Research

# Melnikov theory for weakly coupled nonlinear RLC circuits

Flaviano Battelli1* and Michal Fečkan23

Author Affiliations

1 Department of Industrial Engineering and Mathematical Sciences, Marche Polytecnic University, Via Brecce Bianche 1, Ancona, 60131, Italy

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

3 Mathematical Institute of Slovak Academy of Sciences, Štefánikova 49, Bratislava, 814 73, Slovakia

For all author emails, please log on.

Boundary Value Problems 2014, 2014:101  doi:10.1186/1687-2770-2014-101

 Received: 18 December 2013 Accepted: 1 April 2014 Published: 7 May 2014

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 credited.

### 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

(1)

has been studied. It is assumed that and are smooth functions with at least quadratic at the origin and satisfying suitable assumptions. Setting the equation reads

(2)

It is assumed that, for some , we have and . So for (2) has the Hamiltonian

passing through . Clearly and the Hessian of ℋ at is

so that the condition means that is a saddle for ℋ. Multiplying the second equation by we get the system

(3)

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

(4)

with . Obviously, vanishes on the line and the condition implies that the line consists of noncritical 0-singularities for (3) (see [[4], 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 [[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 . 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 [1] 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.

It is well known [4,8] that and imply

(5)

and then is equivalent to the fact that .

Let . It has been proved in [19] 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

(6)

with , , and other assumptions that will be specified below. Let us remark that (6) is a special kind of the general equation (4) with, among other things,

hence satisfies , and . Thus is not a I-point. Multiplying the first equation by and the second by we obtain the system

(7)

We assume that , and satisfy the following assumptions:

(C1) , and the unperturbed equation

(8)

possesses a noncritical singularity at , i.e. and .

(C2) and the spectrum with , and

has a solution homoclinic to , that is, , and for any . Without loss of generality, we may, and will, assume for any . Moreover, , are 1-periodic in t with for any , and ε sufficiently small.

(C3) Let be the eigenvectors of with the eigenvalues , resp. Then (or else ).

From (C2) we see that is a bounded solution of the equation

(9)

and that persists as a singularity of (7). So this paper is a continuation of [1,19], but here we study more degenerate IODE.

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 [1] 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

By following [1,19] we note that since as then is a bounded solution of the linear equation . Hence for some . We get then, for ,

So

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

(10)

and there exist a constant and a solution of such that

Note that since otherwise , contradicting (10). Hence is an eigenvector of the eigenvalue of . We have then

for a suitable constant . As a consequence,

Next

Taking logarithms, dividing by s and letting we get

that is, in (10) can be replaced with . Similarly, changing s with −s:

Next, set

(11)

Since as and as we have then

(12)

and

From (C2), we know for any , so . Hence condition (C3) means that tends transversally to the singular manifold at .

As in [19] it is easily seen that

(13)

and that solves the equation

(14)

So is a bounded solution of (14). Next, setting as in [19]

(15)

and , it is easily seen that satisfies whose linearization along is

i.e.

(16)

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 [19] for a remark concerning the space of bounded solutions of (14) and that of the equation .

We now prove that is the unique solution of equation (14) which is bounded on ℝ. This is a kind of nondegeneracy of .

Lemma 2.1Assume (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 and . Let , then the linear map

has the eigenvalues ±μ; moreover, since

for two positive constants , it follows that is a solution of

(17)

satisfying

for all . Then (17) satisfies the assumptions of [[19], 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.

Incidentally, since the fundamental matrix of (14) is , we note that it satisfies

Using a similar argument in with , and [[19], Theorem 5.4] instead of [[19], 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

and . Since is a solution of (14) bounded on ℝ we deduce that and the result follows. □

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

(18)

Then the following result holds.

Theorem 2.2Suppose the following hold:

(i) Dhas two simple eigenvaluesand all the other eigenvalues ofDhave either real part less thanor greater than;

(ii) ;

(iii) as.

