- Open Access
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.
- implicit ode
- Melnikov method
- RLC circuits
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.
are bounded uniformly in and , , , ε bounded.
adjoint to (29), are bounded as . We let and be any two linearly independent solutions of (32).
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.
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).