On the periodic solutions of a generalized smooth or non-smooth perturbed planar double pendulum

We provide sufficient conditions for the existence of periodic solutions with small amplitude of the non--linear planar double pendulum perturbed by smooth or non--smooth functions.


Introduction and statement of the main results
We consider a system of two point masses m 1 and m 2 moving in a fixed plane, in which the distance between a point P (called pivot) and m 1 and the distance between m 1 and m 2 are fixed, and equal to l 1 and l 2 respectively. We assume the masses do not interact. We allow gravity to act on the masses m 1 and m 2 . This system is called the planar double pendulum.
Here the dot denotes derivative with respect to the time T .
The authors in [8] have studied, in the vicinity of the equilibrium φ 1 = φ 2 = 0, the persistence of periodic solutions of system (1) perturbed smoothly in the particular case when m 1 = m 2 and l 1 = l 2 . Now m 1 , m 2 , l 1 and l 2 can take arbitrary positive values and we shall study the periodic orbits of system (1) which persist under smooth and non-smooth perturbations.
Here the smooth functionsF i andR i for i = 1, 2, 3, 4 define the perturbation. These functions are respectively T Fi -periodic and T Ri -periodic in t and respectively in resonance p Fi :q Fi and p Ri :q Ri with some of the periodic solutions of the linearized unperturbed double pendulum, being p and q relatively prime positive integers for p = p Fi , p Ri , q = q Fi , q Ri and i = 1, 2, 3, 4. We also assume that F i (t, 0, 0, 0, 0) = 0 for i = 1, 2, 3, 4.
Remark 1. For simplicity, we can assume that the functionsF i andR i for i = 1, 2, 3, 4 are T -periodic with T = pT j for some integer p where T j for j = 1, 2 are the periods of the solutions of the linearized unperturbed double pendulum. Indeed, if we take p the least common multiple among p Fi and p Ri for i = 1, 2, 3, 4, then there exists integers n Fi and n Ri such that p = n Fi p Fi = n Ri p Ri for i = 1, 2, 3, 4. Hence For i = 1, 2, 3, 4, and j = 1, 2.
Note that the functionsF i andR i for i = 1, 2, 3, 4, can be taken in a certain way arbitrary, i.e., only assuming some hypotheses. It makes us able to provide, in a physical context, real meaning for these functions. In our case, since we are working with discontinuity in the variables θ 1 and θ 2 , the functionsF 1 ,F 2 ,R 1 andR 2 could model the escapement for the particle m 1 , and the functionsF 3 ,F 4 ,R 3 andR 4 could model the escapement for the particle m 2 . If we work with discontinuity in the variables θ ′ 1 and θ ′ 2 , instead with discontinuity in the variables θ 1 and θ 2 , the respective functions could model the Coulomb Friction. We also can work composing these two phenomena. For more details on physical systems with discontinuous models see, for instance, [1] and [2]. Now, we follow the steps: (i) proceed with the change of variable φ 1 = εθ 1 and φ 2 = εθ 2 ; (ii) expand in Taylor series, for ε = 0, the expressions ofθ 1 andθ 2 ; (iii) Take a new time t given by the rescaling t = α τ , with α = l 1 m 1 /(g m 2 ); (iv) and finally, denote Hence, we obtain the following equations of motion for the double pendulum (3) , where now the prime denotes derivative with respect to the new time t. Here the functions F i , for i = 1, 2, 3, 4, are linear in the spatial variables, and are given by and K i (τ ) = α 2R i (ατ, 0, 0, 0). Observe that, for i, j = 1, 2, 3, 4, d j i (τ ) and K i (τ ) are T /αperiodic functions.
The objective of this paper is to provide a system of equations whose simple zeros provide periodic solutions (see Figure 2) of the perturbed planar double pendulum (2).
In order to present our results we need some preliminary definitions and notations. The unperturbed system (3) has a unique singular point, the origin with eigenvalues ±ω 1 i, ±ω 2 i, where with ∆ = (a − b) 2 + 4b > 0. Consequently this system in the phase space (θ 1 , θ ′ 1 , θ 2 , θ ′ 2 ) has two planes filled with periodic solutions except the origin. The periods of such periodic orbits are These periodic orbits live into the planes 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 Figure 2. Periodic solution of the perturbed system (2) converging to the origin, when ε → 0.
system (2) when the parameter ε is sufficiently small and the functions of perturbationF i and R i for i = 1, 2, 3, 4 have period either pαT 1 , or pαT 2 , with p positive integer.