Then there are as many solutionsof (18) satisfying

(19)

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

(20)

for suitable constants, as the dimension of the space of the generalized eigenvectors of the matrixDwith 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 have real parts less than and those of have real parts greater than or equal to . So the system reads

(21)

where and are matrices (or vectors) of suitable orders. Setting we get

(22)

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,

(23)

where and . Note that since and and , are bounded, we can interpret (22) as a fixed point theorem in the Banach space of bounded function on :

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

and then

So, for any let be such that and set . We have

with . Taking the limit as we get

Since as , from the above it follows that . Similarly we get

where and then . As a consequence we obtain and then

So, provided we take we see that eventually (i.e. for , for some )

and the existence of such that

for all follows from the fact that cannot vanish in any bounded interval. Finally since we get, for ,

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 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 [26].

### 3 Solutions asymptotic to the fixed point

It follows from (11)-(12) that as then, since we obtain . Furthermore, from (13) we also get:

As a consequence

Since it follows that is a strictly increasing diffeomorphism (see (15) for the definition of ). Then satisfies (8) on the interval and

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

and plug in (7). We get

(24)

Since we are looking for solutions of (7) tending to at the same rate as , in (24) we make the change of variables

(25)

where is the bounded function . Since

(26)

for a suitable constant and any , we get, using (C3), (26):

(27)

for large and sufficiently small. Then (27) and imply the existence of and so that

for any and . Now plugging (25) into (24) we derive the equations

(28)

From (11) it follows that

Next we note that from it follows that the quantities

are bounded uniformly in and , , , ε bounded.

The linearization of (28) at , is

(29)

Taking the limit as we get the systems

(30)

Similarly taking the limit as we get the systems

(31)

From the proof of Lemma 2.1 (see also [[1], 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 [19]) all solutions of the system

(32)

adjoint to (29), are bounded as . We let and be any two linearly independent solutions of (32).

### 4 Melnikov function and the original equation

In this section we will give a condition for solving (28) for near the solution of the same equation with . Writing

(33)

and

(34)

we look for solutions of

(35)

in the Banach space of -functions on ℝ, bounded together with their derivatives and with small norms. We observe that and equation reads

(36)

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.

Theorem 4.1LetY, Xbe Banach spaces, be a small parameter and. Let, , be-functions such that

(a) ;

(b) ;

(c) there existsuch that.

Setby

(37)

and suppose there existssuch thatand the derivativeis invertible. Then there existand unique-function, defined in a neighborhood ofsuch that

(38)

and for, , (35) has a unique solutionsatisfying

Moreover, for-functionsand we have

(39)

Proof We look for solutions of (35) that are close to . Let be the projection such that . Note . From the implicit function theorem, we solve the projected equations

for unique such that

provided is sufficiently small and η in a fixed closed ball with . Note that are -smooth. Setting , we need to solve the bifurcation equations:

(40)

Observe that

(41)

Then and so

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

(42)

where

Note that are -functions of and that uniformly with respect to η, so

is in . By (41), system (42) is equivalent to

(43)

Because of the assumptions we can apply the implicit function theorem to (43) to obtain a -function defined in a neighborhood of satisfying (43) and such that (38) holds. Setting

we see that , are bounded -solutions of (35) with such that , . Then we can write , ) for continuous where

Clearly (39) follows differentiating the above equality at . The proof is complete. □

Remark 4.2 Note that, because of , (39) is equivalent to

which has the unique solution

(44)

Now we apply Theorem 4.1 to (28) with , , as in (33), (34) and

where is the Banach space of -functions bounded together with their derivatives with the usual sup-norm.

We already observed that and . Moreover,

where have been defined in the previous Section 3. So and . We recall, from [19], that where are solutions of the adjoint equation of (16):

(45)

or passing to time :

(46)

A direct application of Theorem 4.1 gives the following.

Theorem 4.3Letandbe given as in (46) where, are two independent bounded solutions (on ℝ) of the adjoint equation (45). Suppose thatandexist so that

(47)

Then there exist, , unique-functionsandwithand, defined for, and a unique solutionof (24) with, , , such that

(48)

Moreover,

Remark 4.4 (i) Equation (48) implies

for -functions with . Then we have

with , so . Hence

(49)

which gives a first order approximation of . Next, can be computed using (44) adapted to this case. Hence are bounded solutions of

Since (36) has exponential dichotomies on both (with projection ) and (with projection ) it follows that

(50)

is the fundamental solution of (36). Note formulas (50) are well defined at , i.e., , due to the first assumption of (47). Next, passing to time and taking , we get

for and

for . Note that solves

with , and is a fundamental solution of (16).

