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
Note that (2) falls in the class of implicit differential equations (IODE) like
with . Obviously, vanishes on the line and the condition implies that the line consists of noncritical 0-singularities for (3) (see [, p.163]). Let denote the kernel of the linear map L and its range. Then is the subspace having zero first component and then the right hand side of (3) belongs to if and only if . So all the singularities with are impasse points while is a so called I-point (see [, 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.
where . Here we note that 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  are , and , for some (that is, is an I-point for (4)) and the existence of a solution in a bounded interval J tending to as t tends to the endpoints of J.
Let . It has been proved in  that (5) implies that is at most 2. So, if , with then cannot be hyperbolic for the map .
In this paper we study coupled IODEs such as
The objective of this paper is to give conditions, besides (C1)-(C3), assuring that for , the coupled equations (7) has a solution in a neighborhood of the orbit and reaching is a finite time. Our approach mimics that in  and uses Melnikov methods to derive the needed conditions. Let us briefly describe the content of this paper. In Section 2 we make few remarks concerning assumptions (C1)-(C3). Then, in Section 3, we change time to reduce equation (7) to a smooth perturbation of (9) whose unperturbed part has the solution . Next, in Section 4 we derive the Melnikov condition. Finally Section 5 is devoted to the application of our result to coupled equations of the form (1) for RLC circuits, while some computations are postponed to the appendix.
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
From [, Theorem 4.3, p.335 and Theorem 4.5, p.338] it follows that
As in  it is easily seen that
So is a bounded solution of (14). Next, setting as in 
Note, then, that (14) is derived from (16) with the change . This fact should clarify why we need to consider the linear system (14) instead of . However, see  for a remark concerning the space of bounded solutions of (14) and that of the equation .
Proof From [, Lemma 3.1] it follows that the linear map:
has the eigenvalues ±μ; moreover, since
for all . Then (17) satisfies the assumptions of [, Theorem 5.3] and hence its conclusion with , that is, the fundamental matrix of (17) satisfies
where . However, it is well known (see [23-25]) that is the space of initial conditions for the bounded solutions on of (17) that, then, tend to zero as at the exponential rate . As a consequence a solution of (17) is bounded on if and only if is a bounded solution of (14). Then we conclude that the space of solutions of (14) that are bounded on is one dimensional.
Using a similar argument in with , and [, Theorem 5.4] instead of [, Theorem 5.3] with we see that (14) has at most a one dimensional space of solutions bounded in ℝ. More precisely, with , and a projection on exists such that
We conclude this section with a remark about condition (c) in [, Theorem 5.3]. Consider a system in such as
Then the following result holds.
Theorem 2.2Suppose the following hold:
as the dimension of the space of the generalized eigenvectors of the matrixDwith real parts less than or equal to; hereare two suitable positive constants. Similarly there are as many solutions of (18) such that
Proof We prove the first statement concerning (19). By a similar argument (20) is handled. Changing variables we may assume that
Now we observe that is a solution of (22) bounded at +∞ if and only if is a solution of (21) which is bounded on ℝ when multiplied by . Moreover, since , , and belong to , the limit exists for any solution of (22) bounded on . So, let us fix and take . If is a solution of (22) bounded at +∞ it must be, by the variation of constants formula,
with the obvious norm. Since we see that the map (23) is a contraction on B, provided is sufficiently large, and then, for any given , it has a unique solution . Note that a priori is defined only on but of course we can extend it to going backward with time. We now prove that positive constants exist such that fox any . Let . We have
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 is a simple eigenvalue of D and all the others either have real parts less than or (i.e. we do not need that is simple). Similarly inequalities of (20) hold if is a simple eigenvalue of D and all the others either have real parts greater than or (i.e. we do not need that is simple).
(ii) Note that a result related to Theorem 2.2 has been proved in .
3 Solutions asymptotic to the fixed point
As a consequence
Moreover (see (13))
Hence is not an I-point of (8). In this paper we want to look for solutions of the coupled equation (7) that belong to a neighborhood of , they are defined in the interval , for some , and tend to at the same rate as . To this end we first perform a change of the time variable as follows. Set
From (11) it follows that
From the proof of Lemma 2.1 (see also [, Lemma 3.1]) we know that (30) has the positive simple eigenvalues and , and (31) has the negative simple eigenvalues and . From the roughness of exponential dichotomies it follows that both equations in (29) have an exponential dichotomy on both and with projections, resp. and . Hence (see also ) all solutions of the system
4 Melnikov function and the original equation
In Section 3 (see also [1,19]) we have seen that (36) has an exponential dichotomy on both and with projections . So the only bounded solution of is . In other words . So we are lead to prove the following.
uniformly with respect to η. We conclude that (40) can be written as
which has the unique solution
where have been defined in the previous Section 3. So and . We recall, from , that where are solutions of the adjoint equation of (16):
A direct application of Theorem 4.1 gives the following.
Remark 4.4 (i) Equation (48) implies
Summarizing, we obtain the following corollary of Theorem 4.3.
5 Applications to RLC circuits
In this section we study the coupled equations
bounded on ℝ. Hence
So now (16) has the form
possessing the solution . Following [, p.327], the second solution of (58) is given by
Consequently a fundamental matrix solution of (57) has the form
with the fundamental matrix solution
where is the sine integral function, is the cosine integral function [, p.886] and Γ is the Euler constant. Similarly
Consequently, the Melnikov function is now
Summarizing we see that for any fixed χ and ϖ satisfying
the Melnikov function (63) has a simple zero given by (64) and (67), and , , . Hence in the region given by (69) we apply Corollary 4.5 to (51) with parameters , , λ near , , determined by (64) and (67), i.e.,
for any fixed ϖ and χ satisfying (69). Summarizing, we get the following.
Theorem 5.1For any fixedϖ, χsatisfying (69) and then, , , given by (67) and (70), there is anand smooth functionswith, , , , such that for any, system (51) with, , , possesses a unique solutiononsuch that
and hence, using (74) and (72),
Vice versa, if (75) and the first of (72) hold, then (73) gives
Hence (71) and (72) are equivalent. The proof is complete. □
Summarizing we arrive at
So we obtain (61). Similarly,
which implies the second formula of (62).
The authors declare that they have no competing interests.
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.
BF is partially supported by PRIN-MURST ‘Equazioni Differenziali Ordinarie e Applicazioni’ (Italy). MF is partially supported by the Grants VEGA-MS 1/0071/14, VEGA-SAV 2/0029/13 and Marche Polytechnic University, Ancona (Italy).
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