Melnikov theory for weakly coupled nonlinear RLC circuits

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.

In [1], motivated by [2, 3], the equation modeling nonlinear RLC circuits

has been studied. It is assumed that f(u) and h(t,u,\epsilon ) are smooth functions with f(u) at least quadratic at the origin and satisfying suitable assumptions. Setting v={(u+f(u))}^{\prime } the equation reads

It is assumed that, for some u_0\in \mathbb{R}, we have f^{\prime }(u_0)+1=0 and . So for \epsilon =0 (2) has the Hamiltonian

passing through (u_0,0). Clearly \mathrm{\nabla }\mathcal{H}(u_0,0)=\left(\begin{array}{c}0\\ 0\end{array}\right) and the Hessian of ℋ at (u_0,0) is

so that the condition means that (u_0,0) is a saddle for ℋ. Multiplying the second equation by 1+f^{\prime }(u) we get the system

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

with A(u,v)=\left(\begin{array}{cc}1+f^{\prime }(u)& 0\\ 0& 1\end{array}\right). Obviously, detA(u,v)=1+f^{\prime }(u) vanishes on the line (u_0,v) and the condition f^{″}(u_0)\ne 0 implies that the line u=u_0 consists of noncritical 0-singularities for (3) (see [[4], p.163]). Let \mathcal{N}L denote the kernel of the linear map L and \mathcal{R}L its range. Then \mathcal{R}A(u_0,0) is the subspace having zero first component and then the right hand side of (3) belongs to \mathcal{R}A(u_0,0) if and only if v=0. So all the singularities (u_0,v) with v\ne 0 are impasse points while (u_0,0) 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 A^a(x):

where \omega (x)=detA(x). 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 \omega (x_0)=0, {\omega }^{\prime }(x_0)\ne 0 and A^a(x_0)f(x_0)=0, A^a(x_0)h(t,x_0,\epsilon ,\kappa )=0 for some x_0 (that is, x_0 is an I-point for (4)) and the existence of a solution x(t) in a bounded interval J tending to x_0 as t tends to the endpoints of J.

It is well known [4, 8] that \omega (x_0)=0 and {\omega }^{\prime }(x_0)\ne 0 imply

and then A^a(x_0)f(x_0)=0 is equivalent to the fact that f(x_0)\in \mathcal{R}A(x_0).

Let F(x):=A^a(x)f(x). It has been proved in [19] that (5) implies that rank{F}^{\prime }(x_0) is at most 2. So, if x\in {\mathbb{R}}^n, with n>2 then x=x_0 cannot be hyperbolic for the map x\mapsto F^{\prime }(x_0)x.

In this paper we study coupled IODEs such as

with x_1,x_2\in {\mathbb{R}}^2, det{A}_0(x_0)=0\ne {(det{A}_0)}^{\prime }(x_0), f(x_0),g_j(t,x_0,x_0,\epsilon ,\kappa )\in \mathcal{R}{A}_0(x_0) 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 detA(x)=det{A}_0(x_1)det{A}_0(x_2) satisfies detA(x_0,x_0)=0, {(detA)}^{\prime }(x_0,x_0)=0 and {(detA)}^{″}(x_0,x_0)\ne 0. Thus (x_0,x_0) is not a I-point. Multiplying the first equation by A_0^a(x_1) and the second by A_0^a(x_2) we obtain the system

We assume that \omega (x):=det{A}_0(x), F(x) and G_j(x_1,x_2,t,\epsilon ,\kappa ) satisfy the following assumptions:

1. (C1)

   F\in C^2({\mathbb{R}}^2,{\mathbb{R}}^2), \omega \in C^2({\mathbb{R}}^2,\mathbb{R}) and the unperturbed equation

   \omega (x)x^{\prime }=F(x)

   (8)

   possesses a noncritical singularity at x_0, i.e. \omega (x_0)=0 and {\omega }^{\prime }(x_0)\ne 0.

2. (C2)

   F(x_0)=0 and the spectrum \sigma (F^{\prime }(x_0))=\{{\mu }_{\pm }\} with , and

   x^{\prime }=F(x)

   has a solution \gamma (s) homoclinic to x_0, that is, {lim}_{s\to \pm \mathrm{\infty }}\gamma (s)=x_0, and \omega (\gamma (s))\ne 0 for any s\in \mathbb{R}. Without loss of generality, we may, and will, assume \omega (\gamma (s))>0 for any s\in \mathbb{R}. Moreover, G_i\in C^2({\mathbb{R}}^{6+m},{\mathbb{R}}^2), i=1,2 are 1-periodic in t with G_i(x_0,x_0,t,\epsilon ,\kappa )=0 for any t\in \mathbb{R}, \kappa \in {\mathbb{R}}^m and ε sufficiently small.

