Melnikov theory for weakly coupled nonlinear RLC circuits

R E S E A R C H

Open Access

Melnikov theory for weakly coupled

nonlinear RLC circuits

Flaviano Battelli

_{and Michal Feˇckan}

Dedicated to Professor Ivan Kiguradze

*_{Correspondence:} battelli@dipmat.univpm.it 1_{Department of Industrial} Engineering and Mathematical Sciences, Marche Polytecnic University, Via Brecce Bianche 1, Ancona, 60131, Italy

Full list of author information is available at the end of the article

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: Primary 34A09; secondary 34C23; 37G99

Keywords: implicit ode; perturbation; Melnikov method; RLC circuits

1 Introduction

In [], motivated by [, ], the equation modeling nonlinear RLC circuits

u+f(u)+εγu+f(u)+u+εh(t+α,u,ε) =  ()

has been studied. It is assumed thatf(u) andh(t,u,ε) are smooth functions withf(u) at least quadratic at the origin and satisfying suitable assumptions. Settingv= (u+f(u))the equation reads

 +f(u)u=v,

v= –u–εh(t+α,u,ε) +γv.

()

It is assumed that, for someu∈R, we havef(u) +  =  anduf(u) < . So forε=  () has the Hamiltonian

H(u,v) =v+ 

u

u

σ +f(σ)dσ

passing through (u, ). Clearly∇H(u, ) =

 

and the Hessian ofHat (u, ) is

H_H(u, ) =

uf(u) 

 

,

so that the conditionuf(u) <  means that (u, ) is a saddle forH. Multiplying the second equation by  +f(u) we get the system

 +f(u)

u v

=

v

–( +f(u)){u+ε[h(t+α,u,ε) +γv]}

. ()

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

A(x)x=f(x) +εh(t,x,ε,κ), (ε,κ)∈R×Rm, ()

withA(u,v) =+f(u) 

 

. Obviously,detA(u,v) =  +f(u) vanishes on the line (u,v) and the conditionf(u)=  implies that the lineu=uconsists of noncritical -singularities for () (see [, p.]). LetNLdenote the kernel of the linear mapLandRLits range. ThenRA(u, ) is the subspace having zero first component and then the right hand side of () belongs toRA(u, ) if and only ifv= . So all the singularities (u,v) withv=  are impasse points while (u, ) is a so calledI-point (see [, pp.-]). Quasilinear implicit differential equations, such as (), find applications in a large number of physical sciences and have been studied by several authors [–]. On the other hand, there are many other works on implicit differential equations [–] dealing with more general implicit differential systems by using analytical and topological methods.

Passing from () to (), in the general case, it corresponds to multiplying () by the ad-jugate matrixAa_(x):

ω(x)x=Aa(x)f(x) +εh(t,x,ε,κ),

whereω(x) =detA(x). Here we note thatAandxmay 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 forAandx. Basic assumptions in [] areω(x) = ,ω(x)=  andAa(x)f(x) = ,Aa(x)h(t,x,ε,κ) =  for somex(that is,x is anI-point for ()) and the existence of a solutionx(t) in a bounded intervalJtending to xasttends to the endpoints ofJ.

It is well known [, ] thatω(x) =  andω(x)=  imply

dimNA(x) = , RAa(x) =NA(x), and NAa(x) =RA(x), ()

and thenAa_(x

)f(x) =  is equivalent to the fact thatf(x)∈RA(x).

LetF(x) :=Aa_{(x)f(x). It has been proved in [] that () implies thatrankF(x}

) is at most . So, ifx∈Rn_{, withn>  thenx=x}

cannot be hyperbolic for the mapx→F(x)x. In this paper we study coupled IODEs such as

A(x)x=f(x) +εg(t,x,x,ε,κ),

A(x)x=f(x) +εg(t,x,x,ε,κ),

()

general equation () with, among other things,

A(x) =

A(x)   A(x)

, x= (x,x)

hence detA(x) =detA(x)detA(x) satisfies detA(x,x) = , (detA)(x,x) =  and (detA)(x,x)= . Thus (x,x) is not aI-point. Multiplying the first equation byAa(x) and the second byAa_(x) we obtain the system

ω(x)x=F(x) +εG(x,x,t,ε,κ),

ω(x)x=F(x) +εG(x,x,t,ε,κ).

()

We assume thatω(x) :=detA(x),F(x) andGj(x,x,t,ε,κ) satisfy the following assump-tions:

(C) F∈C_(R_,R_),ω∈C_(R_{,R)and the unperturbed equation}

ω(x)x=F(x) ()

possesses a noncritical singularity atx,i.e.ω(x) = andω(x)= . (C) F(x) = and the spectrumσ(F(x)) ={μ±}withμ–<  <μ+, and

x=F(x)

has a solutionγ(s)homoclinic tox, that is,lims→±∞γ(s) =x, andω(γ(s))= for anys∈R. Without loss of generality, we may, and will, assumeω(γ(s)) > for any s∈R. Moreover,Gi∈C(R+m,R),i= , are-periodic intwith

Gi(x,x,t,ε,κ) = for anyt∈R,κ∈Rmandεsufficiently small. (C) Letγ_±be the eigenvectors ofF(x)with the eigenvaluesμ∓, resp. Then

∇ω(x),γ±> (or elseω(x)γ±> ).

