On the periodic solutions of perturbed 4D non-resonant systems

We provide sufficient conditions for the existence of periodic solutions of a 4D non-resonant system perturbed by smooth or non-smooth functions. We apply these results to study the small amplitude periodic solutions of the non-linear planar double pendulum perturbed by smooth or non-smooth function.

Observe that system (1) with ε = 0 is simply the unperturbed system. Otherwise we have the perturbed system. Our goal is to provide an algorithm, via tools in advanced averaging theory, for detecting the periodic orbits of the perturbed system which emerge from the set of periodic orbits of the unperturbed system. It is to be noted that a very similar scenario holds for a model of the double pendulum in which our main results will be applied.

Background and historical facts
The first serious proofs in the averaging theory for differential systems can be traced back to the works of Fatou [6] in 1928, Krylov and Bogoliubov [3] in the 1930s and Bogoliubov [2] in 1945. For a modern point of view on this theory see the book of Sanders and Verhulst [13].
The method of averaging plays an important role in the study of nonlinear systems related to complex behavior patterns such as bifurcation and stability of the periodic solutions of such systems. Here we need to deal with systems in the normal form of the averaging theory given bẏ see the basic results on this theory in Sect. 3. One of the basic problems for applying the averaging theory is to write the system under study in the normal form (23). Another tool used in this paper is the regularization process for discontinuous differential systems introduced by Sotomayor and Teixeira in [14]. In this process a discontinuous vector field Z (t, x) is approximated by an one-parameter family of continuous vector fields Z δ (t, x).
As far as we know, the two methods cited above has been firstly used together in [10] by Llibre and Teixeixa where it was studied the stable limit cycle of a weightdriven pendulum clock. In this paper we use the ideas presented in [10] to study the periodic solutions of (1).

Setting the problem
The objective of this paper is to provide a system of equations whose simple zeros provide periodic solutions of (1). In order to present our results we need some preliminary definitions and notations.
The unperturbed system (1) has the origin as its unique singular point with eigenvalues ±ω 1 i, ±ω 2 i. If the non-resonant condition ω 1 /ω 2 ∈ R\Q is satisfied then this system has, in the phase space (x, y, z, w), 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 system (1) when the parameter ε is sufficiently small and the functions of perturbation are either T 1 or T 2 periodic. Let ϕ 1 (t, u, v) be the periodic function Define F 1 (x, y): and F 2 (x, y): sign(h 1 (t, ϕ 1 (t, x, y)))dt.
Now let ϕ 2 (t, u, v) be the periodic function Define F 1 (z, w): and F 2 (z, w): A zero (x * , y * ) of the system of functions such that is called a simple zero of (9). Similarly, we define a simple zero of the system of functions Remark 1 For p ∈ D, let ϕ(t, p) be the solution of (1) such that ϕ(0, p) = p. We say that the Crossing Hypothesis is satisfied if there exists a compact set V ⊂ D such that the curve t → (t, ϕ(t, p)) reaches the set at points of crossing regions (see Appendix) for every p ∈ V and t ∈ [0, T ].

Statement of results
Our main result on the periodic solutions of the perturbed system (1) which bifurcate from the periodic solutions of the unperturbed system with period T 1 is the following.
For ε > 0 sufficiently small the solution ϕ(t, ε) of Theorem A is close to the plane defined by the eigenvectors of the eigenvalues ±iω 1 .
With a change of variables we can also state a result on the periodic solutions of perturbed system (1) which bifurcate from the periodic solutions of the unperturbed system with period T 2 .
Again, for ε > 0 sufficiently small the solution ϕ(t, ε) of Corollary 1 is close to the plane defined by the eigenvectors of the eigenvalues ±iω 2 .
Theorem A and Corollary 1 are proved in Section 4. Its proof is based in the averaging theory for computing periodic solutions, see Sect. 3.

Remark 2
The theory could be developed for a more general system by considering the smooth functions P for i = 1, 2, 3, 4, and j = 1, 2. Here to make the notation simpler we assume that all functions have the same period.

