Periodic solutions induced by an upright position of small oscillations of a sleeping symmetrical gyrostat

The aim of this paper is to provide sufficient conditions for the existence of periodic solutions emerging from an upright position of small oscillations of a sleeping symmetrical gyrostat with equations of motion being α and β parameters satisfying Δ=α2−4β>0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta-\frac{\alpha^{2}}{2}\pm \frac{\alpha \sqrt{\varDelta }}{2}<0$\end{document}, ε a small parameter and, F1 and F2 smooth periodic maps in the variable t in resonance p:q with some of the periodic solutions of the system for ε=0, where p and q are positive integers relatively prime. The main tool used is the averaging theory.


Introduction and statement of the main results
The equations of motion of a sleeping gyrostat on the upright position under the influence of small periodic momenta arë x + αẏ − βx = εF 1 (t, x,ẋ, y,ẏ), y − αẋ − βy = εF 2 (t, x,ẋ, y,ẏ), (1) for more details on them see Appendix 2 or papers [6] and [9]. Here the dot denotes derivative with respect to the time t. The parameter ε is small and the smooth functions F 1 and F 2 define the perturbed torques which, in general, are periodic functions in the variable t and in resonance p:q with some of the periodic solutions of the sleeping symmetrical gyrostat for ε = 0, being p and q positive integers relatively prime.
Recall that a gyrostat is a mechanical system S composed by a rigid solid S 1 to which other bodies S 2 are connected; these other bodies may be variable or rigid, but the key property is that they must not be rigidly connected to S 1 , so that the movements of S 2 with respect to S 1 do not modify the distribution of mass within the compound system S. For instance, we can consider a rigid main body S 1 , designated as the platform, supporting additional bodies S 2 , which possess axial symmetry and are designated as rotors. These rotors may rotate with respect to the platform in such a way that the mass distribution within the system as a whole is not altered; this will produce an internal angular momentum, designated as gyrostat momentum, which will be normally regarded as a constant. Note that when this constant vector is zero, the motion of the system is reduced to the motion of a rigid solid, see for instance Fig. 1 where a gyrostat in the frame of the three body problem is presented, and [3,4] or [5] for more details on this class of mechanical systems.
The objective of this paper is to provide, using the averaging theory, a system of nonlinear equations whose simple zeros provide periodic solutions of the differential system (1). In order to present our results we need some preliminary definitions and notation. Periodic orbits of other kind of gyrostat have been studied for several authors, see for instance the article [1] and the references therein.
The unperturbed system with two differential equations of second order written as a differential system of first order with four in the variables (x,ẋ, y,ẏ) has a unique singular point at the origin with eigenvalues which are the roots of the polynomial ω 4 − (α 2 − 2β)ω 2 + β 2 when the parameters α and β verifies that Δ = α 2 − 4β > 0. The frequencies ω i are given by , As usual we define the ratio of the two frequencies to be non-resonant with π if ω 2 π/ω 1 is not a rational number.
System (2) in the phase space (x,ẋ, y,ẏ) has two planes passing through the origin filled of periodic solutions with the exception of the origin. These periodic solutions have periods T 1 = 2π/ω 1 and T 2 = 2π/ω 2 , according they belong to the plane associated to the eigenvectors with eigenvalues ±ω 1 i or ±ω 2 i, respectively. We shall study which of these periodic solutions persist for the perturbed system (1) when the parameter ε is sufficiently small and the perturbed functions F i for i = 1, 2 have period either pT 1 /q, or pT 2 /q, where p and q are positive integers relatively prime.
We define the constants Φ 1 and Φ 2 by and the functions: where and such that is called a simple zero of system (4). The statement of our main result on the periodic solutions of the differential system (1) which bifurcate from the periodic solutions of period T 1 of the unperturbed system traveled p times is the following.
Theorem 1 Let p and q be positive integers relatively prime and assume that the smooth functions F 1 and F 2 of the equations of motion of (1) are periodic in the variable t of period pT 1 /q. We assume that the ratio of the frequencies ω 2 /ω 1 is not resonant with π . Then for ε = 0 sufficiently small and for every simple zero (X * 0 , Y * 0 ) = (0, 0) of the nonlinear system (4), the perturbed system (1) has a periodic solution (x(t, ε), y(t, ε)) tending to the periodic solution Theorem 1 is proved in Sect. 2. Its proof is based in the averaging theory for computing periodic solutions, see Appendix 1 for more details on this technique.
An application of Theorem 1 is presented in the following corollary, which will be proved in Sect. 3.