3. (C3)

   Let {\gamma }_{\pm } be the eigenvectors of F^{\prime }(x_0) with the eigenvalues {\mu }_{\mp }, resp. Then 〈\mathrm{\nabla }\omega (x_0),{\gamma }_{\pm }〉>0 (or else {\omega }^{\prime }(x_0){\gamma }_{\pm }>0).

From (C2) we see that \mathrm{\Gamma }(s):=\left(\begin{array}{c}\gamma (s)\\ \gamma (s)\end{array}\right) is a bounded solution of the equation

and that x_0 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 |\epsilon |\ll 1, the coupled equations (7) has a solution in a neighborhood of the orbit \{\mathrm{\Gamma }(s)\mid s\in \mathbb{R}\} and reaching (x_0,x_0) 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 \mathrm{\Gamma }(s). 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.

By following [1, 19] we note that since \gamma (s)\to x_0 as |s|\to \mathrm{\infty } then {\gamma }^{\prime }(s) is a bounded solution of the linear equation x^{\prime }=F(\gamma (s))x. Hence |{\gamma }^{\prime }(s)|\le k{\mathrm{e}}^{-\mu |s|} for some \mu >0. We get then, for s\ge 0,

So

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

and there exist a constant \delta >0 and a solution {\gamma }_{+}{\mathrm{e}}^{{\mu }_{-}s} of x^{\prime }=F^{\prime }(x_0)x such that

Note that {\gamma }_{+}\ne 0 since otherwise \gamma (s)-x_0=O({\mathrm{e}}^{({\mu }_{-}-\delta )s}), contradicting (10). Hence {\gamma }_{+} is an eigenvector of the eigenvalue {\mu }_{-} of F^{\prime }(x_0). We have then

for a suitable constant c_1\ge 0. As a consequence,

Next

Taking logarithms, dividing by s and letting s\to \mathrm{\infty } we get

that is, in (10) {lim\hspace{0.17em}sup}_{s\to \mathrm{\infty }} can be replaced with {lim}_{s\to \mathrm{\infty }}. Similarly, changing s with −s:

Next, set

Since \frac{\phi (s)}{{\mathrm{e}}^{{\mu }_{-}s}}\to 1 as s\to \mathrm{\infty } and \frac{\phi (s)}{{\mathrm{e}}^{{\mu }_{+}s}}\to 1 as s\to -\mathrm{\infty } we have then

and

From (C2), we know \omega (\gamma (s))>0 for any s\in \mathbb{R}, so 〈\mathrm{\nabla }\omega (x_0),{\gamma }_{\pm }〉\ge 0. Hence condition (C3) means that \gamma (s) tends transversally to the singular manifold {\omega }^{-1}(0) at x_0.

As in [19] it is easily seen that

and that \frac{{\gamma }^{\prime }(s)}{\omega (\gamma (s))} solves the equation

So \frac{{\gamma }^{\prime }(s)}{\omega (\gamma (s))} is a bounded solution of (14). Next, setting as in [19]

and x_h(t)=\gamma ({\theta }^{-1}(t)), it is easily seen that x_h(t) satisfies \omega (x)x^{\prime }=F(x) whose linearization along x_h(t) is

i.e.

Note, then, that (14) is derived from (16) with the change x(s)=z(\theta (s)). This fact should clarify why we need to consider the linear system (14) instead of x^{\prime }=F^{\prime }(\gamma (s))x. However, see [19] for a remark concerning the space of bounded solutions of (14) and that of the equation x^{\prime }=F^{\prime }(\gamma (s))x.

We now prove that \frac{{\gamma }^{\prime }(s)}{\omega (\gamma (s))} is the unique solution of equation (14) which is bounded on ℝ. This is a kind of nondegeneracy of \gamma (s).

Lemma 2.1 Assume (C2) and (C3) hold. Then, up to a multiplicative constant, \frac{{\gamma }^{\prime }(s)}{\omega (\gamma (s))} 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 {\mu }_{+}-{\mu }_{-} and -{\mu }_{-}. Let \mu :=\frac{{\mu }_{+}}{2}, then the linear map

has the eigenvalues ±μ; moreover, since

for two positive constants , it follows that {\gamma }_0(s):=\frac{{\gamma }^{\prime }(s)}{\omega (\gamma (s))}{\mathrm{e}}^{-\mu s} is a solution of

satisfying

for all 0\le s_1\le s_2. Then (17) satisfies the assumptions of [[19], Theorem 5.3] and hence its conclusion with rank{P}_{+}=1, that is, the fundamental matrix X_{+}(s) of (17) satisfies

where . However, it is well known (see [23–25]) that \mathcal{R}{P}_{+} is the space of initial conditions for the bounded solutions on [0,\mathrm{\infty }[ of (17) that, then, tend to zero as s\to \mathrm{\infty } at the exponential rate {\mathrm{e}}^{-\mu s}. As a consequence a solution u(s) of (17) is bounded on [0,\mathrm{\infty }[ if and only if u(s){\mathrm{e}}^{\mu s} is a bounded solution of (14). Then we conclude that the space of solutions of (14) that are bounded on [0,\mathrm{\infty }[ is one dimensional.

