of isochronous centers for differential systems of the type
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jde.2014.11.013 0022-0396/© 2014 Elsevier Inc. All rights reserved.
x˙ = P(x, y) y˙ = Q(x, y),
(1)
1792
A. Sfecci / J. Differential Equations 258 (2015) 1791–1800
presents a wide literature (see e.g. [8,11] and the references therein). In particular, in this paper, we consider Hamiltonian systems of the type
−y˙ = V (x) x˙ = T (y),
(2)
where T and V are C 2 -functions, with associated Hamiltonian function E(x, y) = T (y) + V (x).
(3)
The simple case T (y) = 12 y 2 is related to the scalar second order differential equation x(t) ¨ + V x(t) = 0,
(4)
where V is a C 2 -function with a strict local minimum point, where V is zero. This equation has been studied widely in literature, see e.g. [4,17,19,21,25,26]. In particular, all the periodic solutions have the period √ τ0 = 2 √ x2
x1
dx , E − V (x)
(5)
where E is the energy of the solution and x1 < x2 are the extremals of the orbit. The study of functions V providing a constant period function τ leads to the following definition. Definition 1.1 (Isochronous potential). A real valued C 2 -function A = A(u), defined on an open interval I , is said to be an isochronous potential if the equation u(t) ¨ + A u(t) = 0
(6)
has a unique equilibrium point and all the other solutions such that u(t) ∈ I for every t ∈ R, are periodic of the same period. In 1961, Urabe [25] proved that the harmonic potential V (x) = 12 k(x − x0 )2 (with k = 0) is the unique isochronous analytic potential, such that V is odd. Later on, it has been proved that it is also the unique isochronous potential among polynomials [2,9]. A different approach, introducing strict involutions, has been introduced in [11]. See also [1,10,14–18,23,26] for other results. There are several well-known isochronous potentials A(u), for instance, besides the harmonic one au2 , we have the anharmonic one au2 + bu−2 , with u > 0, where the constants are assumed to be positive. It has been recently proved in [7], that all the rational isochronous potentials have one of these forms up to constants and shifts. An√example of non-rational isochronous potential has been given by Urabe [25]: A(u) = 1 + u − 1 + 2u for u > −1/2. Other examples can be given where the isochronicity property only holds locally, i.e. in a neighborhood of an equilibrium point. The results established below for the global case could be extended to the local case, as well. For briefness, we will not enter into such details. The following statement, quoting [6] (see also [3,5,11]), gives a geometrical characterization of the graph of an isochronous potential:
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1793
“a C 2 -potential is isochronous if and only if its graph arises by horizontally shearing the graph of a parabola, by a shear which preserves monotonicity of the two sides” Let us start with a simple observation: the elastic potential V (x) = 12 kx 2 has the same form of the kinetic energy function T (y) = 12 my 2 , so that both are isochronous potentials in the sense of Definition 1.1. The natural question is the following: can the isochronicity of the potentials T and V be a necessary and sufficient condition for the isochronicity of the planar system (2)? This question has already been faced in [10,27], where a partial answer was given assuming the functions T and V to be analytic. In this paper, just assuming these functions to be of class C 2 , we will prove that, if one of the two potentials is isochronous, then system (2) is isochronous if and only if the other potential is isochronous as well. The paper is organized as follows: in Section 2 we will evaluate the periods of all the periodic orbits of system (2). In particular, when T is assumed to be an isochronous potential, we will show that there exists a constant CT > 0, determined by T , for which all the periodic solutions to system (2) have the period √ τ = CT 2 √ x2
x1
dx , E − V (x)
where E is the constant energy of the orbit, and x1 < x2 are the extremals of the orbit. In the classical case T (y) = 12 y 2 we will consistently obtain CT = 1. As a consequence we will obtain, in Theorem 2.5, that system (2) is isochronous when T and V are isochronous potentials. In Section 3 we provide an example of application: from the estimate of the period of the isochronous system related to the Steen’s equation x¨ −
c + μx = 0, x3
where x > 0,
we obtain that the system (all the constants are assumed to be positive) ⎧ c ⎪ ⎪ −y˙ = − x 3 + μx ⎪ ⎨ d x˙ = − 3 + νy ⎪ ⎪ y ⎪ ⎩ x > 0, y > 0, which we denominate Steen’s planar system, is an isochronous system. 2. Main results In this section we assume that system (2) has a center at a certain point (x0 , y0 ). We can assume without loss of generality that V (x0 ) = T (x0 ) = 0. Let us introduce some notations and results referring to [6]. Definition 2.1 (Normal potential). A real valued C 2 -function A = A(u), defined in (a, b) with −∞ ≤ a < 0 < b ≤ +∞, is said to be a normal potential if it satisfies the following properties
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Fig. 1. An example of the orbit of energy level E = E0 .