Corollary 2 Assume that
and that the ratio of the frequencies ω 2 /ω 1 is not resonant with π . Then the system (1) for ε = 0 sufficiently small has two periodic solutions (x(t, ε), y(t, ε)) tending to the two periodic solutions (x(t), y(t)) =

Now we define the functions
with and

Consider the nonlinear system
The statement of our main result on the periodic solutions of the differential equations (1) which bifurcate from the periodic solutions of the unperturbed system (2) with period T 2 traveled p times is the following.

Theorem 3
Let p and q be positive integers relatively prime and assume that the smooth functions F 1 and F 2 of the equations of motion of (1) are periodic in the variable t of period pT 2 /q. Assume that the ratio of the frequencies ω 2 /ω 1 is not resonant with π . Then for ε = 0 sufficiently small and for every simple zero (Z * 0 , W * 0 ) = (0, 0) of the nonlinear system (6), the perturbed system (1) has a periodic solution (x(t, ε), y(t, ε)) tending to the periodic so- Theorem 3 will be proved in Sect. 2. In the next corollary an application of Theorem 3 is given.

Proof of Theorems 1 and 3
Introducing the variables (X 1 , Y 1 , Z 1 , W 1 ) = (x,ẋ, y,ẏ) we can write the differential system (1) as a firstorder differential system defined in R 4 in the following form: Note that the differential system (7) when ε = 0 is equivalent to the differential system (2), called simply in what follows the unperturbed system. When ε = 0 we called it a perturbed system.
writes the linear part of the differential system (7) in its real Jordan normal form, and this system in the new variables (X, Y, Z, W ) becomeṡ where A, B, C, D), Now, in the following lemma we characterize the periodic orbits of the unperturbed system as a first step for proving Theorems 1 and 3.
Proof Since (9) for ε = 0 is a linear differential system the proof follows easily.
Proof of Theorem 1 Assume that the functions F 1 and F 2 of (1) are periodic in t of period pT 1 /q with p and q positive integers relatively prime. Then, we can consider that the differential system (9) and the periodic solutions (10) have the same period pT 1 . We apply Theorem 6 of Appendix 1 to the differential system (9), and we use the notation introduced there. Note that system (9) can be written in the form of system (12) taking Now we shall study what periodic solutions of the unperturbed system (9) with ε = 0 of the type (10) persist as periodic solutions for the perturbed one for ε = 0 sufficiently small.
We start with the description of the different elements which appear in the statement of Theorem 6 for the particular case of the differential system (9). Thus, we have Ω = R 4 , k = 2 and n = 4. Now, let r 1 > 0 be arbitrarily small and let r 2 > 0 be arbitrarily large. Let V be the open and bounded subset of the plane Z = W = 0 of the form V = {(X 0 , Y 0 , 0, 0) ∈ R 4 : r 1 < X 2 0 + Y 2 0 < r 2 }. As usual Cl(V ) denotes the closure of V . If α = (X 0 , Y 0 ), then we identify V with the set {α ∈ R 2 : r 1 < α < r 2 }, being · the Euclidean norm in R 2 . The function β : Cl(V ) → R 2 is β(α) = (0, 0). Therefore, for our system we have We consider for each z α ∈ Z the periodic solution x(t, z α ) = (X(t), Y (t), 0, 0) given by (10) of period pT 1 .
Computing the fundamental matrix M z α (t) of the linear differential system (9) with ε = 0 associated to the pT 1 -periodic solution z α = (X 0 , Y 0 , 0, 0) such that M z α (0) be the identity of R 4 , we get Note that the matrix M z α (t) does not depend on the particular periodic solution x(t, z α , 0). The matrix satisfies the assumptions of statement (ii) of Theorem 6 because the determinant because the ratio of the frequencies is non-resonant with π . In short, all the assumptions of Theorem 6 are satisfied by the system (9). For our system the map ξ : R 4 → R 2 has the form ξ(X, Y, Z, W ) = (X, Y ). Calculating the function where the functions G k for k = 1, 2 are the ones given in (3). Then, by Theorem 6 we find that for every simple zero (X * 0 , Y * 0 ) ∈ V of the system of nonlinear equations (4) we have a periodic solution (X, Y, Z, W )(t, ε) of system (9) such that (X, Y, Z, W )(0, ε) → X * 0 , Y * 0 , 0, 0 when ε → 0.
Going back through the change of coordinates (8) we get a periodic solution (X 1 , Y 1 , Z 1 , W 1 )(t, ε) of system (9) such that Consequently we obtain a periodic solution (x, y)(t, ε) of system (1) such that This completes the proof of the theorem.

