### Abstract

We apply dynamical system methods and Melnikov theory to study small amplitude perturbation of some coupled implicit differential equations. In particular we show the persistence of such orbits connecting singularities in finite time provided a Melnikov like condition holds. Application is given to coupled nonlinear RLC system.

**MSC: **
34A09, 34C23, 37G99.

##### Keywords:

implicit ode; perturbation; Melnikov method; RLC circuits### 1 Introduction

In [1], motivated by [2,3], the equation modeling nonlinear RLC circuits

has been studied. It is assumed that

It is assumed that, for some

passing through

so that the condition

Note that (2) falls in the class of implicit differential equations (IODE) like

with
*L* and
*I*-point (see [[4], pp.163-166]). Quasilinear implicit differential equations, such as (4), find applications
in a large number of physical sciences and have been studied by several authors [4-12]. On the other hand, there are many other works on implicit differential equations
[13-18] dealing with more general implicit differential systems by using analytical and topological
methods.

Passing from (2) to (3), in the general case, it corresponds to multiplying (4) by
the adjugate matrix

where
*A* and *x* may have different dimensions in this paper depending on the nature of the equation
but the concrete dimension is clear from that equation, so we do not use different
notations for *A* and *x*. Basic assumptions in [1] are
*I*-point for (4)) and the existence of a solution
*J* tending to
*t* tends to the endpoints of *J*.

It is well known [4,8] that

and then

Let

In this paper we study coupled IODEs such as

with

hence
*I*-point. Multiplying the first equation by

We assume that

(C1)

possesses a noncritical singularity at
*i.e.*

(C2)

has a solution
*t* with
*ε* sufficiently small.

(C3) Let

From (C2) we see that

and that

The objective of this paper is to give conditions, besides (C1)-(C3), assuring that
for

We emphasize the fact that Melnikov technique is useful to predict the existence of transverse homoclinic orbits in mechanical systems [20,21] together with the associated chaotic behavior of solutions. However, the result in this paper is somewhat different in that we apply the method to show existence of orbits connecting a singularity in finite time.

### 2 Comments on the assumptions

By following [1,19] we note that since

So

From [[22], Theorem 4.3, p.335 and Theorem 4.5, p.338] it follows that

and there exist a constant

Note that

for a suitable constant

Next

Taking logarithms, dividing by *s* and letting

that is, in (10)
*s* with −*s*:

Next, set

Since

and

From (C2), we know

As in [19] it is easily seen that

and that

So

and

*i.e.*

Note, then, that (14) is derived from (16) with the change

We now prove that

**Lemma 2.1***Assume* (C2) *and* (C3) *hold*. *Then*, *up to a multiplicative constant*,
*is the unique solution of* (14) *which is bounded on* ℝ.

*Proof* From [[19], Lemma 3.1] it follows that the linear map:

has the simple eigenvalues

has the eigenvalues ±*μ*; moreover, since

for two positive constants

satisfying

for all

where

Incidentally, since the fundamental matrix of (14) is

Using a similar argument in

and

We conclude this section with a remark about condition (c) in [[19], Theorem 5.3]. Consider a system in

Then the following result holds.

**Theorem 2.2***Suppose the following hold*:

(i) *D**has two simple eigenvalues*
*and all the other eigenvalues of**D**have either real part less than*
*or greater than*

(ii)

(iii)
*as*

*Then there are as many solutions*
*of* (18) *satisfying*

*as the dimension of the space of the generalized eigenvectors of the matrix**D**with real parts less than or equal to*
*here*
*are two suitable positive constants*. *Similarly there are as many solutions of* (18) *such that*

*for suitable constants*
*as the dimension of the space of the generalized eigenvectors of the matrix**D**with real parts greater than or equal to*

*Proof* We prove the first statement concerning (19). By a similar argument (20) is handled.
Changing variables we may assume that

and the eigenvalues of

where

Now we observe that

where

with the obvious norm. Since
*B*, provided
*a priori*

and then

So, for any

with

Since

where

So, provided we take
*i.e.* for

and the existence of

for all

*i.e.*

The proof is complete. □

**Remark 2.3** (i) It follows from the proof of Theorem 2.2 that inequalities of (19) also hold
replacing (i) with the weaker assumption that
*D* and all the others either have real parts less than
*i.e.* we do not need that
*D* and all the others either have real parts greater than
*i.e.* we do not need that

(ii) Note that a result related to Theorem 2.2 has been proved in [26].

### 3 Solutions asymptotic to the fixed point

It follows from (11)-(12) that

As a consequence

Since

Moreover (see (13))

Hence
*I*-point of (8). In this paper we want to look for solutions of the coupled equation
(7) that belong to a neighborhood of

and plug

Since we are looking for solutions of (7) tending to

where

for a suitable constant

for

for any

From (11) it follows that

Next we note that from

are bounded uniformly in
*ε* bounded.

The linearization of (28) at

Taking the limit as

Similarly taking the limit as

From the proof of Lemma 2.1 (see also [[1], Lemma 3.1]) we know that (30) has the positive simple eigenvalues

adjoint to (29), are bounded as

### 4 Melnikov function and the original equation

In this section we will give a condition for solving (28) for

and

we look for solutions

in the Banach space of

In Section 3 (see also [1,19]) we have seen that (36) has an exponential dichotomy on both

**Theorem 4.1***Let**Y*, *X**be Banach spaces*,
*be a small parameter and*
*Let*
*be*
*functions such that*

(a)

(b)

(c) *there exist*
*such that*

*Set*
*by*

*and suppose there exists*
*such that*
*and the derivative*
*is invertible*. *Then there exist*
*and unique*
*function*
*defined in a neighborhood of*
*such that*

*and for*
*has a unique solution*
*satisfying*

*Moreover*,
*for*
*functions*
*and we have*

*Proof* We look for solutions

for unique

provided
*η* in a fixed closed ball

Observe that

Then

uniformly with respect to *η*. We conclude that (40) can be written as

where

Note that
*η*, so

is

Because of the assumptions we can apply the implicit function theorem to (43) to
obtain a

we see that

Clearly (39) follows differentiating the above equality at

**Remark 4.2** Note that, because of

which has the unique solution

Now we apply Theorem 4.1 to (28) with

where

We already observed that

where

and

or passing to time

A direct application of Theorem 4.1 gives the following.

**Theorem 4.3***Let*
*and*
*be given as in* (46) *where*
*are two independent bounded solutions* (*on* ℝ) *of the adjoint equation* (45). *Suppose that*
*and*
*exist so that*

*Then there exist*
*unique*
*functions*
*and*
*with*
*and*
*defined for*
*and a unique solution*
*of* (24) *with*
*such that*

*Moreover*,

**Remark 4.4** (i) Equation (48) implies

for

with

which gives a first order approximation of

Since (36) has exponential dichotomies on both

*i.e.*,

for

for

with

(ii) Using (49), the functions