(ii) Using (49), the functions are bounded solutions of (7) in the interval such that

Summarizing, we obtain the following corollary of Theorem 4.3.

Corollary 4.5Letandbe given as in (46) where, are two independent bounded solutions (on ℝ) of the adjoint equation (45). Suppose thatandexist so that

Then there exist, , unique-functionsandwithand, defined for, and a unique solutionof (7) with, , , such that

Moreover,

### 5 Applications to RLC circuits

In this section we study the coupled equations

(51)

which is motivated by [2,3]. Note that (51) is obtained by coupling two equations modeling nonlinear RLC circuits as in (1). Here , , λ, χ and ϖ are positive parameters. Setting , , (51) reads

(52)

By solving the second and fourth equations of (52) for and , we get:

(53)

provided . Since , to write the system (52) in the form (7) with parameter and fixed, we have to multiply the second and fourth equation by , respectively, and we obtain the system

(54)

with (uncoupled) unperturbed equation for (see (8)):

(55)

Neglecting left multiplicators () in (55), we obtain the following system (see condition (C2)):

(56)

Clearly, condition (C1) is satisfied with and

The equation has the prime integral . A solution satisfying has to satisfy with the solution

bounded on ℝ. Hence

Note . From

we get and . Since , we derive , and condition (C3) holds as well. Next, we have

hence and assumption (C2) is also verified. Here , . Furthermore by (15)

so and

Thus

So now (16) has the form

(57)

which has the solution . In other words, we deal with

(58)

possessing the solution . Following [[27], p.327], the second solution of (58) is given by

Consequently a fundamental matrix solution of (57) has the form

Note . The adjoint system of (57) is (see (45))

(59)

with the fundamental matrix solution

Note . Thus we take

and

We now compute . We have (see the appendix)

(60)

where is the sine integral function, is the cosine integral function [[28], p.886] and Γ is the Euler constant. Similarly

(61)

and

(62)

Note

Consequently, the Melnikov function is now

(63)

The equation is equivalent to

(64)

So having and such that the equation

(65)

has a simple zero with , and , formulas (64) give a simple zero of (63) with positive , , , and Corollary 4.5 can be applied to (51). If , then (65) is equivalent to

(66)

Hence assuming and , the right hand side of (66) is negative, and then

(67)

satisfies and . Since satisfies (65) and , condition is equivalent to . Then using (66), we derive

(68)

When , then (68) is not satisfied, since . So we take and (68) gives also . Clearly implies .

Summarizing we see that for any fixed χ and ϖ satisfying

(69)

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.,

(70)

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

(71)

Proof We apply Corollary 4.5 to (53). Since, in this case, , according to Corollary 4.5 (53) has a solution , , such that

(72)

Now, and , where . So, taking ,

and then

(73)

or, equivalently,

(74)

Since we see that, for ε sufficiently small,

and hence, using (74) and (72),

(75)

Vice versa, if (75) and the first of (72) hold, then (73) gives

Hence (71) and (72) are equivalent. The proof is complete. □

Of course, solutions given by Theorem 5.1 vary smoothly with respect satisfying (69).

Remark 5.2 Missed in the above analysis is the second possibility when . Then and (66) is negative if

(76)

so we get and . Then inequality of (68) is satisfied since . So we conclude that the result of Theorem 5.1 is valid also for

### Appendix

Let be the second component of . Note that is an even function and then

(77)

Similarly,

(78)

Now, using and integration by parts of the second integral,

Furthermore, since

we compute

and

Next

Furthermore,

and

Hence

Clearly . Next, using , as , we derive

Setting we see , so , and using , we obtain

Summarizing we arrive at

So inserting the above computations into (77) and (78), we get (60) and the first formula of (62). Next, since the second component of is odd, we get

But

and

So we obtain (61). Similarly,

which implies the second formula of (62).

