We approximate impact systems in arbitrary finite dimensions with fast-slow dynamics represented by regular ODE on one side of the impact manifold and singular ODE on the other. Lyapunov-Schmidt method leading to Poincaré-Melnikov function is applied to study bifurcations of periodic solutions. Several examples are presented as illustrations of abstract theory.
MSC: 34C23, 34C25, 37G15, 70K50.
Keywords:fast-slow dynamics; impact systems; bifurcations; periodic solutions; Poincaré-Melnikov method
Non-smooth differential equations when the vector field is only piecewise smooth, occur in various situations: in mechanical systems with dry frictions or with impacts, in control theory, electronics, economics, medicine and biology (see [1-6] for more references). One way of studying non-smooth systems is a regularization process consisting on approximation of the discontinuous vector field by a one-parametric family of smooth vector fields, which is called a regularization of the discontinuous one. The main problem then is to preserve certain dynamical properties of the original one to the regularized system. According to our knowledge, the regularization method has been mostly used to differential equations with non-smooth nonlinearities, like dry friction nonlinearity (see  and a survey paper ). As it is shown in [7,8], the regularization process is closely connected to a geometric singular perturbation theory [9,10]. On the other hand, it is argued in  that a harmonic oscillator with a jumping non-linearity with the force field nearly infinite in one side is a better model for describing the bouncing ball, rather then its limit version for an impact oscillator. This approach is used also in  when an impact oscillator is approximated by a one-parametric family of singularly perturbed differential equations, but as discussed in , the geometric singular perturbation theory does not apply.
In this paper, we continue in a spirit of  as follows. Let be an open subset and a -function, such that for any . Then S is a smooth hyper-surface of Ω that we call impact manifold, (or hyper-surface). We set and consider the following regular-singular perturbed system:
where , , and and . Moreover, is in α, and the maps and give smooth () parameterizations of the manifold S in small neighborhoods of and of . Then the map , is -smooth. In this paper, we study the problem of existence of a -periodic solution of the singular problem (1.1) in a neighborhood of the set
As a matter of fact, in the time interval , resp. , the periodic solutions will stay close to , resp. to , and hence it will pass from the point of S near to the point of S near in a very short time (of the size of ). So, we may say that the behavior of the periodic solutions of (1.1) in the interval is quite well simulated by the solution of the perturbed impact system
It is now clear that our study has been mostly motivated by the paper , where a similar problem on planar perturbed harmonic oscillators is studied. However arguments in  are mainly based on averaging methods whereas, in this paper, we investigate a general higher-dimensional singular equation such as (1.1) by using the Lyapunov-Schmidt reduction. We focus on the existence of periodic solutions and do not check their local asymptotic properties as, for example, stability or hyperbolicity. This could be also done by following our approach but we do not go into detail in this paper.
Our results (see Theorems 3.1 and 5.1) state that if a certain Poincaré-Melnikov-like function has a simple zero then the above problem has an affirmative answer. The proof of this fact is accomplished in several steps. In Section 2, we show, for any α in a neighborhood of , the existence of a unique continuous solution of (1.1) near the set which is defined in , and such that , for some τ, and , belong to . Moreover, and are close to and then and give parameterizations of S in neighborhoods of . Hence, gives a Poincaré-like map and a -periodic solution is found by solving the equations
Thus, the bifurcation equation is obtained by putting conditions , and the fact that the points and belong to S together. Then, in Section 3, we use the Lyapunov-Schmidt method to prove that the above equations can be solved for as functions of small provided a certain Poincaré-Melnikov-like function has a simple zero. We will first study the case, that we call non-degenerate, when
In Section 4, we define the adjoint system to the linearization of the impact system
Section 5 is devoted to the extension of the result to the case (that we call degenerate) where for any . We will see that our results can be easily extended provided one of the following two conditions hold:
Section 6 is devoted to the construction of some planar examples, although our results are given for an arbitrary finite dimension. Finally, the Appendix contains some technical proofs.