Remark 2.
We say that the Crossing Hypothesis is satisfied if there exists a compact set D ⊂ R 4 such that every orbit starting in D reaches the set of discontinuity only at its crossing regions (see Appendix A).
Let X X0,Y0 (τ ) be the periodic function then we define the non-smooth function F 1 (X 0 , Y 0 ) by A zero (X * 0 , Y * 0 ) of the system of the non-smooth functions is called a simple zero of system (7).
Our main result on the periodic solutions of the non-smooth perturbed double pendulum (2) which bifurcate from the periodic solutions of the unperturbed double pendulum (1) with period T 1 traveled p times is the following.
Theorem 1 is proved in section 2. Its proof is based in the averaging theory for computing periodic solutions, see the Appendix B.
Note that the periodic solution given in Theorem 1 is a periodic solution bifurcating at ε = 0 from the equilibrium of system (2) localized at the origin of coordinates. For |ε| > 0 sufficiently small this orbits is close to the plane defined by the eigenvectors of the eigenvalues ±iω 1 .
We provide an application of Theorem 1 in the following corollary, which will be proved in section 3.
Theorem 3. Assume that the functionsF i andR i of the non-smooth perturbed double pendulum (2) are periodic in t of period pαT 2 with p positive integer. Also assume that the Crossing Hypothesis (see Remark 2) is satisfied. Then for ε > 0 sufficiently small and for every simple zero (Z * 0 , W * 0 ) = (0, 0) of the non-smooth system (10) such that the orbits pass by D, the non- Theorem 3 is also proved in section 2. Again the periodic solution given in Theorem 3 is a periodic solution bifurcating at ε = 0 from the equilibrium of system (2) localized at the origin of coordinates. For |ε| > 0 sufficiently small this orbits is close to the plane defined by the eigenvectors of the eigenvalues ±iω 2 .
We provide an application of Theorem 3 in the following corollary, which will be proved in section 3.

Proofs of Theorems 1 and 3
Introducing the variables (x, y, z, w) = (θ 1 , θ ′ 1 , θ 2 , θ ′ 2 ) we write the differential system of the non-smooth perturbed double pendulum (3) as a first-order differential system defined in R 4 . Thus we have the differential system (11) x ′ = y, System (11) with ε = 0 is equivalent to the unperturbed double pendulum system (3), called in what follows simply by the unperturbed system. Otherwise we have the perturbed system.
where s δ (x) is the smooth function defined in Figure 5, such that lim δ→0 s δ (x) = sgn(x). We shall write system (12) in such a way that the linear part at the origin of the unperturbed system will be in its real normal Jordan form. Then, doing the change of variables (τ, x, y, z, w) → (τ, X, Y, Z, W ) given by whereF i (t, X, Y, Z, W ) = F i (t, A, B, C, D) for i = 1, 2, 3, 4 with . Note that the linear part of the differential system (14) at the origin is in its real normal Jordan form.
Lemma 5. The periodic solutions of the differential system (14) with ε = 0 are Proof. Since system (14) with ε = 0 is a linear differential system, the proof follows easily.
Proof of Theorem 1. Assume that the functionsF i andR i of the non-smooth perturbed double pendulum with equations of motion (3) are periodic in t of period pαT 1 with p positive integer. Thus K i and F i are pT 1 -periodic functions, i.e., the differential system (3) and the periodic solutions (15) have the same period pT 1 .
It is well known that a Poincaré map defined in a smooth differential system is smooth. So the Poincaré maps associated to the periodic orbits of the differential system (12) are smooth.
The Poincaré maps, restricted at D, associated to the periodic solutions of the non-smooth differential system (11), which are perturbations of the periodic solutions (15) are also smooth. Since the orbits starting in D reaches the discontinuity set only at the of crossing region (see Appendix A), such Poincaré maps are compositions of smooth Poincaré maps. In a similar way it follows that the Poincaré maps, restricted at D, associated to the periodic solutions of the non-smooth differential system (11), which are perturbations of the periodic solutions (16) are also smooth.
We can use Theorem 6 (see the Appendix B) for computing some periodic solutions of the smooth systems. The periodic solutions are zeros of the displacement function, which is the Poincaré map associated to periodic solutions minus the identity. In fact, the non-linear function (25) whose zeros can provide periodic solutions, is the first term of order ε of the displacement function. See for more details the proof of Theorem 6 in [3].
Since the Poincaré maps associated to periodic solutions of system (11), coming from the perturbed periodic solutions (15) or (16), are smooth and these Poincaré maps are the limit of the Poincaré maps associated to the smooth system (12), for which we can use Theorem 6, it follows that we also can use Theorem 6 for computing some of the periodic solutions of the non-smooth system (11). In other words, we can apply Theorem 6 to the smooth systems (12) and then pass to the limit, when δ → 0, the function (25) for obtaining a function whose zeros can give periodic solutions of the non-smooth system (11).
We shall apply Theorem 6 of the Appendix B to differential system (14). We note that system (14) can be written as system (22) taking We shall study which periodic solutions (15) of the unperturbed system (14) with ε = 0 can be continued to periodic solutions of the perturbed system (14) for ε = 0 sufficiently small.
Computing the fundamental matrix M zα (τ ) of the linear differential system (14) with ε = 0 associated to the T -periodic solution z α = (X 0 , Y 0 , 0, 0) such that M zα (0) be the identity of R 4 , we get that M (τ ) = M zα (τ ) is equal to Note that the matrix M zα (τ ) does not depend on the particular periodic solution x(τ, z α ). Since the matrix satisfies the assumptions of statement (ii) of Theorem 6 because the determinant So we can apply Theorem 6 to system (14).
Proof of Theorem 3. This proof is completely analogous to the proof of Theorem 1.