• A(0) = A (0) = 0, A (0) = 2, • A (u) > 0 when u > 0 and A (u) < 0 when u < 0, • limu→a A(u) = limu→b A(u) = ∈ (0, +∞]. Given an isochronous potential, it is possible to obtain, by shifts of coordinates and scaling time, a normal potential. Let us consider a normal potential A. It is possible to find two local −1 −1 inverse functions A−1 + : [0, ) → [0, b) and A− : [0, ) → (a, 0] such that A(A± (ξ )) = ξ for every ξ ∈ [0, ). Defining −1 A (ξ ) = A−1 + (ξ ) − A− (ξ ),
(7)
we have the following proposition, cf. [3,5,6,10,11]. Proposition 2.2. A normal potential is isochronous if and only if
A (ξ ) = 2 ξ . This result describes the property that an isochronous potential is obtained from a parabola by shearing, as quoted in the introduction. There are normal isochronous potentials such that the interval (a, b) is bounded and < ∞: for example Urabe’s normal potential A(u) = 2[1 + u − √ 1 + 2u], defined in (−1/2, 3/2), with = 1. See [6] for other examples. We are now ready to state our main result for system (2). Theorem 2.3. Suppose that T is a normal isochronous potential. Then every non-trivial closed orbit of (2) corresponds to a periodic solution with the period x2 τ= x1
dx , √ E0 − V (x)
where E0 is the energy of the solutions and x1 < x2 are its horizontal extremals, as in Fig. 1.
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Proof. Consider a non-trivial closed orbit of energy E0 as in the assumption of the theorem, with horizontal extremals x1 < x2 . So, the orbit is the image of a solution (x(t), y(t)) of system (2) which reaches the point (x1 , 0) such that V (x1 ) = E0 at a certain time t0 . Denote with t1 > t0 the smallest time in which the solution reaches the point (x2 , 0), and by t2 > t1 the smallest time in which the solution reaches again (x1 , 0). See Fig. 1. Hence, τ = t2 − t0 . By the second equation in (2) one has, using the previously introduced notation, and setting φ(x(t)) = E0 − V (x(t)), −1 T (T+ (φ(x(t)))) if y(t) > 0 x(t) ˙ = T y(t) = T (T−−1 (φ(x(t)))) if y(t) < 0 [(T+−1 ) (φ(x(t)))]−1 if y(t) > 0 = [(T−−1 ) (φ(x(t)))]−1 if y(t) < 0.
Hence, we have the estimate of the period t1 τ=
−1 φ x(t) dt + x(t) ˙ T+
t0
=
t2
−1 φ x(t) dt x(t) ˙ T−
t1
x2
T+−1
φ(x) − T−−1 φ(x) dx
x1
=
x2
T+−1 − T−−1
φ(x) dx.
x1
Using the notation in (7), we have by Proposition 2.2,
T+−1 (ξ ) − T−−1 (ξ ) = T (ξ ) = 2 ξ , with derivative (when ξ > 0)
1 T+−1 − T−−1 (ξ ) = √ , ξ
thus giving us x2 τ= x1
dx = √ φ(x)
x2 x1
dx . √ E0 − V (x)
2
Let us consider now a system of the type (2) where the function T is an isochronous potential which is not normal. Let y0 be the unique non-degenerate minimum of T with T (y0 ) = 0. With the change of variable w = y − y0 , scaling the time variable with t˜ = kt and indicating with † the derivative with respect t˜, we obtain
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(x) −w † = V † x = T (w),
(8)
where
(x) = 1 V (x) and T (w) = 1 T (y0 + w), V k k with 1 k = T (y0 ). 2 With such a procedure we have a new potential T which is normal isochronous, thus permitting us to apply Theorem 2.3, obtaining the estimate of the period τ of a periodic orbit of system (8), where the associated energy function is
w) = T (w) + V
(x) = 1 E(x, y0 + w). E(x, k Hence, the period τ of periodic orbits of system (2) is 1 1 τ= τ= k k
x2 x1
1 =√ k
x2 x1
dx
(x)
E0 − V
dx . √ E0 − V (x)
We have just proved the following corollary of Theorem 2.3. Corollary 2.4. Suppose that T is an isochronous potential with minimum in y0 . Then every non-trivial closed orbit of system (2) corresponds to a periodic solution with the period √ τ = CT 2 √ x2
x1
dx , E0 − V (x)
where E0 is the energy of the solution, x1 < x2 are its horizontal extremals (see Fig. 2), and −1/2 CT = T (y0 ) . In the next result we focus our attention on systems where both the functions T and V are isochronous potentials. We thus obtain a generalization of [27, Theorem 1.3], where the potentials are required to be analytic.