2 The bifurcation equation
that can be written as
Since describes , we consider (2.2) with the initial condition . Let be the fundamental solution of , such that . Then is the fundamental solution of , with . Let be near . By the variation of constants formula, the solution of (2.2) with the initial condition satisfies
has a fixed point whose sup-norm inis smaller thanρ. To show that (2.3) has a fixed point of norm less than ρ, we set , and note that is a fixed point of (2.3) of norm less than ρ, with , if and only if is a fixed point of norm less than ρ of the map:
the map is a -contraction on the Banach space of bounded continuous functions on whose sup-norm is less than or equal to ρ provided ρ is sufficiently small, is near , is small, and . Let be the -solution of the fixed point (2.4). We emphasize the fact that ε may also be non-positive. Then is a fixed point of (2.3) and
Again by the variation of constants formula we get the integral formula:
Now, we want to solve the equation
Of course, when (2.9) holds, then (2.7) is equivalent to
So, our task reduces to solve the system formed by equations (2.9), (2.10) together with the period equation:
In this section, we will give a criterion to solve equation for in terms of ε for small . We will use a Crandall-Rabinowitz type result (see also [, Theorem 4.1]) concerning the existence of a solution of a nonlinear equation having a manifold of fixed point at a certain value of a parameter.
Our result is as follows. Consider the linear system
We will prove that if (1.4) holds, system (3.1) has a unique solution, up to a multiplicative constant, and the following result holds:
and similarly, using
and hence (3.8) has the trivial solution if and only if the non-degenerateness condition (1.4) holds. We emphasize the fact that, assuming condition (1.4), equation has the manifold of fixed points and the linearization of ℱ at these points is Fredholm with index zero with the one-dimensional kernel . Hence, there is a unique vector, up to a multiplicative constant, such that , i.e.,
We recall that our purpose is to solve the equation for and that has the one-dimensional manifold of solutions and its linearization along the points of this manifold is Fredholm with the one-dimensional kernel . Hence, we are in position of applying the following result that has been more or less proved in .
Theorem 3.2Let, X, Ybe Banach spaces anda-map such thathas a, d-dimensional, manifold of solutions. Assume that for anyμin a neighborhood ofthe linearizationhas the null space. Assume further thatis Fredholm with index zero and leta projection ofYonto the range of. Then if the Poincaré-Melnikov function
Actually the statement in [, Theorem 4.1] is slightly different from the above. Hence, we give a proof of Theorem 3.2 in the Appendix.
We apply Theorem 3.2 to the map with . Then is independent of τ, and hence so is . Next where and , , , is any vector satisfying (3.1). To apply Theorem 3.2, we look at the derivative of with respect to ε at . First, we have:
We obtain then
and then the Poincaré-Melnikov function is:
The conclusion of Theorem 3.1 now easily follows from (3.9) and Theorem 3.2. □
4 Poincaré-Melnikov function and adjoint system
In this section, we want to give a suitable definition of the adjoint system of the linearization of (1.6) along in such a way that the Poincaré-Melnikov function (3.2) can be put in relation with the solutions of such an adjoint system.
Next, recalling (1.1), we consider a perturbed impact system of (4.1) (see also (2.8)) of the form
has a solution . Let us comment on equation (4.4) (and similarly on (4.2)) that condition only involves the derivative of on the tangent space since , . So, it is independent of any extension we take of to a neighborhood of . We also note that for simplicity we denote again by the value of the linear functional in (4.4).
So, if equation (4.4) has a solution, we must necessarily have
Next, we define two Hilbert spaces:
Next, we prove the following result.
Proof Before starting with the proof we observe that, because of , ψ is not uniquely determined by equation (4.5) since changing it with , , the equation remains the same. So, in equation (4.5), we look for ψ in a subspace of which is transverse to . It turns out that the best choice, from a computational point of view, is to take ψ so that (see equation (3.1)).
