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:
for small. We assume that the system
has a continuous periodic solution crossing transversally the impact manifold S, given by
and , . By transversal crossing, we mean that
We set and assume that are -periodic in t.
Transversal crossing implies that (1.2) has a family of continuous solutions , α∈ (an open neighborhood of 0∈) crossing transversally the impact manifold S, given by
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
Condition (1.4) has a simple geometrical meaning. The impact system (1.3) has a -periodic solution if and only if the following condition holds:
Now, suppose there is a sequence , as such that (1.5) holds. Possibly passing to a subsequence we can suppose that , . Then, taking the limit in the equalities:
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 .
In Section 4, we define the adjoint system to the linearization of the impact system
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
We set , and
and that is a continuous periodic solution, of period , of the piecewise continuous singular system:
Obviously, extends to a solution of the following impact system:
that can be written as
Our purpose is to find a -periodic solution of system (1.1), which is orbitally close to for some , as that is such that
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).
To this end, we first set in equation . Then satisfies
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
Thus, we conclude that for and near equation has a solution such that if and only if the map given by
has a fixed point whose sup-norm in is 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:
. Note that
and hence in the fixed-point equation (2.4), we may also take . Then since , is a -map and
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
is in all parameters and t.
Writing in place of t in (2.4) and using (2.5) we see that
. We have, by definition, and
if and only if (recall that )
We remark that equation (2.7) has meaning also when but its relevance for our problem is only when .
As second step we consider the solution of the differential equation on :
which is close to on , . Let be the fundamental solution of the linear system such that . Setting we see that (for ) satisfies the equation:
Again by the variation of constants formula we get the integral formula:
which, as before, has a unique solution of norm less than a given, small, , with . At the solution of (2.8) takes the value:
Now, we want to solve the equation
that is [again using and ]:
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:
that is the equation where:
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.
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:
Theorem 3.1Assume condition (1.4) holds and let be the unique (up to a multiplicative constant) solution of (3.1). If the Poincaré-Melnikov function
has a simple zero at , then system (1.1) has a -periodic solution satisfying (2.1).
Proof To start with, we make few remarks on the functions . First we note that when equation (2.8) reads
which has the (unique) solution . Thus,
Next, differentiating equation (2.8) with respect to ε we see that satisfies the equation:
Next, by the definition and differentiating equation (2.6) with respect to ε at and using the equalities:
So, equation (2.9) at and becomes
which is satisfied for . Now we look at equation (2.10). Since , we see that when and the equality is satisfied. As a consequence, we get
and . Next we look at derivatives of ℱ with respect to , , α and ε at the point . We have
and similarly, using
So, the Jacobian matrix L of ℱ at the point is
and belongs to the kernel of L if and only if
From , we get
thus, on account of the transversality condition , (3.4) is equivalent to
Next, from , we get
then subtracting (3.5) from (3.7) and using we obtain:
So, if satisfies (3.6), we see that
and then, on account of transversality, . Summarizing, we have seen that, if then , and satisfies
On the other hand, if satisfies (3.8), then belongs to . So if and only if system (3.8) has the trivial solution only. But (3.8) is equivalent to
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.,
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 .
Theorem 3.2Let, X, Ybe Banach spaces and a -map such that has a , d-dimensional, manifold of solutions . Assume that for anyμin a neighborhood of the linearization has the null space . Assume further that is Fredholm with index zero and let a projection ofYonto the range of . Then if the Poincaré-Melnikov function
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.
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:
whereas differentiating (2.10) with respect to ε at we get
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.
Let be the -map defined in Introduction and recall the impact equation (1.6):
For , (4.1) has the solution , . We let denote the solution of the impact system (4.1) on . Then its derivative with respect to α at satisfies the linearized equation:
Next, recalling (1.1), we consider a perturbed impact system of (4.1) (see also (2.8)) of the form
where is defined as follows: and is the solution of
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 , .
To study the problem of existence of solutions of system (4.3), we are then led to find conditions on , d and so that the non-homogeneous linear equation:
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).
Since , we get
for any and then
So, if equation (4.4) has a solution, we must necessarily have
Next, we define two Hilbert spaces:
Note Y is a Hilbert space and X is a closed subspace of a Hilbert space . Then (4.4) can be written as
Lemma 4.1The range is closed.
Proof Indeed, let . Then
and is closed, then and there exists so that
we derive . The proof is finished. □
Next, we prove the following result.
Proposition 4.1Let . Then the inhomogeneous system (4.4) has a solution if and only if equation
holds for any solution of the adjoint system
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)).
First, we prove necessity. Assume that (4.4) can be solved for and let , , be a solution of equation (4.6). Then
Plugging these equalities in the left-hand side of (4.5) and integrating by parts, (4.5) reads
because of the definition of and the fact that satisfies (4.6).
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. □
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 4.2The 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 ).
Proof Indeed let be a solution of (4.6) then
being the fundamental matrix of the linear equation . Then, taking the two remaining condition in (4.6) read:
that can be written as
or else, on account of :
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
is we see that
As for the first term in the above equality, we can show it is related to the impact . Indeed, from Section 2 we know that the solution of the singular equation
can be written as
with as in equation (2.6). Thus, and
for some . Then