Application: double pendulum model
The planar double pendulum model consists in 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 g 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 .
The position of the double pendulum is determined by the two angles φ 1 and φ 2 shown in Fig. 1. The corresponding Lagrange equations of motion are (11) where g is the acceleration of the gravity. For more details on these equations of motion see [8]. Here the dot denotes derivative with respect to the time τ .
The authors in [9] have studied, in the vicinity of the equilibrium φ 1 = φ 2 = 0, the persistence of periodic solutions of system (11) perturbed smoothly in the particular case when m 1 = m 2 and l 1 = l 2 . In this paper stronger generalizations are considered.
Roughly speaking, the functions F i and G i for i = 1, 2, 3, 4, can be taken in a certain way arbitrary. It makes us able to provide, in a physical context, the real meaning of these functions. In our case, since we are working with discontinuities in the variables φ 1 and φ 2 , the functions F 1 , F 2 , G 1 and G 2 could model the escapement Periodic solution of the perturbed system (12) converging to the origin when ε → 0 for the particle m 1 , and the functions F 3 , F 4 , G 3 and G 4 could model the escapement for the particle m 2 . If discontinuities in the variablesφ 1 andφ 2 are considered instead discontinuities in the variables φ 1 and φ 2 , the respective functions could model the Coulomb Friction. We can also work by composing these two phenomena. For more details on physical systems with discontinuous models see, for instance, [1,5].
The objective of this section is to provide a system of equations whose simple zeros provide periodic solutions (see Fig. 2) of the perturbed planar double pendulum (12). In order to present our results we need some preliminary definitions and notations.
In what follows we define the real parameters the frequencies with = (a − b) 2 + 4b > 0, and the periods We shall study the persistence of periodic solutions for the perturbed system (12) when the parameter ε is sufficiently small and the functions F i and G i for i = 1, 2, 3, 4 have period either γ T 1 , or γ T 2 . Now letF with and letG We define the functions: for i = 1, 2, 3, 4 with and π 1 is the projection onto the first coordinate. We also define the functions: where , and π 3 is the projection onto the third coordinate. Consider the systems and In the next proposition we state a result on the existence of periodic solutions of the non-smooth perturbed double pendulum (12). (12) are γ T 1 -periodic in the variable t. Also assume that the crossing hypothesis is satisfied for ε ∈ (0, ε 0 ) with ε 0 > 0 and (x, y, z, w) ∈ V . Then for ε > 0 sufficiently small and for every simple zero (x * , y * , 0, 0) ∈ V of the non-smooth system (21) such that the orbits pass by D, the non-smooth perturbed double pendulum (12) has a γ T 1 -periodic solution ϕ(t, ε) such that ϕ(0, ε) → (0, 0, 0, 0) when ε → 0.

Proposition 2 Assume that F i and G i of
In a similar way we can also state a result on the periodic solutions of the nonsmooth perturbed double pendulum (12) which bifurcate from the periodic solutions of the unperturbed one with period T 2 .
We provide now an application of Propositions 2 and 3.

Corollary 4 Suppose that F
with f 1 , f 3 , g 2 , g 4 , F 2 , F 4 , G 1 , and G 3 being γ T i -periodic functions in the variable τ having no linear terms and no constant terms in relation with the spatial variables. Then the differential system (12) for |ε| > 0 sufficiently small has two γ T i -periodic solution bifurcating from the origin. Here i = 1, 2.

Basic results on averaging theory
For proving the main results of this paper we present the basic result from the averaging theory that we need here. Consider the differential systems of the forṁ with |ε| > 0 sufficiently small, where the functions G 0 , G 1 : R × → R n and G 2 : R × × (−ε 0 , ε 0 ) → R n are C 2 functions, T -periodic in the variable t, and is an open subset of R n . Assume that the T -periodic solutions of the unperturbed systemẋ form a submanifold Z of dimension k in R n . In the study of the periodic solutions the objective of the averaging theory is to detect which periodic solutions of the unperturbed system (24) persist as periodic solutions in the perturbed differential system (23) for |ε| > 0 sufficiently small. We denote by x(t, z, ε) the solution of system (24) such that x(0, z, ε) = z. The first variational equation of system (24) on the periodic solution x(t, z, 0) iṡ , x(t, z, 0))y, where y is a n × n matrix. From now on let M z (t) be the fundamental matrix of system The T -periodic solutions of the perturbed system (24) coming from the unperturbed system (23) for |ε| > 0 sufficiently small can be computed using the following result.

Theorem 5 We denote by V an open and bounded subset of R k containing the submanifold Z, and we denote by
If there is a ∈ V such that G(a) = 0 and det ((dG/dα) (a)) = 0, therefore there exists a T -periodic solution ϕ(t, ε) of system (23) satisfying ϕ(0, ε) → z a as ε → 0.

Proofs of Theorem A and Corollary 1
The averaging theory we shall use here (see Appendix) deals with smooth system. So, first of all, instead of working with the discontinuous differential system (1) we shall work with the smooth differential system 1δ (t, x, y, z, w, ε), x, y, z, w, ε), where where s δ (x) is the smooth function defined in Fig. 3, such that lim δ→0 s δ (x) = sgn(x). (3) and (6) are periodic solutions of the unperturbed system (27) respectively with periods T 1 and T 2 .

