New 1:1:1 periodic solutions in 3 -dimensional galactic-type Hamiltonian systems

Applying the averaging theory, we prove the existence of new families of periodic orbits for 3-dimensional type-galactic Hamiltonian systems.

In order to apply the averaging theory to system (3), we introduce a small parameter ε doing the rescaling (q 1 , q 2 , q 3 , p 1 , p 2 , p 3 √ ε p Z ). Since this change of coordinates is ε −1symplectic (see for more details [10]), the Hamiltonian function (1)- (2) in these new variables writes and system (3) becomeṡ The periodic orbits are the most simple non-trivial solutions of a differential system. Their study is of particular interest because the motion in their neighborhood can be determined by their kind of stability. In general, it is very difficult to study analytically the existence of periodic orbits and their kind of stability for a given Hamiltonian system. In this work, we use the averaging theory of first order for computing periodic orbits and their kind of stability, see Appendix 1 for a summary of this theory. The averaging theory allows to find analytically periodic orbits of this galactic model (5) in any positive Hamiltonian level. Roughly speaking, this method reduces the problem of finding periodic solutions of some differential system to the one of finding zeros of some convenient finite-dimensional function.
Our main result about the periodic orbits of the Hamiltonian system (5) is summarized as follows.

Moreover, on each Hamiltonian level H = h, the 16 periodic solutions are unstable.
Theorem 1 is proved in Sect. 2 using the averaging theory of first order. Note that the change of variables is only a scale transformation; for ε > 0, the original system (3) and the transformed systems (5) have essentially the same phase portrait. Note that system (5) for ε > 0 sufficiently small is close to an integrable one.
In [4], the Hamiltonian was also studied (1) with The Hamiltonian system associated to (1)- (7) iṡ In order to apply the averaging method of second order to system (8), we introduce a small parameter ε by the change of the variables (q 1 , q 2 , q 3 , p 1 , p 2 , p 3 ) = (ε X, εY, εZ , εp X , εp Y , εp Z ). Since this change of coordinates is ε −2 -symplectic, the Hamiltonian (1)- (7) in these new variables takes the form and system (8) becomeṡ Our main result about the periodic orbits of the Hamiltonian system (9) is summarized as follows.
Theorem 2 For ε > 0 sufficiently small at every positive Hamiltonian level H = h, the galactictype Hamiltonian system (9) has 8 periodic solutions given by (6) where r * , ρ * , R * , α * , β * are given by The proof of Theorem 2 is given in Sect. 3. Caranicolas in [4] using his semi-numerical method found 2 families of rectilinear periodic solutions on the invariant sets x = y = ±z/2. These two families correspond to the two families (i) of Theorem 2. Additionally, we obtain 6 new families of periodic solutions parametrized by the Hamiltonian level h.
As we shall see one of the main problems for applying the averaging theory for studying, the periodic orbits of a given differential system are to find the changes of variables which allow to write the original differential system in the normal form for applying the averaging theory. For more details in this direction, see the book [11].
In [15][16][17], dynamics aspects have been studied in Hamiltonian systems in 2D and 3D; in particular, the existence of periodic orbits escapes, and chaos was considered.

