Research

# Fast-slow dynamical approximation of forced impact systems near periodic solutions

Flaviano Battelli1 and Michal Fečkan2*

Author Affiliations

1 Dipartimento di Ingegneria Industriale e Scienze Matematiche, Marche Polytecnic University, Ancona, 60100, Italy

2 Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, Bratislava, 842 48, Slovakia

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Boundary Value Problems 2013, 2013:71  doi:10.1186/1687-2770-2013-71

 Received: 14 October 2012 Accepted: 21 March 2013 Published: 3 April 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

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

### 1 Introduction

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 [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 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 [12] when an impact oscillator is approximated by a one-parametric 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 hyper-surface of Ω that we call impact manifold, (or hyper-surface). We set and consider the following regular-singular perturbed system:

(1.1)

for small. We assume that the system

(1.2)

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

(1.3)

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 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

(1.4)

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:

(1.5)

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

(1.6)

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:

eitherorfor 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

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

(2.1)

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

(2.2)

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

(2.3)

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:

(2.4)

. Note that

and hence in the fixed-point 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 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

(2.5)

is in all parameters and t.

Writing in place of t in (2.4) and using (2.5) we see that

(2.6)

. We have, by definition, and

if and only if (recall that )

(2.7)

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:

(2.8)

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

that is [again using and ]:

(2.9)

Of course, when (2.9) holds, then (2.7) is equivalent to

(2.10)

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 .

### 3 Solving

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 [[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

(3.1)

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 Poincaré-Melnikov function

(3.2)

has a simple zero at, then system (1.1) has a-periodic 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

(3.3)

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

(3.4)

From , we get

(3.5)

thus, on account of the transversality condition , (3.4) is equivalent to

(3.6)

Next, from , we get

(3.7)

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

(3.8)

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 [13].

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

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 Poincaré-Melnikov function is:

(3.9)

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):

(4.1)

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:

(4.2)

Next, recalling (1.1), we consider a perturbed impact system of (4.1) (see also (2.8)) of the form

(4.3)

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:

(4.4)

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

with

and .

Lemma 4.1The rangeis closed.

Proof Indeed, let . Then

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

(4.5)

holds for any solutionof the adjoint system

(4.6)

and.

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

or

(4.7)

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

(4.8)

and

(4.9)

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

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

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

for some . Then

and then, using again we see that

i.e.

(4.10)

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 [[14], 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

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 Poincaré-Melnikov 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

Now, from we get, for any :

and hence

which, in turn, is equivalent to because of transversality and the fact that .

Hence, we conclude that .

Now we consider the second condition. The Poincaré-Melnikov function (vector) , can be written as

(5.1)

where is a matrix whose rows are left eigenvectors of zero eigenvalue of the matrix , that is,

(5.2)

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

(5.3)

Moreover, the adjoint variational system along is defined as

(5.4)

where satisfy equation (5.2). Then the Poincaré-Melnikov vector can be written as

(5.5)

or else

(5.6)

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 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 [15] we can show existence results.

### 6 Examples

We consider a second-order 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 ):

(6.1)

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 Poincaré-Melnikov function (5.6) 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)

Note for this case. Since

(6.2) is equivalent to

(6.3)

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

For example, taking , where is the time the solution of equation , , takes to reach the discontinuity manifold , we get

which has a simple zero at .

To conclude the example we need to find a second-order 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,

Setting , we have

(6.4)

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 a-periodic 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

(6.5)

and hence

(6.6)

We observe that is the first positive time such that where is the solution of

hence

(6.7)

More related results are derived in the Appendix.

Then equations (6.2) have to be changed to

(6.8)

But

and (6.8) easily follows. Now we compute the variational equation and the Poincaré-Melnikov 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

The first three equations give the boundary value problem

possessing the unique solution (up to a multiplicative constant) which gives

and, since , the Poincaré-Melnikov function is

(6.9)

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 a-periodic, function and suppose that the function (6.9) has a simple zero at. Then, for, sufficiently small the singularly perturbed system

has a-periodic solution orbitally near the set.

As an example of the second situation, we consider the case where , , , i.e., we take

(6.10)

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 two-dimensional family of solutions . We take the two independent solutions: and with the corresponding vectors:

With (which implies is independent of ε) the Poincaré-Melnikov vector is then

Then we obtain the following corollary.

Corollary 6.3Letbe a-periodic, function and suppose thathas a simple zero at, . Then the singularly perturbed system (6.10) has a-periodic solution orbitally near the set.

### Appendix

#### A.1 Further properties of the solution of (6.5)

From the identity

we derive

(7.1)

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,

We are interested in . Then

and hence

(7.2)

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

(7.3)

for any and

(7.4)

Solving (7.3), we obtain

where am is the Jacobi amplitude function. Solving (7.4), we obtain

(7.5)

for

(7.6)

where cn is the Jacobi elliptic function. Formulas (7.5) and (7.6) give explicit solution . For , we derive

(7.7)

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 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:

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

with respect to μ at to get

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

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).

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