1. bookVolume 6 (2021): Issue 1 (January 2021)
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Periodic solutions for differential systems in ℝ3 and ℝ4

Published Online: 31 Dec 2020
Page range: 373 - 380
Received: 27 Sep 2019
Accepted: 02 Feb 2020
Journal Details
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01 Jan 2016
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Languages
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Abstract

We provide sufficient conditions for the existence of periodic solutions for the differential systems x=y,y=z,z=yεF(t,x,y,z),andx=y,y=xεG(t,x,y,z,u),z=u,u=zεH(t,x,y,z,u), \matrix{{x' = y,\;\;\;y' = z,\;\;\;z' = - y - \varepsilon F(t,x,y,z),\;\;\;{\rm{and}}} \cr {x' = y,\quad y' = - x - \varepsilon G(t,x,y,z,u),\quad z' = u,\quad u' = - z - \varepsilon H(t,x,y,z,u),} \hfill \cr } where F, G and H are 2π–periodic functions in the variable t and ɛ is a small parameter. These differential systems appear frequently in many problems coming from the sciences and the engineering.

Keywords

MSC 2010

Introduction and statement of the main results

In the study of the dynamics of the differential systems after the analysis of their equilibrium points, we must study the existence or not of their periodic orbits. This paper is dedicated to study the periodic orbits of two kind of differential systems which appear frequently in many problems coming from the physics, chemist, economics, engineering, ...

First we shall provide sufficient conditions for the existence of periodic orbits in the differential systems in ℝ3 of the form x=y,y=z,z=yεF(t,x,y,z), x' = y,\quad y' = z,\quad z' = - y - \varepsilon F(t,x,y,z), where F is a 2π–periodic function in the variable t, ɛ is a small parameter, and the prime denotes derivative with respect to the variable t. These differential systems usually come when we write as a first–order differential system in ℝ3 the third–order differential equation x+x+εF(t,x,x,x)=0, {x^{'''}} + {x^\prime} + \varepsilon F(t,x,{x^\prime},{x^{''}}) = 0, taking y = x′ and z = x″. Third–order differential equations have been studied by many authors, see for instance [1, 3, 8, 9, 12, 14, 15, 17,18,19]. But as far as we know we are presenting in the next theorem the more general sufficient conditions up to know for determining the existence of periodic orbits of the differential equation (2), or equivalently of the differential system (1).

Theorem 1

We define 1(x0,y0,z0)=12π02πF(t,A(t),B(t),C(t))(cost1)dt,2(x0,y0,z0)=12π02πF(t,A(t),B(t),C(t))sintdt,3(x0,y0,z0)=12π02πF(t,A(t),B(t),C(t))costdt, \matrix{ {{{\cal F}_1}({x_0},{y_0},{z_0}) = {1 \over {2\pi }}\int_0^{2\pi } F(t,A(t),B(t),C(t))(\cos t - 1)dt,} \hfill & {} \hfill \cr {{{\cal F}_2}({x_0},{y_0},{z_0}) = {1 \over {2\pi }}\int_0^{2\pi } F(t,A(t),B(t),C(t))\sin t{\kern 1pt} dt,} \hfill & {} \hfill \cr {{{\cal F}_3}({x_0},{y_0},{z_0}) = - {1 \over {2\pi }}\int_0^{2\pi } F(t,A(t),B(t),C(t))\cos t{\kern 1pt} dt,} \hfill & {} \hfill \cr } where A(t) = x0 + y0 sint + z0(1 − cost), B(t) = y0 cost + z0 sint and C(t) = −y0 sint + z0 cost. If the function F(t,x,x′,x″) is 2π-periodic in the variable t, then for every (x0,y0,z0) (x_0^ * ,y_0^ * ,z_0^ * ) solution of the system ℱk(x0,y0,z0) = 0 for k = 1,2,3, satisfying det((1,2,3)(x0,y0,z0)|(x0,y0,z0)=(x0,y0,z0))0, \det \left( {{{\left. {{{\partial ({{\cal F}_1},{{\cal F}_2},{{\cal F}_3})} \over {\partial ({x_0},{y_0},{z_0})}}} \right|}_{({x_0},{y_0},{z_0}) = (x_0^ * ,y_0^ * ,z_0^ * )}}} \right) \ne 0, the differential equation (2) has a 2π–periodic solution x(t,ɛ) which when ɛ → 0 tends to the 2π–periodic solution x0(t) given by x0(t)=x0+y0sint+z0(1cost) {x_0}(t) = x_0^ * + y_0^ * \sin t + z_0^ * (1 - \cos t) of x‴ + x′ = 0.