Proof of Theorem 1
For proving Theorem 1, we shall apply Theorem 3 to the Hamiltonian system (5). Generically, the periodic orbits of a Hamiltonian system with more than one degree of freedom are on cylinders filled of periodic orbits parametrized by the value h of Hamiltonian level. Therefore, we cannot apply directly Theorem 3 to a Hamiltonian system, because the Jacobian of the function f defined in the Appendix 1 is always zero. Then, we must apply Theorem 3 to every Hamiltonian level where the periodic orbits generically are isolated. For more details on the families of periodic solutions of Hamiltonian systems, see [1].
First, we change the Hamiltonian (4) and the equations of motion (5) to convenient generalized polar coordinates in such a way that for ε = 0, we have a pair of harmonic oscillators. Thus, we consider the change of variables, with (r, θ, ρ, α, R, β) ∈ R + ×S 1 ×R + ×S 1 ×R + ×S 1 . Note that this is a change of variables when r, ρ, and R are positive, and that this change of coordinates is not canonical. So, we lose the Hamiltonian structure of the differential equations. Moreover, doing this change of variables appear in the system the angular variables θ, α and β. Later on, the variable θ will be used for obtaining the periodicity necessary for applying the averaging theory.
The fixed value of the Hamiltonian level H = h in polar coordinates is and the equations of motion are given bẏ r = 2εr sin θ cos θ Note that the derivatives on the left-hand side of these equations are with respect to the time t, and that the system is not periodic in t. For ε > 0 sufficiently small in a neighborhood of r = ρ = R = 0, we have thatθ > 0.
In such a neighborhood, we take θ as the new independent variable, and we denote by a prime the derivative with respect to θ . The angular variables α, β cannot be used as the independent variable because the new differential system would not have the normal form (23) for applying the averaging theory given in Theorem 3. By fixing the value of the first integral H at h such that 2h − r 2 − R 2 > 0, solving Eq. (11) with respect to ρ, and expanding ρ in Taylor series of ε, we obtain We write the differential system (r , R , α , β ), substitute it in the expression of ρ given in (12), and expanding in Taylor series in ε, we obtain the differential system Clearly, system (13) satisfies the assumptions of Theorem 3, and it has the normal form (23) of the averaging theory with The function F 1 is analytical. Furthermore, it is 2πperiodic in the variable θ , the independent variable of system (13). In order to apply the averaging theory of first order, we must calculate the averaged functions of We compute the real solutions (r * , R * , α * , β * ) of the system f j (r, R, α, β) = 0 for j = 1, 2, 3, 4. It is important to remember that at order 0 in ε, we have ρ = √ 2h − r 2 − R 2 , so we must have to present that r, R, and ρ cannot be simultaneously zero. Solving the first equation of (14), we obtain the following possibilities. The first case, r = 0; the second case, sin(2α) = 0 and r = csc(2α) (2h − R 2 ) sin(2α) + R 2 sin(2β) ; the third case, sin(2α) = 0 and R = 0; and finally, the fourth case sin(2α) = 0 and sin(2β) = 0. (14), we obtain h[cos(2β) + 2]/2, then r = 0, and R = √ 2h is not solution.
Then, equations f 2 and f 4 of (14) become Solving this system, we verify that k must be even, i.e., k = 2l and the solutions are R * = √ h, β * = ±2π/3, ±π/3, α * = lπ + β. Note that in this case, ρ * = √ h, and the corresponding Jacobian satisfies These families of periodic solutions can be reduced only to two periodic solutions, because they are the same in first-order approximation, so we have proved item (i).
and sin(2α) = 0. The function f 3 takes the form . Substituting these values of r and R in equations f 2 and f 4 , we obtain the system Under the restriction sin α sin β sin(α − β) = 0, we obtain the solutions r . It is verified that the Jacobian at these solutions satisfies Therefore, Theorem 3 guarantees the existence of 8 periodic solutions, and then we have proved part of (iv). Subcase II. 3 Here, we have β = kπ/2 with k ∈ Z. Then, r = √ 2h − R 2 and substituting this value in equation f 4 , we arrive to the equation Next, we substitute the value of β and R in equation f 2 , and we obtain that k must be even, i.e., k = 2l and then α = ±2π/3, ±π/3, β = lπ . Therefore, r = √ h. Moreover, the Jacobian at these solutions satisfies These families of periodic solutions can be reduced only to two periodic solutions, because they are the same in first-order approximation, so we have proved item (ii). Case III sin(2α) = and R = 0, or equivalently, α = kπ/2 with k ∈ Z. Then, equation f 2 writes (2h − 2r 2 )[cos(2α) + 2]/4 = 0. So then r = √ h. Equation for f 4 implies −h −2(−1) k sin 2 β+ cos(2β)+2 /4 = 0, where we get that k must be even, i.e., k = 2l with l ∈ Z, and β = ±2π/3, ±π/3 and α = lπ . We verify that the Jacobian at these solutions satisfies By the same arguments as in the previous cases, we conclude the proof of (iii).
Case IV sin(2α) = 0 and sin(2β) = 0, or equivalently, α = kπ/2 and β = mπ/2 with k, m ∈ Z. Therefore, f 3 = 0, and f 2 , f 3 take the form Solving this system, we get that k = 2l, m = 2n, so α = lπ , β = nπ with l, n ∈ Z, and r * = R * = √ 2h/3. In any of these points, we have that the Jacobian at this critical point satisfies These periodic solutions can be reduced to only four periodic solutions, so we conclude the proof of(iv).
The proof of the second part of the theorem is as follows. For the family (1), we have the characteristic polynomial of the linearization of the averaged system (18) at the points (r * = 0, . We verify that one of the roots is Thus, the two periodic orbits of (i) are unstable.
The characteristic polynomial for any solution in Since p 3 (0) < 0 and p 3 (λ) → +∞ as λ → +∞, it follows that p 3 (λ) as at least one positive root; thus, the periodic orbits of (iii) are unstable.

