Abstract
We approximate impact systems in arbitrary finite dimensions with fastslow dynamics represented by regular ODE on one side of the impact manifold and singular ODE on the other. LyapunovSchmidt method leading to PoincaréMelnikov function is applied to study bifurcations of periodic solutions. Several examples are presented as illustrations of abstract theory.
MSC: 34C23, 34C25, 37G15, 70K50.
Keywords:
fastslow dynamics; impact systems; bifurcations; periodic solutions; PoincaréMelnikov method1 Introduction
Nonsmooth 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 [16] for more references). One way of studying nonsmooth systems is a regularization process consisting on approximation of the discontinuous vector field by a oneparametric 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 nonsmooth nonlinearities, like dry friction nonlinearity (see [7] and a survey paper [8]). 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 [11] that a harmonic oscillator with a jumping nonlinearity 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 [12] when an impact oscillator is approximated by a oneparametric family of singularly perturbed differential equations, but as discussed in [12], the geometric singular perturbation theory does not apply.
In this paper, we continue in a spirit of [12] as follows. Let be an open subset and a function, such that for any . Then S is a smooth hypersurface of Ω that we call impact manifold, (or hypersurface). We set and consider the following regularsingular 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 [12], where a similar problem on planar perturbed harmonic oscillators is studied. However arguments in [12] are mainly based on averaging methods whereas, in this paper, we investigate a general higherdimensional singular equation such as (1.1) by using the LyapunovSchmidt 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 PoincaréMelnikovlike 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 Poincarélike 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 LyapunovSchmidt method to prove that the above equations can be solved for as functions of small provided a certain PoincaréMelnikovlike function has a simple zero. We will first study the case, that we call nondegenerate, 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 PoincaréMelnikov 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:
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
Note that
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
where
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 equationhas a solutionsuch thatif and only if the mapgiven by
has a fixed point whose supnorm 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:
and hence in the fixedpoint equation (2.4), we may also take . Then since , is a map and
where
the map is a contraction on the Banach space of bounded continuous functions on whose supnorm 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 nonpositive. Then is a fixed point of (2.3) and
Writing in place of t in (2.4) and using (2.5) we see 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:
where
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
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:
According to the smoothness properties of and , it results that is .
3 Solving
In this section, we will give a criterion to solve equation for in terms of ε for small . We will use a CrandallRabinowitz type result (see also [[13], 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 letbe the unique (up to a multiplicative constant) solution of (3.1). If the PoincaréMelnikov function
has a simple zero at, then system (1.1) has aperiodic solutionsatisfying (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:
Hence,
Next, by the definition and differentiating equation (2.6) with respect to ε at and using the equalities:
we get
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
we get
Next
and
So, the Jacobian matrix L of ℱ at the point is
and belongs to the kernel of L if and only if
thus, on account of the transversality condition , (3.4) is equivalent to
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 nondegenerateness 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 onedimensional 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 onedimensional manifold of solutions and its linearization along the points of this manifold is Fredholm with the onedimensional kernel . Hence, we are in position of applying the following result that has been more or less proved in [13].
Theorem 3.2Let, X, Ybe Banach spaces andamap such thathas a, ddimensional, 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 PoincaréMelnikov function
has a simple zero at, there existsand a unique mapsuch that. Moreover, is an isomorphism for.
Actually the statement in [[13], 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 PoincaréMelnikov function is:
The conclusion of Theorem 3.1 now easily follows from (3.9) and Theorem 3.2. □
4 PoincaréMelnikov 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 PoincaréMelnikov 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 nonhomogeneous 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).
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
with
and
Since
and is closed, then and there exists so that
By taking
we derive . The proof is finished. □
Next, we prove the following result.
Proposition 4.1Let. Then the inhomogeneous system (4.4) has a solutionif and only if equation
holds for any solutionof 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 lefthand side of (4.5) and integrating by parts, (4.5) reads
or
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 HahnBanach theorem, there is an such that
and
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 ifsatisfy 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
The proof is finished. □
We conclude this section giving another expression of the PoincaréMelnikov 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
so:
Then
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
and then, using again we see that
i.e.
When is autonomous, then R is independent of τ, and the expression (4.10) of the PoincaréMelnikov function should be compared with the one given in [[14], Theorem 4.2] where a PoincaréMelnikov 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
In this section we assume that for any α in (an open neighborhood of in) . Hence, from (3.3), we see that
We distinguish the two cases: and for all α in (an open neighborhood of in) . First, we assume that
Then a , dimensional submanifold of (an open neighborhood of in) exists such that for any . So, for , has the dimensional manifold of solutions
So, we are in position to apply Theorem 3.2. First, we have to verify that the kernel equals the tangent space , , and then that the PoincaréMelnikov function (vector):
has a simple zero at . Note that
From (3.3), we get:
Note that does not depend on τ. Using and for any we easily see that
for any . On the other hand, assume that
for some and . Then and satisfies
that, on account of is equivalent to
and hence
which, in turn, is equivalent to because of transversality and the fact that .
Now we consider the second condition. The PoincaréMelnikov function (vector) , can be written as
where is a matrix whose rows are left eigenvectors of zero eigenvalue of the matrix , that is,
Note that does not depend on τ since so does . Then (5.1) reads:
Arguing as in Section 3, equation (5.2) is equivalent to
Moreover, the adjoint variational system along is defined as
where satisfy equation (5.2). Then the PoincaréMelnikov vector can be written as
or else
being the solution of (5.4) and the fundamental matrix of the linear equation
Of course the only difference between the cases and for all is that in the first case the PoincaréMelnikov 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 addimensional space of solutions whereoraccording to which of the two conditionsorholds. Moreover, if the PoincaréMelnikov function (5.5) (or (5.6)) has a simple zero atthen system (1.1) has aperiodic solutionsatisfying (2.1).