Theorem 1 is proved in section 3. Its proof is based in the averaging theory for computing periodic orbits, see section 6.

An application of Theorem 1 is the following.

Corollary 2

Consider the generalized memory oscillators given by the differential equation (2) with F(t,x,x′,x″) = x″ − (b0 + b1x + b2x2) sint. If b2 ≠ 0, 4+3b1212b0b1>0 4 + 3b_1^2 - 12{b_0}{b_1} > 0 and 4+b124b0b2>0 - 4 + b_1^2 - 4{b_0}{b_2} > 0 , then this differential equation has four periodic solutions xk (t,ɛ) for k = 1,2,3,4, tending when ɛ → 0 to the periodic solutions xk(t) where x1,2(t)=b124+b124b0b22b22b2sint±4+b124b0b2b2(1cost),x3,4(t)=b12b22±4+3b1212b0b23b2sint, \matrix{ {{x_{1,2}}(t) = } \hfill & {{{ - {b_1} \mp 2\sqrt { - 4 + b_1^2 - 4{b_0}{b_2}} } \over {2{b_2}}} - {2 \over {{b_2}}}\sin t \pm {{\sqrt { - 4 + b_1^2 - 4{b_0}{b_2}} } \over {{b_2}}}(1 - \cos t),} \hfill \cr {{x_{3,4}}(t) = } \hfill & { - {{{b_1}} \over {2{b_2}}} - {{2 \pm \sqrt {4 + 3b_1^2 - 12{b_0}{b_{_2}}} } \over {3{b_2}}}\sin t,} \hfill \cr } of x‴ + x′ = 0.

Corollary 2 is proved in section 4.

Our second result is on the periodic orbits of the differential system in ℝ4 of the form x=y,y=xεG(t,x,y,z,u),z=u,u=zεH(t,x,y,z,u). x' = y,\quad y' = - x - \varepsilon G(t,x,y,z,u),\quad z' = u,\quad u' = - z - \varepsilon H(t,x,y,z,u). where G and H are a 2π–periodic functions in the variable t and ɛ is a small parameter. These systems are a perturbation of the harmonic oscillator in ℝ4 and these kind of perturbations appear frequently in the study of the dynamics of a galaxy, in atomic physic, ... see for instance [2, 4, 6, 7, 10, 11].

Theorem 3

We define 1(x0,y0,z0,u0)=12π02πG(t,A(t),B(t),C(t),D(t))sintdt,2(x0,y0,z0,u0)=12π02πG(t,A(t),B(t),C(t),D(t))costdt,3(x0,y0,z0,u0)=12π02πH(t,A(t),B(t),C(t),D(t))sintdt,4(x0,y0,z0,u0)=12π02πH(t,A(t),B(t),C(t),D(t))costdt, \matrix{ {{{\cal F}_1}({x_0},{y_0},{z_0},{u_0}) = {1 \over {2\pi }}\int_0^{2\pi } G(t,A(t),B(t),C(t),D(t))\sin t{\kern 1pt} dt,} \hfill & {} \hfill \cr {{{\cal F}_2}({x_0},{y_0},{z_0},{u_0}) = - {1 \over {2\pi }}\int_0^{2\pi } \;G(t,A(t),B(t),C(t),D(t))\cos t{\kern 1pt} dt,} \hfill & {} \hfill \cr {{{\cal F}_3}({x_0},{y_0},{z_0},{u_0}) = {1 \over {2\pi }}\int_0^{2\pi } H(t,A(t),B(t),C(t),D(t))\sin t{\kern 1pt} dt,} \hfill & {} \hfill \cr {{{\cal F}_4}({x_0},{y_0},{z_0},{u_0}) = - {1 \over {2\pi }}\int_0^{2\pi } H(t,A(t),B(t),C(t),D(t))\cos t{\kern 1pt} dt,} \hfill & {} \hfill \cr } where A(t) = x0 cost + y0 sint, B(t) = −x0 sint + y0 cost, C(t) = z0 cost + u0 sint and D(t) = −z0 sint + u0 cost. Then for every (x0,y0,z0,u0*) (x_0^ * ,y_0^ * ,z_0^ * ,u_0^*) solution of the system ℱk(x0,y0,z0,u0) = 0 for k = 1,2,3,4, satisfying det((1,2,3,4)(x0,y0,z0,u0)|(x0,y0,z0,u0)=(x0,y0,z0,u0*))0, \det \left( {{{\left. {{{\partial ({{\cal F}_1},{{\cal F}_2},{{\cal F}_3},{{\cal F}_4})} \over {\partial ({x_0},{y_0},{z_0},{u_0})}}} \right|}_{({x_0},{y_0},{z_0},{u_0}) = (x_0^ * ,y_0^ * ,z_0^ * ,u_0^*)}}} \right) \ne 0, the differential system (3) has a 2π–periodic solution (x(t, ɛ), y(t, ɛ), z(t, ɛ), u(t, ɛ)) which when ɛ → 0 tends to the 2π–periodic solution (x0(t),y0(t),z0(t),u0(t)) given by x0(t)=x0*cost+y0*sint {x_0}(t) = x_0^*\cos t + y_0^*\sin t , y0(t)=x0*sint+y0*cost {y_0}(t) = - x_0^*\sin t + y_0^*\cos t , z0(t)=z0*cost+u0*sint {z_0}(t) = z_0^*\cos t + u_0^*\sin t , u0(t)=z0*sint+u0*cost {u_0}(t) = - z_0^*\sin t + u_0^*\cos t , of the unperturbed system (3) with ɛ = 0.