Lemma 6 The functions
Proof Since system (27) with ε = 0 is a linear differential system, the proof follows easily.
Proof of Theorem A 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 (27) are smooth. The Poincaré maps, restricted at V , associated to the periodic solutions of the nonsmooth differential system (1), which are perturbations of the periodic solutions (3) are also smooth, indeed, since the orbits starting in V reach the discontinuity set only at points of crossing region (see Appendix), such Poincaré maps are compositions of smooth Poincaré maps. In a similar way it follows that the Poincaré maps, restricted at V , associated to the periodic solutions of the non-smooth differential system (1), which are perturbations of the periodic solutions (6) are also smooth.
We can use Theorem 5 (see Sect. 3) 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 (26) 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 5 in [4].
Since the Poincaré maps associated to periodic solutions of system (1), coming from the perturbed periodic solutions (3) or (6), are smooth and these Poincaré maps are the limit of the Poincaré maps associated to the smooth system (27), for which we can use Theorem 5, it follows that we also can use Theorem 5 for computing some of the periodic solutions of the non-smooth system (1). In other words, we can apply Theorem 5 to the smooth systems (27) and then pass to the limit, when δ → 0, the function (26) for obtaining a function whose zeros can give periodic solutions of the non-smooth system (1). We note that system (27) can be written as system (23) taking We shall describe the different elements which appear in the statement of Theorem 5 in the particular case of the differential system (27). Thus we have that = R 4 , k = 2 and n = 4. Let r 1 > 0 be arbitrarily small and let r 2 > 0 be arbitrarily large.
As usual Cl(V ) denotes the closure of V . If α = (x, y), then we can identify V with the set here || · || denotes the Euclidean norm of R 2 . The function β : (0, 0). Therefore, in our case the set Clearly for each z α ∈ Z we can consider the periodic solution x(t, z α ) = ϕ 1 (t, x, y) given by (3) with period T 1 .
Note that system (30) is equivalent to system (9), because both equations only differs in a non-zero multiplicative constant. Hence Theorem A is proved.
Proof of Corollary 1 This proof follows immediately from Theorem A.

Proof of Proposition 3
Computing the functions (7) and (8) to the differential system (34) we obtain the functions given in (20). Consequently, the system of functions (10) is equivalent to the system of functions (22). Then, by Corollary 1 we have that for every simple zero (z * , w * ) of the system of functions (22) there exists a periodic solution (x, y, z, w)(t, ε) of (34) such that From here, the proof follows analogously to the proof of Proposition 2.
The study of changes of the sign of the function π 3 • ϕ 2 (t, z, w) for t ∈ [0, T 2 ] and zw = 0 is completely analogous.
Proof of Corollary 4 Firstly, we have to check the crossing hypothesis for the system (12) or equivalently for the system (32). Note that we have four different vector fields defined in four different regions (see Fig. 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 i j (see Appendix), we have to compute Lie derivative of the functions π 1 and π 3 with respect to the vector fields X i for i = 1, 2, 3, 4, i.e.
where π j is the projection onto the jth coordinate.
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.
In the coordinates defined in (33), we have that Observe that the periodic orbits given by Lemma 6 filling the planes {(x, y, 0, 0)} and {(0, 0, z, w)}, except the origin, do not reach the set T . Thus, for |ε| > 0 sufficiently small, there exists a neighborhood of {(x, y, 0, 0)}\(0, 0, 0, 0) and {(0, 0, z, w)}\(0, 0, 0, 0) such that the orbits cross the set of discontinuity. In other words, the crossing hypothesis is satisfied for every ε ∈ (−ε 0 , ε 0 ) for some ε 0 > 0. Now assume that the function are γ T 1 -periodic in the variable τ . By studying the changes of the sign of the function π 1 • ϕ 1 (t, x, y) for t ∈ [0, T 1 ] we conclude that the functions (18) and (20) are given by if y < 0, So the system F 1 (x, y) = 0 and F 2 (x, y) = 0 has the following simple solutions: and (x * 2 , y * 2 ) = 0 , −4 Hence, by Theorem A we have two γ T 1 -periodic solution of the non-smooth perturbed double pendulum. The argument in the case when the functions are γ T 2 -periodic is completely analogous. So we have conclude the proof of corollary.

Appendix: Basic concepts on Filippov systems
We say that a vector field X : D ⊂ R n → R n is piecewise continuous if its domain of definition D can be partitioned in a finite collection of connected, open and disjoint sets D i , i = 1, · · · , k, such that ∪D i = D, and 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 ⊂ S X is a manifold of codimension one, then can be decomposed as the union of the closure of the following three kind of regions (see  For p ∈ e ∪ s we define the Sliding Vector Field as Consider the following equationẋ where X : D ⊂ R n → R n is a piecewise continuous vector field. The local solution of the equation (36) passing through a point p ∈ is given by the Filippov convention: (i) for p ∈ 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; (ii) for p ∈ e ∪ s such that Z s ( p) = 0, ϕ Z (t, p) = ϕ Z s (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 [7].