Proofs of corollaries
To obtain the expression of the functions given in (4) and (5) we have to study the changes of sign of the functions X X0,Y0 (τ ) and Z Z0,W0 (τ ) respectively for t ∈ [0, pT 1 ] and t ∈ [0, pT 2 ].
The study of changes of the sign of the function Z Z0,W0 (τ ) for t ∈ [0, pT 2 ] and Z 0 W 0 = 0 is completely analogous.
Proof of Corollary 2. Firstly, we have to check the Crossing Hypothesis (see Remark 2) to the system (3).
Note that we have four different vector fields defined in four different regions (see Figure 4).
In the region R 1 = {x > 0 and z > 0} we have In the region R 2 = {x < 0 and z > 0} we have In the region R 3 = {x < 0 and z < 0} we have Finally, in the region R 4 = {x > 0 and z < 0} we have To study the types of the sets M ij (see Appendix A), we have to compute Lie derivative of the functions g 1 and g 2 with respect to the vector fields X i for i = 1, 2, 3, 4, i.e.
(L Xi )(g j )(x, y, z, w) = ∇g j , X i (x, y, z, w). Proceeding with these calculations we have Hence we can conclude that in the set the flow is tangent to the discontinuous set, and in any other point the flow cross the set of discontinuity.
Using the coordinates defined in (13), we have that Observe that the periodic orbits given by Lemma 5 filling the planes {(X, Y, 0, 0)} and {(0, 0, X, Y )}, except the origin, do not reach the set T . Thus, for |ε| > 0 sufficiently small, there exists a neighborhood of the planes {(X, Y, 0, 0)} and {(0, 0, X, Y )} such that the orbits cross the set of discontinuity. In other words, the Crossing Hypothesis is satisfied. Studying the changes of the sign of the function X X0,Y0 (τ ) for t ∈ [0, T 1 ] we conclude that the non-smooth functions (4) and (5) are given by This system has all solutions inside a periodic orbit of the unperturbed systems passing through It is easy to check that this solution are simple. So, by Theorem 1 we have one periodic solution of the non-smooth perturbed double pendulum. This completes the proof of the corollary.
Proof of Corollary 4. This proof is completely analogous to the proof of Corollary 2.

Appendix A: Basic concepts on Filippov systems
We say that a vector field X : D ⊂ R n → R n is Piecewise Continuous if the domain D can be partitioned in a finite collection of connected, open and disjoint sets D i , i = 1, · · · , k, such that, the vector field X D i is continuous for i = 1, · · · , k.
We denote by S X ⊂ ∂D 1 ∪· · ·∪∂D k the set of points where the vector field X is discontinuous. By assumptions, the set S X has measure zero.
If M ⊂ S X is a manifold of codimension one, then M can be decomposed as the union of the closure of the regions (see Figure 5   Consider the following equation where X : D ⊂ R n → R n is a piecewise continuous vector field. The local solution of the equation (21) passing through a point p ∈ M is given by the Filippov convention: (i) for p ∈ Σ c such that (Xh)(p), (Y h)(p) > 0 and taking the origin of time at p, the trajectory is defined as ϕ Z (t, p) = ϕ Y (t, p) for t ∈ I p ∩ {t < 0} and ϕ Z (t, p) = ϕ X (t, p) for t ∈ I p ∩ {t > 0}. For the case (Xh)(p), (Y h)(p) < 0 the definition is the same reversing time; (i) for p ∈ Σ e ∪ Σ s such that Z s (p) = 0, ϕ Z (t, p) = ϕ Zs (t, p) for t ∈ I p ⊂ R.
Here ϕ W denotes the flow of a vector field W . For more details about discontinuous differential equation see Filippov's book [4].