Theorem 3 is proved in section 5.

An application of Theorem 3 is the following.

Corollary 4

Consider the differential system (3) with G(t,x,y,z,u) = (−1 − x2 + z2) sint and H(t,x,y,z,u) = (1 − x2) cost. Then this differential system has 8 periodic solutions (xk(t, ɛ), yk(t, ɛ), zk(t, ɛ), uk(t, ɛ)) for k = 1,...,8, tending when ɛ → 0 to the periodic solutions (xk(t), yk(t), zk(t), uk(t)) where (x1,2(t),y1,2(t),z1,2(t),u1,2(t))=(±2sint,±2cost,4cost,4sint),(x3,4(t),y3,4(t),z3,4(t),u3,4(t))=(23cost,±23sint,43sint,43cost),(x5,6(t),y5,6(t),z5,6(t),u5,6(t))=(±2sint,±2cost,43sint,43cost),(x7,8(t),y7,8(t),z7,8(t),u7,8(t))=(23cost,±23sint,43cost,43sint), \matrix{ {({x_{1,2}}(t),{y_{1,2}}(t),{z_{1,2}}(t),{u_{1,2}}(t)) = ( \pm 2\sin t, \pm 2\cos t, - 4\cos t,4\sin t),} \hfill & {} \hfill \cr {({x_{3,4}}(t),{y_{3,4}}(t),{z_{3,4}}(t),{u_{3,4}}(t)) = \left( { \mp {2 \over {\sqrt 3 }}\cos t, \pm {2 \over {\sqrt 3 }}\sin t, - {4 \over 3}\sin t, - {4 \over 3}\cos t} \right),} \hfill & {} \hfill \cr {({x_{5,6}}(t),{y_{5,6}}(t),{z_{5,6}}(t),{u_{5,6}}(t)) = \left( { \pm 2\sin t, \pm 2\cos t, - {4 \over {\sqrt 3 }}\sin t, - {4 \over {\sqrt 3 }}\cos t} \right),} \hfill & {} \hfill \cr {({x_{7,8}}(t),{y_{7,8}}(t),{z_{7,8}}(t),{u_{7,8}}(t)) = \left( { \mp {2 \over {\sqrt 3 }}\cos t, \pm {2 \over {\sqrt 3 }}\sin t, - {4 \over {\sqrt 3 }}\cos t,{4 \over {\sqrt 3 }}\sin t} \right),} \hfill & {} \hfill \cr }

of x′ = y, y′ = −x, z′ = u, u′ = −z.

Corollary 4 is proved in section 6.

Averaging theory

We want to study the T–periodic solutions of the periodic differential systems of the form x=F0(t,x)+εF1(t,x)+ε2F2(t,x,ε), {\bf{x}}' = {F_0}(t,{\bf{x}}) + \varepsilon {F_1}(t,{\bf{x}}) + {\varepsilon ^2}{F_2}(t,{\bf{x}},\varepsilon ), with ɛ > 0 sufficiently small, where F0,F1 : ℝ × Ω n and F2 : ℝ × Ω × (−ɛ0, ɛ0) n are 𝒞2 functions, T–periodic in the variable t, and Ω is an open subset of ℝn. We denote by x(t,z,ɛ) the solution of the differential system (4) such that x(0,z,ɛ) = z. We assume that the unperturbed system x=F0(t,x), {\bf{x}}' = {F_0}(t,{\bf{x}}), has an open set V with Cl(V ) ⊂ Ω such that for each z ∈ Cl(V ), x(t, z, 0) is T–periodic.