Proof of Theorem 2
We continue using the polar coordinates given in (10), and we observe that in the Hamiltonian level H = h of the Hamiltonian (1)-(7) in polar coordinates, we have and the equations of motion writė r = −ε R 2 sin θ cos 2 (β + θ), ρ = −ε R 2 sin(α + θ) cos 2 (β + θ), R = ε R sin(2(β + θ ))(r cos θ + ρ cos(α + θ )), In order to write this system as a periodic differential system, we take the variable θ as the new independent variable, and we use a prime to denote the derivative with respect to θ . The angular variables α and β cannot be used as independent variable because the new differential system would not be in the normal form for applying the averaging theory described in Theorem 3. Of course, the new system has now only five equations because we do not need theθ equation. Writing it in Taylor series of ε, we get Therefore, system (16) is 2π -periodic in the variable θ . In order to apply Theorem 3, we fix the value of the first integral at h > 0, and by solving equation (15) for ρ and expanding it in Taylor series of ε, we obtain − ε 2 R 4 cos 4 (β + θ) −2h + r 2 + R 2 cos 2 (α + θ) + r 2 cos 2 θ 2 2h − r 2 − R 2 3/2 Using this value of ρ in equations (16), we obtain the following 4-dimensional differential system r = −ε R 2 sin θ cos 2 (β + θ) −ε 2 1 r R 4 sin θ cos θ cos 4 +r cos θ −2h + 2r 2 + R 2 cos(α + θ) Using the notation x = (r, R, α, β) ∈ D = (0, √ 2h) × (0, √ 2h) × R × R and t = θ , system (18) has the normal form of the averaging of Theorem 3.
Since ρ ∈ (0, √ 2h), the zeros of ρ in that interval of the polynomial (21) are ρ * , and it is not difficult to check that the polynomial of degree 20 in the variable ρ which appears as factor in (21) has no real roots in the interval (0, √ 2h). From (21) and (22), we obtain that sin β = 0, i.e., β = 0, π, for the solutions ρ * j for j = 1, 2, 3. Now, we consider the third polynomial of the Groebner basis given in Appendix 1, and we substitute it in the value of ρ * 4 obtaining that sin β = ±1, i.e., β = ±π/2. For each value of ρ * j , we compute R = 2h − r 2 − ρ 2 , and substituting it in the equation of f 11 , we obtain for j = 1, 2, 3, 4 that sin α = 0, i.e., α = 0, π. Continuing the analysis as in the proof of Theorem 1, we get the solutions (i)-(viii) given in the statement of Theorem 2.
The characteristic polynomial of (i) and (ii) is whose roots are ±i4 √ 5h/3 and ±i4 √ 5/3h/3. So, the periodic solutions of (i) and (ii) are linearly stable. For Again, this polynomial has two real solutions (one positive and one negative) and two pure imaginary pure; therefore, the periodic solutions of (vii) and (viii) are unstable. This completes the proof of Theorem 2.

Conclusions
The objective of this work was to prove the existence of periodic orbits and its type of stability, in the galactic Hamiltonian of three degrees of freedom with 1 : 1 : 1 resonance where H 1 is either −(q 2 1 q 2 2 + q 2 1 q 2 3 + q 2 2 q 2 3 ), or −(q 1 + q 2 )q 2 3 . The families of periodic solutions of these two 3dimensional galactic-type Hamiltonian systems started to be studied in [4].
We have used an important tool from the area of dynamical systems, the averaging theory for studying the existence of periodic orbits and their stability, and we have applied it for studying the families of periodic solutions of the Hamiltonian systems defined by the two previous Hamiltonians. Our main results are summarized in Theorems 1 and 2.
In Theorem 1, we have recuperated the 4 families of straight line periodic orbits found by Caranicolas in [4], but we also have obtained 12 new families of periodic solutions parametrized by the value of the Hamiltonian, and we also have proved that these 16 families of periodic orbits are unstable in each Hamiltonian level.
In Theorem 2, we again reobtained the 2 families of straight line periodic orbits found by Caranicolas

Appendix 1
Averaging theory of first and second order In this section, we recall the results of the averaging theory that we have used for proving the results of this paper. For a general introduction to the averaging theory, see Chapter 11 of the book [13], and mainly the book [11]. The averaging theory up to second order stated in what follows, with the weak assumptions used, was proved in [3].
Theorem 3 Consider the differential systeṁ for all t ∈ R, F 1 , F 2 , R and D x F 1 are locally Lipschitz with respect to x, and R is differentiable with respect to ε. We define f 1 , f 2 : Then, for |ε| > 0 sufficiently small, there exists a T −periodic solution x(t, ε) of the system such that x(0, ε) → a when ε → 0.
As usual, we have denoted by d B ( f 1 + ε f 2 ), the Brouwer degree of the function f 1 + ε f 2 : V → R n at its zero a; for more details on the Brouwer degree, see [2]. A sufficient condition for showing that the Brouwer degree of a function f at its zero a is non-zero is that the Jacobian of the function f at a (when it is defined) is non-zero, see for more details [6].
If the function f 1 is not identically zero, then the zeros of f 1 + ε f 2 are essentially the zeros of f 1 for ε sufficiently small. In this case, Theorem 3 provides the so-called averaging theory of first order.
If the function f 1 is identically zero and f 2 is not identically zero, then the zeros of f 1 + ε f 2 are the zeros of f 2 . In this case, Theorem 3 provides the so-called averaging theory of second order.
In the case of the averaging theory of first order, we consider in D the averaged differential equatioṅ where Then, Theorem 3 gives information about the stability or instability of the periodic solution x(t, ε). In fact, it is given by the stability or instability of the equilibrium point p of the averaged system (24). In fact, the singular point p has the stability behavior of the Poincaré map associated to the periodic solution x(t, ε). In the case of the averaging theory of second order, i.e., f 1 ≡ 0 and f 2 non-identically zero, we have that the stability and instability of the limit cycle ϕ(t, ε) coincide with the type of stability or instability of the equilibrium point p of the averaged systeṁ i.e., it is the same that the singular point p associated to the Poincaré map of the periodic solution ϕ(t, ε), see for instance the Chapter 11 of [13].

Appendix 2
Third factor in the basis of Groebner