Incidentally, since the fundamental matrix of (14) is X(s)=X_{+}(s){\mathrm{e}}^{\mu s}, we note that it satisfies

Using a similar argument in {\mathbb{R}}_{-}=]-\mathrm{\infty },0] with , and [[19], Theorem 5.4] instead of [[19], Theorem 5.3] with {\mu }^{\ast }=-\mu we see that (14) has at most a one dimensional space of solutions bounded in ℝ. More precisely, \tilde{\mu } with , and a projection P_{-} on {\mathbb{R}}^2 exists such that

and dim\mathcal{N}{P}_{-}=1. Since \frac{{\gamma }^{\prime }(s)}{\omega (\gamma (s))} is a solution of (14) bounded on ℝ we deduce that \mathcal{R}{P}_{+}=\mathcal{N}{P}_{-}=span\{\frac{{\gamma }^{\prime }(0)}{\omega (\gamma (0))}\} and the result follows. □

We conclude this section with a remark about condition (c) in [[19], Theorem 5.3]. Consider a system in {\mathbb{R}}^n such as

Then the following result holds.

Theorem 2.2 Suppose the following hold:

1. (i)

   D has two simple eigenvalues and all the other eigenvalues of D have either real part less than {\mu }_{\ast } or greater than {\mu }^{\ast };

2. (ii)

   ;

3. (iii)

   A(s)\to 0 as s\to \mathrm{\infty }.

Then there are as many solutions x(t) of (18) satisfying

as the dimension of the space of the generalized eigenvectors of the matrix D with real parts less than or equal to {\mu }_{\ast }; here k_1,k_2>0 are two suitable positive constants. Similarly there are as many solutions of (18) such that

for suitable constants {\tilde{k}}_1,{\tilde{k}}_2>0, as the dimension of the space of the generalized eigenvectors of the matrix D with real parts greater than or equal to {\mu }^{\ast }.

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 D_{-} have real parts less than {\mu }_{\ast } and those of D_{+} have real parts greater than or equal to {\mu }^{\ast }. So the system reads

where a_{11}(t)\in \mathbb{R} and A_{ij}(t) are matrices (or vectors) of suitable orders. Setting y_i(t)={\mathrm{e}}^{-{\mu }_{\ast }t}{x}_i(t) we get

Now we observe that y(t) is a solution of (22) bounded at +∞ if and only if x(t) is a solution of (21) which is bounded on ℝ when multiplied by {\mathrm{e}}^{-{\mu }_{\ast }t}. Moreover, since |a_{11}(t)|, |A_{12}(t)|, and |A_{13}(t)| belong to L^1(\mathbb{R}), the limit {lim}_{t\to +\mathrm{\infty }}{y}_1(t) exists for any solution y(t) of (22) bounded on {\mathbb{R}}_{+}. So, let us fix t_0>0 and take t\ge t_0. If y(t) is a solution of (22) bounded at +∞ it must be, by the variation of constants formula,

where y_2^0=y_2(t_0) and y_1^{\mathrm{\infty }}={lim}_{t\to +\mathrm{\infty }}{y}_1(t). Note that since and \sigma (D_{+}-{\mu }_{\ast }\mathbb{I})\subset \{\lambda \in \mathbb{C}\mid \mathrm{\Re }\lambda >0\} and a_{11}(t), A_{ij}(t) are bounded, we can interpret (22) as a fixed point theorem in the Banach space of bounded function on [t_0,+\mathrm{\infty }[:

with the obvious norm. Since a_{11}(t),A_{ij}(t)\in L^1({\mathbb{R}}_{+}) we see that the map (23) is a contraction on B, provided t_0 is sufficiently large, and then, for any given (y_1^{\mathrm{\infty }},y_2^0), it has a unique solution y(t,y_1^{\mathrm{\infty }},y_2^0)\in B. Note that a priori y(t,y_1^{\mathrm{\infty }},y_2^0) is defined only on [t_0,+\mathrm{\infty }[ but of course we can extend it to [0,+\mathrm{\infty }[ going backward with time. We now prove that positive constants exist such that c_1\le |y(t,y_0)|\le c_2 fox any t\ge 0. Let . We have

and then

So, for any \delta >0 let t_1 be such that {sup}_{t\ge t_1}|A_{ij}(t)|\le \delta and set {sup}_{t\ge t_1}|y_j(t)|={\overline{y}}_j. We have

with . Taking the limit as t\to +\mathrm{\infty } we get

Since \delta \to 0 as t_1\to +\mathrm{\infty }, from the above it follows that {lim}_{t\to \mathrm{\infty }}|y_2(t)|=0. Similarly we get

where and then {lim}_{t\to \mathrm{\infty }}|y_3(t)|=0. As a consequence we obtain {lim}_{t\to \mathrm{\infty }}|y(t)|-|y_1(t)|=0 and then