We consider the variational equation y=DxF0(t,x(t,z,0))y, {\bf{y}}' = {D_{\bf{x}}}{F_0}(t,{\bf{x}}(t,{\bf{z}},0)){\bf{y}}, of the unperturbed system on the periodic solution x(t,z,0), where y is an n × n matrix. Let Mz(t) be the fundamental matrix of the linear differential system (6) such that Mz(0) is the n × n identity matrix. The next result is due to Malkin [13] and Roseau [16], for a shorter and easier proof see [5].

Theorem 5

Consider the function ℱ : Cl(V ) n (z)=1T0TMz1(t)F1(t,z(t,z,0))dt. {\cal F}({\bf{z}}) = {1 \over T}\int_0^T M_{\bf{z}}^{ - 1}(t){F_1}(t,{\bf{x}}(t,{\bf{z}},0))dt. If there exists αV with ℱ (α) = 0 and det ((dℱ/dz)(α)) ≠ 0, then there exists a T–periodic solution x(t, ɛ) of system (4) such that when ɛ → 0 we have that x(0,ɛ) → α.

Proof of Theorem 1

For ɛ = 0 all singular points of the differential system (1) are in the x–axis, i.e. (x,0,0) are the singular points of system (1). The eigenvalues of the linearized system at these singular points are ± i, 0. The 2π–periodic solutions (x(t), y(t), z(t)) of the unperturbed system (i.e. system (1) with ɛ = 0) such that (x(0), y(0), z(0)) = (x0,y0,z0) are (x0+y0sint+z0(1cost),y0cost+z0sint,y0sint+z0cost). ({x_0} + {y_0}\sin t + {z_0}(1 - \cos t),{y_0}\cos t + {z_0}\sin t, - {y_0}\sin t + {z_0}\cos t).

Using the notion introduced in section 2, we have that x = (x, y, z), z = (x0, y0, z0), F0 (x, t) = (y, z,y), F1 (x,t) = (0,0,−F) and F2 (x,t,ɛ) = (0,0,0). The fundamental matrix solution Mz(t) is independent of z and we shall denote it by M(t). An easy computation shows that M(t)=(1sint1cost0costsint0sintcost). M(t) = \left( {\matrix{ 1 & {\sin t} & {1 - \cos t} \cr 0 & {\cos t} & {\sin t} \cr 0 & { - \sin t} & {\cos t} \cr } } \right).

According to Theorem 5 we study the zeros α = (x0,y0,z0) of the three components of the function (α) given in (7). More precisely we have (α) = (1(α),2(α), 3(α)), such that 1(α)=12π02πF(t,x(t),y(t),z(t))(cost1)dt,2(α)=12π02πF(t,x(t),y(t),z(t))sintdt,3(α)=12π02πF(t,x(t),y(t),z(t))costdt, \matrix{ {{{\cal F}_1}(\alpha ) = {1 \over {2\pi }}\int\limits_0^{2\pi } F(t,x(t),y(t),z(t))(\cos t - 1)dt,} \hfill \cr {{{\cal F}_2}(\alpha ) = {1 \over {2\pi }}\int\limits_0^{2\pi } F(t,x(t),y(t),z(t))\sin t{\kern 1pt} dt,} \hfill \cr {{{\cal F}_3}(\alpha ) = - {1 \over {2\pi }}\int\limits_0^{2\pi } F(t,x(t),y(t),z(t))\cos t{\kern 1pt} dt,} \hfill \cr } where x(t), y(t), z(t) are given by (8). Now the rest of the proof of Theorem 1 follows directly from the statement of Theorem 5.