Finally, we note that when we can show that a Brouwer degree of a PoincaréMelnikov function from either Theorem 3.1 or 5.1 is nonzero then by following [15] we can show existence results.
6 Examples
We consider a secondorder equation
with the line as discontinuity manifold (i.e., with ). We write with (i.e. and ). We also write so that
i.e., we take
in the plane coordinates . According to equation (5.4), the adjoint variational system reads, with :
which can be written as (with and ):
Note that (when ) the last three equation are actually the definitions of , and in terms of the unique (up to a multiplicative constant) bounded solution of the boundary value problem:
and the PoincaréMelnikov function (5.6) reads:
whereas (4.10) reads:
As an example, we take that is we consider the equation
The unperturbed equation with the condition has the solutions:
and . Note that, to have for we need .
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
Then it is easily seen that system (6.1), with , reads
Solving , we get and the boundary condition reads: . So, we can take . Since , then
and the PoincaréMelnikov function reads
For example, taking , where is the time the solution of equation , , takes to reach the discontinuity manifold , we get
To conclude the example we need to find a secondorder equation such that (6.3) holds. We consider
with and . It has the solution and . So, we take and then . Note is equivalent to and . Then , so we take . Furthermore,
Since and , we obtain and . Clearly, (6.4) has a solution . Then the second solution is
Hence,
This implies
Consequently, if
then . So, we conclude with the following.
Corollary 6.1Letand, befunctions such that, and
Suppose, also, that the function
has a simple zero at. Then, for, sufficiently small the singularly perturbed system
has aperiodic solution orbitally near the set.
To get a second example, we change the above as follows: we take
with equations:
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 .
The unperturbed equation on has the solutions:
with and . Then is the value of the solution of
at the time where . Since the equation has the Hamiltonian , we see that satisfies
and hence
We observe that is the first positive time such that where is the solution of
hence
More related results are derived in the Appendix.
Then equations (6.2) have to be changed to
But
and (6.8) easily follows. Now we compute the variational equation and the PoincaréMelnikov function. From (6.6) and , it follows that we can take
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
Finally, since the equations on can be written as
we get
Putting all together we see that the adjoint variational system reads:
The first three equations give the boundary value problem
possessing the unique solution (up to a multiplicative constant) which gives
and, since , the PoincaréMelnikov function is
We conclude with the following.
Figure 1. The upper parts of homoclinic and periodic orbits of.
Corollary 6.2Letbe as in equation (6.7), be aperiodic, function and suppose that the function (6.9) has a simple zero at. Then, for, sufficiently small the singularly perturbed system
has aperiodic solution orbitally near the set.
As an example of the second situation, we consider the case where , , , i.e., we take
where is a periodic, function. Since
we get for any α in a neighborhood of and . Hence, we are in the degenerate case considered in Section 5. The adjoint variational equation along reads now
The first two equations have the twodimensional family of solutions . We take the two independent solutions: and with the corresponding vectors:
With (which implies is independent of ε) the PoincaréMelnikov vector is then
Then we obtain the following corollary.
Corollary 6.3Letbe aperiodic, function and suppose thathas a simple zero at, . Then the singularly perturbed system (6.10) has aperiodic solution orbitally near the set.
Appendix
A.1 Further properties of the solution of (6.5)
From the identity
we derive
Note that
where
and, for α sufficiently small (in fact for )
Using formula 3.131.5 in [[16], p.254] we know that, for any :
where
and F is the elliptic integral of the first kind.
Next note , for and , . Hence, (7.1) gives
where
So,
and hence
On the other hand, by (6.7), we directly verify that by a numerical integration. But we derived (7.2) to get an explicit formula for and in general for .
Furthermore, the above computations also give
Solving (7.3), we obtain
where am is the Jacobi amplitude function. Solving (7.4), we obtain
for
where cn is the Jacobi elliptic function. Formulas (7.5) and (7.6) give explicit solution . For , we derive
We can also compute the Taylor series of (7.7) integrating by series the equation with , . Setting
we see that the following recurrence condition holds:
where we set and , . Since , we see by the induction that for any (note that in the product one of the two indexes is odd). So,
and
For the first few indexes, we get
so that:
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 [[13], Theorem 4.1].
Proof of Theorem 3.2 The existence part is quite standard so we sketch it and give emphasis to the proof of invertibility of for . Since , we get and, differentiating twice, . As a consequence, and
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, differentiating the equality with respect to μ and to ε at , we get:
Next, for , equation is equivalent to , but the l.h.s. tends, for to which gives the PoincaréMelnikov condition. We conclude that, if the PoincaréMelnikov 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:
On account of we are led to look at the solutions of
with . From the previous remarks, we get:
for any , since . So, the claim to be proved is
where we have replaced z with since . Now we differentiate the equality
Hence,
since, from and we get . The proof of Theorem 3.2 is complete. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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.
Acknowledgements
This article is dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.
BF is partially supported by GNAMPACNR and MURSTgroup ‘Equazioni differenziali ordinarie e applicazioni’ (Italy). MF is partially supported by the grant APVV013410 and Marche Polytechnic University, Ancona (Italy).
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