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Fig. 2. An example of the orbit of energy level E = E0 , when the potentials T and V are not normal potentials.
Theorem 2.5. Suppose that T is an isochronous potential. Then system (2) is isochronous if and only if V is an isochronous potential. In such a case, the period is τ = 2πCT CV , with −1/2 CT = T (y0 )
and
−1/2 CV = V (x0 ) ,
where y0 and x0 are the minimum points of T and V , respectively. Proof. Applying Corollary 2.4, we obtain τ = CT τ0 , with τ0 as in (5). On the other hand, it is well-known that the period of all the periodic solutions of Eq. (4) is constant if and only if V is an isochronous potential, having −1/2 τ0 = 2π V (x0 ) = 2π CV . Therefore, τ = CT τ0 = 2π CT CV .
2
The previous theorem leaves open the problem of characterizing isochronous systems of the type (2), when neither T nor V are isochronous potentials. Let us finally state the following immediate consequence of Theorem 2.5. Corollary 2.6. If T and V are C 2 -potentials obtained by horizontally shearing the graph of a parabola, by a shear which preserves monotonicity of the two sides, then system (2) is isochronous.
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3. An example of application In [6], Bolotin and MacKay present a method to create different isochronous potentials starting from suitable C 2 -functions. In this section we focus our attention to systems where one or both the potentials are anharmonic. Consider the Steen’s equation x¨ −
c + μx = 0, x3
where x > 0,
(9)
with μ > 0 and c > 0, and the associated planar system of the form (2), with V (x) =
√ 1 −2 cx + μx 2 − μc 2
1 and T (y) = y 2 . 2
It is well-known that V is an isochronous potential (see e.g. [22], and [12,13,20,21,24] for related problems), having minimum at x0 = (c/μ)1/4 with V (x0 ) = 4μ. Hence, the period of all the √ periodic solutions is τ = π/ μ, independently of the choice of c. A natural generalization of Steen’s equation (9), is what we call Steen’s planar system ⎧ c ⎪ −y˙ = − 3 + μx ⎪ ⎪ x ⎨ d (10) x˙ = − 3 + νy ⎪ ⎪ y ⎪ ⎩ x > 0, y > 0, where again all the constants are assumed to be positive. The associated potentials V (x) =
√ 1 −2 cx + μx 2 − μc 2
and T (y) =
√ 1 −2 dy + νy 2 − νd 2
are isochronous. Hence, applying Theorem 2.5, we obtain the next result. Proposition 3.1. The Steen’s planar system (10) is an isochronous system of the period π τ= √ , 2 μν independently of the constants c and d. In what follows we want to briefly study the relations between the Steen’s equation (9), the Steen’s planar system (10), assuming for simplicity ν = 1, and the second order differential equation x¨ +√μx = 0. These are three examples of isochronous systems with periods, respectively, √ √ τ2 = π/ 4μ, τ1 = π/ μ and τ0 = 2π/ μ. Let us consider the respective energy functions √ 2 = 1 dy −2 + y 2 + 1 cx −2 + μx 2 − d − √μc, E 2 2 1 1 1 = y 2 + cx −2 + μx 2 − √μc, E 2 2 1 μ 0 = y 2 + x 2 , E 2 2
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2 = 2, with c = 1/2, μ = 1/3, and d ∈ {4, 1, 0.25, 0.01}. When d tends to zero the Fig. 3. Level curves of value E 1 = 2, with c = 1/2, μ = 1/3. branches of level curves go to glue together forming the level curve E
2 = 2 with μ = 1/3, and c = d ∈ {4, 1, 0.25, 0.01}. When both c and d tend to zero the Fig. 4. Level curves of value E 0 = 2, with μ = 1/3. branches of level curves go to glue together forming the level curve E
2 tend to the as defined on their full domain. Fig. 3 shows how the branches of level curves of E x-axis when the constant d goes to zero, and seem to glue together forming the branches of level 1 . Similarly, in Fig. 4, the branches of the level curves of E 2 tend to the axes when curves of E 0 . both the constants c and d go to zero, and seem to glue together forming the level curves of E It has been proved (cf. [13,24]) that the speed of a solution to (9), or resp. to (10), increase to infinity when it approaches the y-axis, or resp. one of the axes. Hence, the largest part of the period of a solution is spent when it is far from the y-axis, or resp. the axes. Consistently, we have τ0 = 4 τ2 and τ1 = 2 τ2 . Acknowledgments I express my gratitude to Alessandro Fonda for useful discussions. I thank also Jean Mawhin and Rafael Ortega for their suggestions.
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