So, provided we take y_1^{\mathrm{\infty }}\ne 0 we see that eventually (i.e. for t\ge \overline{t}, for some \overline{t}>0)

and the existence of c_1,c_2>0 such that

for all t\ge 0 follows from the fact that |y(t)| cannot vanish in any bounded interval. Finally since |x(t)|=|y(t)|{\mathrm{e}}^{{\mu }_{\ast }t} we get, for 0\le s\le t,

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 {\mu }_{\ast } is a simple eigenvalue of D and all the others either have real parts less than {\mu }_{\ast } or \ge {\mu }^{\ast } (i.e. we do not need that {\mu }^{\ast } is simple). Similarly inequalities of (20) hold if {\mu }^{\ast } is a simple eigenvalue of D and all the others either have real parts greater than {\mu }^{\ast } or \le {\mu }_{\ast } (i.e. we do not need that {\mu }_{\ast } is simple).

1. (ii)

   Note that a result related to Theorem 2.2 has been proved in [26].

It follows from (11)-(12) that \gamma (s)-x_0=O({\mathrm{e}}^{{\mu }_{\mp }s}) as s\to \pm \mathrm{\infty } then, since {\gamma }^{\prime }(s)=F(\gamma (s))=O(\gamma (s)-x_0) we obtain {\gamma }^{\prime }(s)=O({\mathrm{e}}^{{\mu }_{\mp }s}). Furthermore, from (13) we also get:

As a consequence

Since \omega (\gamma (s))>0 it follows that \theta :\mathbb{R}\to ]T_{-},T_{+}[ is a strictly increasing diffeomorphism (see (15) for the definition of \theta (s)). Then x_h(t):=\gamma ({\theta }^{-1}(t)) satisfies (8) on the interval ]T_{-},T_{+}[ and

Moreover (see (13))

Hence x_0 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 ]T_{-}+\alpha ,T_{+}+\alpha [, for some \alpha =\alpha (\epsilon ), and tend to (x_0,x_0) at the same rate as (x_h(t),x_h(t)). To this end we first perform a change of the time variable as follows. Set

and plug z_j(s)=x_j(\alpha +\theta (s)) in (7). We get

Since we are looking for solutions of (7) tending to (x_0,x_0) at the same rate as \gamma (s), in (24) we make the change of variables

where \eta (s) is the bounded function \frac{\gamma (s)-x_0}{\phi (s)}. Since

for a suitable constant K_1>0 and any s\in \mathbb{R}, |y|\le 1 we get, using (C3), (26):

for |s|>0 large and sufficiently small. Then (27) and \omega (\gamma (t))>0 imply the existence of M>0 and \delta >0 so that

for any s\in \mathbb{R} and |y|\le \delta. Now plugging (25) into (24) we derive the equations

From (11) it follows that

Next we note that from G_j(x_0,x_0,t,\epsilon ,\kappa )=0 it follows that the quantities

are bounded uniformly in s\in \mathbb{R} and \kappa \in {\mathbb{R}}^m, y_1, y_2, ε bounded.

The linearization of (28) at y=0, \epsilon =0 is

Taking the limit as s\to +\mathrm{\infty } we get the systems

Similarly taking the limit as s\to -\mathrm{\infty } we get the systems

From the proof of Lemma 2.1 (see also [[1], Lemma 3.1]) we know that (30) has the positive simple eigenvalues {\mu }_{+}-{\mu }_{-} and -{\mu }_{-}, and (31) has the negative simple eigenvalues {\mu }_{-}-{\mu }_{+} and -{\mu }_{+}. From the roughness of exponential dichotomies it follows that both equations in (29) have an exponential dichotomy on both {\mathbb{R}}_{+} and {\mathbb{R}}_{-} with projections, resp. P_{+}=0 and P_{-}=\mathbb{I}. Hence (see also [19]) all solutions of the system

adjoint to (29), are bounded as |s|\to \mathrm{\infty }. We let {\psi }_1(s) and {\psi }_2(s) be any two linearly independent solutions of (32).