Fast-slow dynamical approximation of forced impact systems near periodic solutions
© Battelli and Fečkan; licensee Springer. 2013
Received: 14 October 2012
Accepted: 21 March 2013
Published: 3 April 2013
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.
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.
We set and assume that are -periodic in t.
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.
we see that condition (1.4) does not hold. Thus, (1.4) implies that, in a neighborhood of , there are no other -periodic solutions of (1.3) apart from .
along the solution and relate the Poincaré-Melnikov function obtained in Section 3 with the solutions of such an adjoint 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:
either or for any .
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
for some as . Thus, we may say that, in some sense, the impact periodic solution approximates the periodic solution of the singular perturbed equation (1.1).
is in all parameters and t.
We remark that equation (2.7) has meaning also when but its relevance for our problem is only when .
According to the smoothness properties of and , it results that is .
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.
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:
has a simple zero at , then system (1.1) has a -periodic solution satisfying (2.1).
Writing , , we see that ψ, , satisfy (3.1). This proves the claim before the statement of Theorem 3.1.
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 .
has a simple zero at , there exists and a unique map such that . Moreover, is an isomorphism for .
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.
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.
Note that R is a -map on taking values on and ; moreover, when is autonomous then R is independent of τ, so we may take in its definition. We recall that for simplicity we write instead of , .
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).
Lemma 4.1 The range is closed.
we derive . The proof is finished. □
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)).
because of the definition of and the fact that satisfies (4.6).
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. □
Again we note that equation (4.6) only depends on the derivative on since , where we use or, in other words, it is independent of any -extension we take of to the whole .
We now prove the following proposition.
Proposition 4.2 The adjoint system (4.6) has a solution if and only if satisfy the first and the third equation in (3.1) (and we take the second equation in (3.1) as definition of ).
The proof is finished. □
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
which, in turn, is equivalent to because of transversality and the fact that .
Hence, we conclude that .
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 5.1 Assume that for any α in a neighborhood of , and that either or for any α (in the same neighborhood). Then system (5.3) has a d-dimensional space of solutions where or according to which of the two conditions or holds. Moreover, if the Poincaré-Melnikov function (5.5) (or (5.6)) has a simple zero at then system (1.1) has a -periodic solution satisfying (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.
and . Note that, to have for we need .
which has a simple zero at .
then . So, we conclude with the following.
has a -periodic solution orbitally near the set .
It should be noted that the discontinuity line is the union of the two half lines and which is not . However, all results hold true as long as we remain outside a (small) neighborhood of .