Proof of Corollary 2

We must apply Theorem 1 with F(t, x, y, z) = z − sint (b0 + b1x + b2x2). Computing the function = (1,2,3) of Theorem 1 we get 1(x0,y0,z0))=14(2b1y0+2z0+b2y0(4x0+5z0)),2(x0,y0,z0)=18(4b04y04b1(x0+z0)b2(4x02+3y02+8x0z0+5z02)),2(x0,y0,z0)=14(2+b2y0)z0. \matrix{ {{{\cal F}_1}({x_0},{y_0},{z_0})) = {1 \over 4}(2{b_1}{y_0} + 2{z_0} + {b_2}{y_0}(4{x_0} + 5{z_0})),} \hfill & {} \hfill \cr {{{\cal F}_2}({x_0},{y_0},{z_0}) = {1 \over 8}( - 4{b_0} - 4{y_0} - 4{b_1}({x_0} + {z_0}) - {b_2}(4x_0^2 + 3y_0^2 + 8{x_0}{z_0} + 5z_0^2)),} \hfill & {} \hfill \cr {{{\cal F}_2}({x_0},{y_0},{z_0}) = - {1 \over 4}(2 + {b_2}{y_0}){z_0}.} \hfill & {} \hfill \cr } System 1 = 2 = 2 = 0 has six solutions (x0*,y0*,z0*) (x_0^*,y_0^*,z_0^*) given by (b124+b124b0b22b2,2b2,±4+b124b0b2b2),(b12b2,2±4+3b1212b0b23b2,0),(b1±b124b0b22b2,0,0). \matrix{ {\left( {{{ - {b_1} \mp 2\sqrt { - 4 + b_1^2 - 4{b_0}{b_2}} } \over {2{b_2}}}, - {2 \over {{b_2}}}, \pm {{\sqrt { - 4 + b_1^2 - 4{b_0}{b_2}} } \over {{b_2}}}} \right),} \hfill & {} \hfill \cr {\left( { - {{{b_1}} \over {2{b_2}}}, - {{2 \pm \sqrt {4 + 3b_1^2 - 12{b_0}{b_2}} } \over {3{b_2}}},0} \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} \left( { - {{{b_1} \pm \sqrt {b_1^2 - 4{b_0}{b_2}} } \over {2{b_2}}},0,0} \right).} \hfill & {} \hfill \cr } The last two solutions going to the differential equation x‴ +x′ = 0 provide equilibrium points instead of periodic solutions, so we do not consider them. Since the Jacobian det((1,2,3)(x0,y0,z0)|(x0,y0,z0)=(x0*,y0*,z0*)) \det \left( {{{\left. {{{\partial ({{\cal F}_1},{{\cal F}_2},{{\cal F}_3})} \over {\partial ({x_0},{y_0},{z_0})}}} \right|}_{({x_0},{y_0},{z_0}) = (x_0^*,y_0^*,z_0^*)}}} \right) for these remaining four solutions (x0*,y0*,z0*) (x_0^*,y_0^*,z_0^*) is 18(4+b124b0b2) {1 \over 8}( - 4 + b_1^2 - 4{b_0}{b_2}) and 172(4+3b1212b0b2)(1+124+3b1212b0b2) {1 \over {72}}(4 + 3b_1^2 - 12{b_0}{b_2})(1 + {1 \over 2}\sqrt {4 + 3b_1^2 - 12{b_0}{b_2}} ) respectively, we obtain using Theorem 1 the four periodic solutions given in the statement of the corollary.

Proof of Theorem 3

Consider system (3). Its unperturbed system is the system (3) with ɛ = 0, which has the singular point (x, y, z, u) = (0, 0, 0, 0). The eigenvalues of the linearized system at this singular point are ±i, of multiplicity two. The periodic solutions (x(t), y(t), z(t), u(t)) of the unperturbed system such that (x(0), y(0), z(0), u(0)) = (x0, y0, z0, u0) are (x0cost+y0sint,x0sint+y0cost,z0cost+v0sint,z0sint+v0cost). ({x_0}\cos t + {y_0}\sin t, - {x_0}\sin t + {y_0}\cos t,{z_0}\cos t + {v_0}\sin t, - {z_0}\sin t + {v_0}\cos t). Note that all these solutions are periodic with period 2π.