Proof of Theorem 3
The proof is analogous to the proof of Theorem 1 changing the roles of T 1 for T 2 .

Proof of the two corollaries
Proof of Corollary 2 Under the assumptions of Corollary 2 the nonlinear system (4) becomes This system has the following four solutions: Note that the solutions which differs in a sign are different initial conditions of the same periodic solution of the system (2). Moreover, since these solutions are simple. Finally, by Theorem 1 we only have two periodic solutions for the system of this corollary.

Proof of Corollary 4
Under the assumptions of Corollary 4 the nonlinear system (6) becomes This system only has the two real solutions both providing different initial conditions of the same periodic orbit of the unperturbed system. Since by Theorem 3 we only have one periodic solution of the differential system of this corollary. vertical axis in ascendent sense of the gyrostat, the second one corresponds to rotation in descendent sense.
To derive the equations of motion close to the upright solution we consider the Lagrangian of the system given by We introduce the generalized Eulerian angular variables (θ 1 , θ 2 , ϕ). The following relations are verified: and the Lagrangian of the problem is The variable ϕ is a cyclic coordinate of L and k ϕ = C( dϕ dt + dθ 2 dt sin θ 1 + l) is an integral of the system. The constant k ϕ are given by k ϕ = C r 0 + l + dθ 2 dt t=0 sin θ 1 (0) being dϕ dt (0) = r 0 . For Lagrangians with a cyclic variable it is possible to construct the so called Routh's map given by R = L − k ϕ dθ dt removing the cyclic variable of the motion equations. Thus, performing this Lagrangian reduction to L we obtain the Routhian of our system given by The equations of motion associated to R are A d 2 θ 1 dt 2 + dθ 2 dt 2 cos θ 1 sin θ 1 − k ϕ dθ 2 dt cos θ 1 − V (cos θ 1 cos θ 2 )(sin θ 1 cos θ 2 ) = 0, A d 2 θ 2 dt 2 cos 2 θ 1 − 2θ 1 dθ 2 dt sin θ 1 cos θ 1 + k ϕ dθ 1 dt cos θ 1 − V (cos θ 1 cos θ 2 )(cos θ 1 sin θ 2 ) = 0.
When θ 1 and θ 2 are small via the change of variables θ 1 = y, θ 2 = x and expanding in Taylor series to second order in we see that (16) are reduced to the unperturbed equations (2) x + αẏ − βx = 0, y − αẋ − βy = 0, with α = (Cr 0 + l)/A and β = V (1)/A. Remark 1 We remark that system (2) models the perturbations given by the maps F 11 (x, y, dx dt , dy dt , t) and F 21 (x, y, dx dt , dy dt , t) of the small oscillations around the vertical position of a symmetrical gyrostat.

Remark 2
Our results provides sufficient conditions for the existence of periodic solutions of system (2).