### 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 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).

### References

1. Battelli, F, Fečkan, M: Nonlinear RLC circuits and implicit ode’s. Differ. Integral Equ. (2014)

2. Lazarides, N, Eleftheriou, M, Tsironis, GP: Discrete breathers in nonlinear magnetic metamaterials. Phys. Rev. Lett.. 97, Article ID 157406 (2006)

3. Veldes, GP, Cuevas, J, Kevrekidis, PG, Frantzeskakis, DJ: Quasidiscrete microwave solitons in a split-ring-resonator-based left-handed coplanar waveguide. Phys. Rev. E. 83, Article ID 046608 (2011)

4. Riaza, R: Differential-Algebraic Systems, Analytical Aspects and Circuit Applications, World Scientific, Singapore (2008)

5. Kunkel, P, Mehrmann, V: Differential-Algebraic Equations, Analysis and Numerical Solution, Eur. Math. Soc., Zürich (2006)

6. Medved’, M: Normal forms of implicit and observed implicit differential equations. Riv. Mat. Pura Appl.. 10, 95–107 (1991)

7. Medved’, M: Qualitative properties of generalized vector fields. Riv. Mat. Pura Appl.. 15, 7–31 (1994)

8. Rabier, PJ: Implicit differential equations near a singular point. J. Math. Anal. Appl.. 144, 425–449 (1989). Publisher Full Text

9. Rabier, PJ, Rheinboldt, WC: A general existence and uniqueness theorem for implicit differential algebraic equations. Differ. Integral Equ.. 4, 563–582 (1991)

10. Rabier, PJ, Rheinbold, WC: A geometric treatment of implicit differential-algebraic equations. J. Differ. Equ.. 109, 110–146 (1994). Publisher Full Text

11. Rabier, PJ, Rheinbold, WC: On impasse points of quasilinear differential algebraic equations. J. Math. Anal. Appl.. 181, 429–454 (1994). Publisher Full Text

12. Rabier, PJ, Rheinbold, WC: On the computation of impasse points of quasilinear differential algebraic equations. Math. Comput.. 62, 133–154 (1994)

13. Andres, J, Górniewicz, L: Topological Principles for Boundary Value Problems, Kluwer, Dordrecht (2003)

14. Górniewicz, L: Topological Fixed Point Theory of Multivalued Mappings, Springer, Berlin (2009)

15. Fečkan, M: Existence results for implicit differential equations. Math. Slovaca. 48, 35–42 (1998)

16. Frigon, M, Kaczynski, T: Boundary value problems for systems of implicit differential equations. J. Math. Anal. Appl.. 179, 317–326 (1993). Publisher Full Text

17. Heikkilä, S, Kumpulainen, M, Seikkala, S: Uniqueness and comparison results for implicit differential equations. Dyn. Syst. Appl.. 7, 237–244 (1998)

18. Li, D: Peano’s theorem for implicit differential equations. J. Math. Anal. Appl.. 258, 591–616 (2001). Publisher Full Text

19. Battelli, F, Fečkan, M: Melnikov theory for nonlinear implicit ode’s. J. Differ. Equ.. 256, 1157–1190 (2014). Publisher Full Text

20. Awrejcewicz, J, Holicke, MM: Melnikov’s method and stick-slip chaotic oscillations in very weekly forced mechanical systems. Int. J. Bifurc. Chaos. 9, 505–518 (1999). Publisher Full Text

21. Awrejcewicz, J, Pyryev, Y: Chaos prediction in the Duffing type system with friction using Melnikov’s functions. Nonlinear Anal., Real World Appl.. 7, 12–24 (2006). Publisher Full Text

22. Coddington, EA, Levinson, N: Theory of Ordinary Differential Equations, Tata McGraw-Hill, New Delhi (1972) Reprint (1987)

23. Battelli, F, Lazzari, C: Exponential dichotomies, heteroclinic orbits, and Melnikov functions. J. Differ. Equ.. 86, 342–366 (1990). Publisher Full Text

24. Coppel, WA: Dichotomies in Stability Theory, Springer, Berlin (1978)

25. Palmer, KJ: Exponential dichotomies and transversal homoclinic points. J. Differ. Equ.. 55, 225–256 (1984). Publisher Full Text

26. Calamai, A, Franca, M: Melnikov methods and homoclinic orbits in discontinuous systems. J. Dyn. Differ. Equ.. 25, 733–764 (2013). Publisher Full Text

27. Hartman, P: Ordinary Differential Equations, Wiley, New York (1964)

28. Gradshteyn, IS, Ryzhik, IM: Table of Integrals, Series, and Products, Academic Press, Boston (2007)