Plugging these equalities in the left-hand side of (4.5) and integrating by parts, (4.5) reads
To prove the sufficiency, we show that if does not belong to , then there exists a solution of the variational equation (4.6) such that (4.7) does not hold. So, assume that . By Lemma 4.1 and the Hahn-Banach theorem, there is an such that
where is the usual scalar product on Y. We already noted that we can assume that , and (4.8)-(4.9) remain valid. Repeating our previous arguments, we see that and that (4.8) implies solves the adjoint system (4.6). Summarizing, if there exists a solution of the adjoint system for which (4.6) does not hold. This finishes the proof. □
We now prove the following proposition.
that can be written as
The proof is finished. □
We conclude this section giving another expression of the Poincaré-Melnikov function (3.2) in terms of the solution of the adjoint system (4.6). To this end, let be a solution of the adjoint system (4.6). Since a fundamental matrix of the linear equation
can be written as
When is autonomous, then R is independent of τ, and the expression (4.10) of the Poincaré-Melnikov function should be compared with the one given in [, Theorem 4.2] where a Poincaré-Melnikov function, characterizing transition to chaos, is given for almost periodic perturbations of autonomous impact equations with a homoclinic orbit.
5 The case of a manifold of periodic solutions
From (3.3), we get:
Arguing as in Section 3, equation (5.2) is equivalent to
Of course the only difference between the cases and for all is that in the first case the Poincaré-Melnikov function is defined for while in the second it is defined for for an open neighborhood of . Summarizing, we proved the following result.
Theorem 5.1Assume thatfor anyαin a neighborhood of, and that eitherorfor anyα (in the same neighborhood). Then system (5.3) has ad-dimensional space of solutions whereoraccording to which of the two conditionsorholds. Moreover, if the Poincaré-Melnikov function (5.5) (or (5.6)) has a simple zero atthen system (1.1) has a-periodic solutionsatisfying (2.1).
Finally, we note that when we can show that a Brouwer degree of a Poincaré-Melnikov function from either Theorem 3.1 or 5.1 is non-zero then by following  we can show existence results.
We consider a second-order equation
i.e., we take
and the Poincaré-Melnikov function (5.6) reads:
whereas (4.10) reads:
We assume we are in the first (non degenerate) case that is it holds (1.4), which now has the form
(6.2) is equivalent to
and the Poincaré-Melnikov function reads
Suppose, also, that the function
To get a second example, we change the above as follows: we take
More related results are derived in the Appendix.
Then equations (6.2) have to be changed to
from which we get
Note has a homoclinic solution , so the solution is a part of a periodic solution inside of bounded by the homoclinic one (see Figure 1). Then, since in a neighborhood of we have we get
Putting all together we see that the adjoint variational system reads:
The first three equations give the boundary value problem
We conclude with the following.
Figure 1. The upper parts of homoclinic and periodic orbits of.
Then we obtain the following corollary.
A.1 Further properties of the solution of (6.5)
From the identity
Using formula 3.131.5 in [, p.254] we know that, for any :
and F is the elliptic integral of the first kind.
Furthermore, the above computations also give
Solving (7.3), we obtain
where am is the Jacobi amplitude function. Solving (7.4), we obtain
we see that the following recurrence condition holds:
For the first few indexes, we get
On the other hand, using Mathematica, we can expand (7.7) to get
which coincides with our above analytical expansion.
A.2 Proof of Theorem 3.2
Here, we prove Theorem 3.2. We emphasize the fact that proof mainly follows the idea in [, Theorem 4.1].
Let as in the statement of the theorem. We write , with . Applying the Implicit Function Theorem to the map , we get the existence of a unique -solution of the equation . From uniqueness, we obtain also
Next, for , equation is equivalent to , but the l.h.s. tends, for to which gives the Poincaré-Melnikov condition. We conclude that, if the Poincaré-Melnikov condition is satisfied, for (small) there exists a unique solution of equation , , with .
Now we prove the invertibility of . Since is Fredholm with index zero, it is enough to prove that equation has, for , the unique solution . Although is only with respect to ε, it is linear in z. Thus, we can still apply the existence and uniqueness argument given above. Of course, vanishes on the linear subspace , and clearly . Next so that . Thus, from the existence and uniqueness result it follows that if the following condition is satisfied:
The authors declare that they have no competing interests.
The work presented here was carried out in collaboration between the authors. The authors contributed to every part of this study equally and read and approved the final version of the manuscript.
This article is dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.
BF is partially supported by GNAMPA-CNR and MURST-group ‘Equazioni differenziali ordinarie e applicazioni’ (Italy). MF is partially supported by the grant APVV-0134-10 and Marche Polytechnic University, Ancona (Italy).