Using the notations introduced in section 2, we have that x = (x, y, z, u), z = (x0, y0, z0, u0), F0 (x, t) = (y,x, u,z), F1 (x, t) = (0, −G, 0, −H) and F2 (x, t, ɛ) = (0, 0, 0, 0). The fundamental matrix solution Mz(t) corresponding to system (5) for our system (3) is independent of z and we shall denote it by M(t). An easy computation shows that M(t)=(costsint00sintcost0000costsint00sintcost). M(t) = \left( {\matrix{ {\cos t} & {\sin t} & 0 & 0 \cr { - \sin t} & {\cos t} & 0 & 0 \cr 0 & 0 & {\cos t} & {\sin t} \cr 0 & 0 & { - \sin t} & {\cos t} \cr } } \right). According to Theorem 5 we study the zeros α = (x0, y0, z0, u0) of the four components of the function (α) given in (7). More precisely we have (α) = (1(α), 2(α), 3(α), 4(α)), such that 1(α)=12π02πG(t,x(t),y(t),z(t),u(t))sintdt,2(α)=12π02πG(t,x(t),y(t),z(t),u(t))costdt,3(α)=12π02πH(t,x(t),y(t),z(t),u(t))sintdt,4(α)=12π02πH(t,x(t),y(t),z(t),u(t))costdt, \matrix{ {{{\cal F}_1}(\alpha ) = {1 \over {2\pi }}\int\limits_0^{2\pi } G(t,x(t),y(t),z(t),u(t))\sin t{\kern 1pt} dt,} \hfill \cr {{{\cal F}_2}(\alpha ) = - {1 \over {2\pi }}\int\limits_0^{2\pi } G(t,x(t),y(t),z(t),u(t))\cos t{\kern 1pt} dt,} \hfill \cr {{{\cal F}_3}(\alpha ) = {1 \over {2\pi }}\int\limits_0^{2\pi } H(t,x(t),y(t),z(t),u(t))\sin t{\kern 1pt} dt,} \hfill \cr {{{\cal F}_4}(\alpha ) = - {1 \over {2\pi }}\int\limits_0^{2\pi } H(t,x(t),y(t),z(t),u(t))\cos t{\kern 1pt} dt,} \hfill \cr } where (x(t),y(t),z(t),u(t)) is periodic solution given in (9). Now the rest of the proof of Theorem 3 follows directly from the statement of Theorem 5.

Proof of Corollary 4

We must apply Theorem 1 with G(t, x, y, z, u) = (−1 − x2 + z2) sint and H(t, x, y, z, u) = (1 − x2) cost. Computing the function = (1, 2, 3, 4) of Theorem 5 we obtain 1(x0,y0,z0))=18(4+3u02x023y02+z02),2(x0,y0,z0)=14(x0y0u0z0),3(x0,y0,z0)=14x0y0,4(x0,y0,z0)=18(4+3x02+y02). \matrix{ {{{\cal F}_1}({x_0},{y_0},{z_0})) = {1 \over 8}( - 4 + 3u_0^2 - x_0^2 - 3y_0^2 + z_0^2),} \hfill & {} \hfill \cr {{{\cal F}_2}({x_0},{y_0},{z_0}) = {1 \over 4}({x_0}{y_0} - {u_0}{z_0}),} \hfill & {} \hfill \cr {{{\cal F}_3}({x_0},{y_0},{z_0}) = - {1 \over 4}{x_0}{y_0},} \hfill & {} \hfill \cr {{{\cal F}_4}({x_0},{y_0},{z_0}) = {1 \over 8}( - 4 + 3x_0^2 + y_0^2).} \hfill & {} \hfill \cr } System 1 = 2 = 3 = 4 = 0 has sixteen solutions (x0*,y0*,z0*,u0*) (x_0^*,y_0^*,z_0^*,u_0^*) given by (0,±2,±4,0),(±23,0,±43,0),(0,±2,0,±43),(±23,0,±43,0). \matrix{ {(0, \pm 2, \pm 4,0),\quad \left( { \pm {2 \over {\sqrt 3 }},0, \pm {4 \over 3},0} \right),\quad \left( {0, \pm 2,0, \pm {4 \over {\sqrt 3 }}} \right),\quad \left( { \pm {2 \over 3},0, \pm {4 \over {\sqrt 3 }},0} \right).} \hfill & {} \hfill \cr } Since the Jacobian det((1,2,3),4)(x0,y0,z0,u0)|(x0,y0,z0,u0)=(x0*,y0*,z0*,u0*)) \det \left( {{{\left. {{{\partial ({{\cal F}_1},{{\cal F}_2},{{\cal F}_3}),{{\cal F}_4})} \over {\partial ({x_0},{y_0},{z_0},{u_0})}}} \right|}_{({x_0},{y_0},{z_0},{u_0}) = (x_0^*,y_0^*,z_0^*,u_0^*)}}} \right) for these solutions (x0*,y0*,z0*,u0*) (x_0^*,y_0^*,z_0^*,u_0^*) is 14,112,14,112 {1 \over 4},{1 \over {12}}, - {1 \over 4}, - {1 \over {12}} respectively, we obtain using Theorem 5 sixteen periodic solutions, but only eight of them are different because all periodic solutions appear repeated when we change t → t + π. Hence we obtain the eight periodic solutions given in the statement of the corollary.

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