Melnikov theory for weakly coupled nonlinear RLC circuits
© Battelli and Fečkan; licensee Springer. 2014
Received: 18 December 2013
Accepted: 1 April 2014
Published: 7 May 2014
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: Primary 34A09; secondary 34C23; 37G99.
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.
and then is equivalent to the fact that .
Let . It has been proved in  that (5) implies that is at most 2. So, if , with then cannot be hyperbolic for the map .
- (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.
Let be the eigenvectors of with the eigenvalues , resp. Then (or else ).
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 (C2), we know for any , so . Hence condition (C3) means that tends transversally to the singular manifold at .
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 .
We now prove that is the unique solution of equation (14) which is bounded on ℝ. This is a kind of nondegeneracy of .
Lemma 2.1 Assume (C2) and (C3) hold. Then, up to a multiplicative constant, is the unique solution of (14) which is bounded on ℝ.
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.
and . Since is a solution of (14) bounded on ℝ we deduce that and the result follows. □
Then the following result holds.
D has two simple eigenvalues and all the other eigenvalues of D have either real part less than or greater than ;
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 .
The proof is complete. □
Note that a result related to Theorem 2.2 has been proved in .
3 Solutions asymptotic to the fixed point
are bounded uniformly in and , , , ε bounded.
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 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.
there exist such that .
Clearly (39) follows differentiating the above equality at . The proof is complete. □
where is the Banach space of -functions bounded together with their derivatives with the usual sup-norm.
A direct application of Theorem 4.1 gives the following.
with , and is a fundamental solution of (16).
Summarizing, we obtain the following corollary of Theorem 4.3.
5 Applications to RLC circuits
When , then (68) is not satisfied, since . So we take and (68) gives also . Clearly implies .
for any fixed ϖ and χ satisfying (69). Summarizing, we get the following.
Hence (71) and (72) are equivalent. The proof is complete. □
Of course, solutions given by Theorem 5.1 vary smoothly with respect satisfying (69).