There is a wide literature involving the study of isochronous equations of the type

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From isochronous potentials to isochronous systems

Andrea Sfecci

UniversitàPolitecnicadelleMarche,DipartimentodiIngegneriaIndustrialeeScienzeMatematiche, ViaBrecceBianche12,I-60131Ancona,Italy

Received 25February2014

Abstract

There is a wide literature involving the study of isochronous equations of the type

¨x(t) + V





= 0,

where

is a

-function. In this paper we show how the kinetic energy

=

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

= ax

+ bx

.

©

2014 Elsevier Inc. All rights reserved.

MSC: 34C25

Keywords: Isochronouspotential;Periodicsolution;Planarsystem;Steen’sequation

1. Introduction

The study of isochronous centers for differential systems of the type

 ˙x = P(x,y)

˙y = Q(x, y), (1)

E-mailaddress:sfecci@dipmat.univpm.it.

http://dx.doi.org/10.1016/j.jde.2014.11.013 0022-0396/© 2014ElsevierInc.All rights reserved.

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

-functions, with associated Hamiltonian function

E(x, y) = T (y) + V (x). (3)

The simple case T (y) =

y

is related to the scalar second order differential equation

¨x(t) + V

 x(t) 

= 0, (4)

where V is a C

-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

τ

= √ 2

x₂



x₁

√ dx

E − V (x) , (5)

where E is the energy of the solution and x

< x

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

-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) =

k(x −x

)

(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, introduc- ing 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 au

, we have the anharmonic one au

+ bu

, 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 equilib- rium 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 C

-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) =

kx

has the same form of the kinetic energy function T (y) =

my

, 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

, 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 C

> 0, determined by T , for which all the periodic solutions to system (2) have the period

τ = C

√ 2

x₂



x₁

√ dx

E − V (x) ,

where E is the constant energy of the orbit, and x

< x

are the extremals of the orbit. In the classical case T (y) =

y

we will consistently obtain C

= 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

x

+ μx = 0, where x > 0, we obtain that the system (all the constants are assumed to be positive)

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

− ˙y = − c x

+ μx

˙x = − d y

+ νy 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 (x

, y

). We can assume without loss of generality that V (x

) = T (x

) = 0. Let us introduce some notations and results referring to [6].

Definition 2.1 (Normal potential). A real valued C

-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

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,

• lim

A(u) = lim

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 inverse functions A

: [0, ) → [0, b) and A

: [0, ) → (a, 0] such that A(A

(ξ )) = ξ for every ξ ∈ [0, ).

Defining



(ξ ) = A

(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



(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

τ =

x2



x1

√ dx

E

− V (x) ,

where E

is the energy of the solutions and x

< x

are its horizontal extremals, as in Fig. 1.

Proof. Consider a non-trivial closed orbit of energy E

as in the assumption of the theorem, with horizontal extremals x

< x

. So, the orbit is the image of a solution (x(t), y(t)) of system (2) which reaches the point (x

, 0) such that V (x

) = E

at a certain time t

. Denote with t

> t

the smallest time in which the solution reaches the point (x

, 0), and by t

> t

the smallest time in which the solution reaches again (x

, 0). See Fig. 1. Hence, τ = t

− t

. By the second equation in (2) one has, using the previously introduced notation, and setting φ(x(t)) = E

− V (x(t)),

˙x(t) = T

 y(t) 

=

 T

(T

(φ(x(t)))) if y(t) > 0 T

(T

(φ(x(t)))) if y(t) < 0

=

 [(T

)

(φ(x(t))) ]

if y(t) > 0 [(T

)

(φ(x(t))) ]

if y(t) < 0.

Hence, we have the estimate of the period

τ =

t₁



t₀

˙x(t)  T





φ  x(t) 

dt +

t₂



t₁

˙x(t)  T





φ  x(t) 

dt

=

x2



x1

 T



 φ(x) 

−  T





φ(x)  dx

=

x₂



x₁

 T

− T



 φ(x) 

dx.

Using the notation in (7), we have by Proposition 2.2, T

(ξ ) − T

(ξ ) = 

(ξ ) = 2

ξ ,

with derivative (when ξ > 0)

 T

− T



(ξ ) = 1

√ ξ ,

thus giving us

τ =

x₂



x1

√ dx φ(x) =

x₂



x1

√ dx

E

− 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 y

be the unique non-degenerate minimum of T with T (y

) = 0. With

the change of variable w = y − y

, scaling the time variable with ˜t = kt and indicating with

the derivative with respect ˜t, we obtain

 −w

= V

(x)

x

= T

(w), (8)

where

V (x) = 1

k V (x) and T (w) = 1

k T (y

+ w), with

k = 1 2 T

(y

).

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

E(x, w) = T (w) + V (x) = 1

k E(x, y

+ w).

Hence, the period τ of periodic orbits of system (2) is

τ = 1 k τ = 1

k

x2



x1

dx

E

− V (x)

= 1

√ k

x₂



x₁

√ dx

E

− 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 y

. Then every non-trivial closed orbit of system (2) corresponds to a periodic solution with the period

τ = C

√ 2

x2



x1

√ dx

E

− V (x) ,

where E

is the energy of the solution, x

< x

are its horizontal extremals (see Fig. 2), and C

= 

T

(y

) 

.

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 poten-

tials are required to be analytic.

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 = 2πC

C

, with

C

= 

T

(y

) 

and C

= 

V

(x

) 

,

where y

and x

are the minimum points of T and V , respectively.

Proof. Applying Corollary 2.4, we obtain

τ = C

τ

,

with τ

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

τ

= 2π 

V

(x

) 

= 2πC

. Therefore,

τ = C

τ

= 2πC

C

. 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

-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.

3. An example of application

In [6], Bolotin and MacKay present a method to create different isochronous potentials start- ing from suitable C

-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

x

+ μx = 0, 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



− √

μc and T (y) = 1 2 y

.

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 x

= (c/μ)

with V

(x

) = 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

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

− ˙y = − c x

+ μx

˙x = − d y

+ νy x > 0, y > 0,

(10)

where again all the constants are assumed to be positive. The associated potentials V (x) = 1

2

 cx

+ μx



− √

μc and T (y) = 1 2

 dy

+ νy



− √ νd

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

τ = π

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 equa- tion ¨x + μx = 0. These are three examples of isochronous systems with periods, respectively,

 τ

= π/ √

4μ,  τ

= π/√μ and  τ

= 2π/√μ. Let us consider the respective energy functions E 

= 1

2

 dy

+ y

 + 1

2

 cx

+ μx



− √ d − √

μc,

E 

= 1 2 y

+ 1

2

 cx

+ μx



− √ μc,

 E

= 1

2 y

+ μ

2 x

,

Fig. 3.LevelcurvesofvalueE₂= 2,withc= 1/2,μ= 1/3,andd∈ {4,1,0.25,0.01}.Whendtendstozerothe branchesoflevelcurvesgotogluetogetherformingthelevelcurveE₁= 2,withc= 1/2,μ= 1/3.

Fig. 4.LevelcurvesofvalueE₂= 2 withμ= 1/3,andc= d ∈ {4,1,0.25,0.01}.Whenbothcanddtend tozerothe branchesoflevelcurvesgotogluetogetherformingthelevelcurveE₀= 2,withμ= 1/3.

as defined on their full domain. Fig. 3 shows how the branches of level curves of  E

tend to the x-axis when the constant d goes to zero, and seem to glue together forming the branches of level curves of  E

. Similarly, in Fig. 4, the branches of the level curves of  E

tend to the axes when 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  τ

= 4 τ

and  τ

= 2 τ

.

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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