On the existence of solutions connecting IK singularities and impasse points in fully nonlinear RLC circuits

Higher-dimensional nonlinear and perturbed systems of implicit ordinary differential equations are studied by means of methods of dynamical systems. Namely, the persistence of solutions are studied under nonautonomous perturbations connecting either impasse points with IK-singularities or two impasse points. Important parts of the paper are applications of the theory to concrete perturbed fully nonlinear RLC circuits.

1. Introduction. Motivated by [8,16], a fully nonlinear RLC circuit is described by the second order ordinary differential equation u + L(v) + R(v) = e(t), v = C(u) , (1.1) where L, R are the nonlinear self-inductance and the ohmic resistance, respectively and C is the nonlinear capacitance. u is the potential/voltage, q = C(u) is the charge and v = dq dt is the current intensity. We suppose that L, R, C and e are C 3 -smooth.
We can write equation (1.1) as a system in R 2 : Quasilinear implicit differential equations, such as (1.2), find applications in a large number of physical sciences and have been studied by several authors [7,9,10,11,12,13,14].
In this paper we suppose that equation (1.5) has a solution connecting the fixed point e 3 (at t = −∞) with another point in the manifold ω(v, u) = 0. We want to study persistence of this kind of solutions.
This situation arises, for example, if equation (1.5) has heteroclinic connection between the fixed points e 3 , e 2 crossing the set S := {(v, u) | ω(v, u) = 0}, where we assume that 0 is a regular value of ω. In particular, if this happens, it must be:

EXISTENCE OF SOLUTIONS CONNECTING IK SINGULARITIES 3045
In Section 4 of this paper we will give a couple of examples of such a situation. So let a heteroclinic orbit γ(t) intersect the set S transversally at two points In Section 2, we will study the persistence of the branch {γ(t) | t ≤ t * 1 } to a solution of equation (1.3) tending to the fixed point e 3 as t → −∞ (or +∞ possibly reversing time) and hitting S at finite time. In Section 3, we will prove a result concerning the middle part It is easy to check that e 3 persists as equilibrium in (1.7) after perturbation, but e 2 does not. Next, since ω(v, u) = 0 only on the lines u = u 0 , v = v 0 , the system is regular outside these lines. As in [1] it can be seen that the orbits of equations (1.7) outside the lines u = u 0 and v = v 0 are the same as those of equatioṅ Similarly any solution of (1.7) approaching, as t →t, a point like (v 0 ,ū),ū = 0, u 0 has to satisfy: lim t→t v (t) = ∞.
As we said, in this paper we study the persistence of solutions of the unperturbed system connecting regular points in the manifold ω(v, u) = 0 toward solutions of the perturbed equation (1.7). Furthermore, since the fixed point e 2 disappears it would also be interesting to consider persistence of solutions of equation (1.9) tending to e 2 as t → ±∞. We leave this last study to a forthcoming paper. These problems are different from the ones studied in [1] where equation (1.5) has an hyperbolic equilibrium together with a homoclinic orbit γ(s) = (γ v (s), γ u (s)) to it and the fixed point persists after perturbation. In [1] it has been proved that a change of time scale (depending on the solution) exists so that γ(s) corresponds to a solution Γ(t) of equation (1.9) tending to the fixed point as t → ±T , but ω(Γ(t)) > 0 for any t ∈ (−T, T ). The persistence of such orbits are studied in [1]. Similar problems are studied in [2,3,4].
We also refer to [6] for more examples of electronic nonlinearities like a gyrator circuit as a nonlinear inductor.
In the following we assume that ω(γ(s)) > 0 for any s ∈ (−∞, 1) and (2.5) instead of (C4), as we can always reduce to this situation possibly changing t with −t in equation (2.1).
First we consider equation (2.2) in the interval (−T * , 0], where T * > 0 is given as follows. For s ≤ 0, let In [2, p. 1164] it has been observed that T * is finite. Let k − be the number of eigenvalues of F (x 0 ) with negative real parts counted with multiplicities. Then F (x 0 ) has n − k − eigenvalues with positive real parts counted with multiplicities. According to arguments of [2, p. 1169], the linear systemẏ has an exponential dichotomy on R − with a projection P − such that rank We prove the following Lemma 2.1. Assume conditions (C1)-(C4) hold and that ω(γ(s)) > 0 for any s ≤ 0. Then for any κ ∈ I, for a compact subset I ⊂ R m , η ∈ N P − and ε sufficiently small, equation uniformly for κ in compact sets in R m .

FLAVIANO BATTELLI AND MICHAL FEČKAN
and make in (2.9) the change of variables where η(s) is the bounded function γ(s)−x0 ϕ(s) . So we derive the equatioṅ , by following [2]. It is easy to check that the linearization of (2.11) at y = 0 and ε = 0 is equation (2.7) and we already noted that this equation has an exponential dichotomy on R − with projection of rank k − + 1 or dim N P − = n − k − − 1. Now we rewrite (2.11) as followṡ Clearly H(s, y, ε, κ) = O(|y| 2 + |ε|).
(2.13) Let Y (s) be the fundamental solution of the linear part of (2.12) satisfying Y (0) = I. Then, for any η ∈ N P − , the solutions of (2.12) belonging to C b satisfy the implicit equation (2.14) By using (2.13) and method of [2], given any ρ > 0 and any compact subset I ⊂ R m there is a neighborhood B of (0, 0) in the (ε, η)-space such that for any (ε, η, κ) ∈ B × I there exists a unique solution y(s, ε, η, κ) ∈ C b of (2.14) on s ≤ 0 such that sup s≤0 y(s, ε, η, κ) < ρ. Moreover this solution depends smoothly on its arguments. By setting: the proof is concluded.
If, instead, we take y(s) . e s e s + 3