From (C) we see that(s) :=_γγ₍(_ss₎)is a bounded solution of the equation

x_=F(x),

x_=F(x)

()

and thatxpersists as a singularity of (). So this paper is a continuation of [, ], but here we study more degenerate IODE.

The objective of this paper is to give conditions, besides (C)-(C), assuring that for

|ε| , the coupled equations () has a solution in a neighborhood of the orbit{(s)|s∈

We emphasize the fact that Melnikov technique is useful to predict the existence of transverse homoclinic orbits in mechanical systems [, ] 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 [, ] we note that sinceγ(s)→xas|s| → ∞thenγ(s) is a bounded so-lution of the linear equationx=F(γ(s))x. Hence|γ(s)| ≤ke–μ|s|_{for someμ> . We get} then, fors≥,

γ(s) –x ≤

∞

s

γ(s) ds≤μ–ke–μs_.

So

lim sup

s→∞

log|γ(s) –x|

s ≤–μ< .

From [, Theorem ., p. and Theorem ., p.] it follows that

lim sup

s→∞

log|γ(s) –x|

s =μ–< , ()

and there exist a constantδ>  and a solutionγ+eμ–sofx=F(x)xsuch that

γ(s) –x–γ+eμ–s =O

e(μ––δ)s_{, ass→ ∞.}

Note thatγ+=  since otherwiseγ(s) –x=O(e(μ––δ)s), contradicting (). Henceγ+is an eigenvector of the eigenvalueμ–ofF(x). We have then

γ(s) –x eμ–s –γ+

≤ce–δs

for a suitable constantc≥. As a consequence,

lim

s→∞

γ(s) –x eμ–s =γ+.

Next

|γ+|–ce–δs≤|

γ(s) –x|

eμ–s ≤ |γ+|+ce

–δs_.

Taking logarithms, dividing bysand lettings→ ∞we get

lim

s→∞

log|γ(s) –x| s =μ–,

that is, in ()lim sup_s→∞can be replaced withlims→∞. Similarly, changingswith –s:

lim

s→–∞

Next, set

ϕ(s) := 

e–μ–s_{+ e}–μ+s. ()

Since_eϕμ(–s)s → ass→ ∞and_eϕμ(+s)s→ ass→–∞we have then

lim

s→±∞

γ(s) –x

ϕ(s) =γ±=  ()

and

lim

s→±∞

ω(γ(s)) ϕ(s) =s→±∞lim

ω(x)(γ(s) –x) +o(γ(s) –x)

eμ∓s =

∇ω(x),γ±

.

From (C), we knowω(γ(s)) >  for anys∈R, so∇ω(x),γ± ≥. Hence condition (C)

means thatγ(s) tends transversally to the singular manifoldω–() atx. As in [] it is easily seen that

lim

s→±∞ γ(s) ω(γ(s))=

F(x)γ± ω(x)γ±

= μ∓γ± ω(x)γ±

=  ()

and that _ωγ_(γ(₍s_s)₎₎solves the equation

x=

Fγ(s)–F(γ(s)) ω(γ(s))ω

_{γ(s)x=Fγ(s)x–}ω(γ(s))x

ω(γ(s)) F

γ(s). ()

So _ωγ_(γ(₍s_s)₎₎is a bounded solution of (). Next, setting as in []

θ(s) :=

s



ωγ(τ)dτ ()

andxh(t) =γ(θ–(t)), it is easily seen thatxh(t) satisfiesω(x)x=F(x) whose linearization alongxh(t) is

Fxh(t)

z=x_h(t)ωxh(t)

z+ωxh(t)

z=Fxh(t)

ω(xh(t))z ω(xh(t))

+ωxh(t)

z

i.e.

ωxh(t)

z=Fxh(t)

z–Fxh(t)

ω(xh(t))z ω(xh(t))

. ()

Note, then, that () is derived from () with the changex(s) =z(θ(s)). This fact should clarify why we need to consider the linear system () instead ofx=F(γ(s))x. However, see [] for a remark concerning the space of bounded solutions of () and that of the equationx=F(γ(s))x.

Lemma . Assume(C)and(C)hold.Then,up to a multiplicative constant, _ωγ_(γ(₍s_s)₎₎ is the unique solution of ()which is bounded onR.

Proof From [, Lemma .] it follows that the linear map:

x→

F(x) – μ–γ+ ω(x)γ+

ω(x) –μ–I

x

has the simple eigenvaluesμ+–μ–and –μ–. Letμ:=μ_+, then the linear map

x→

F(x) – μ–γ+ ω(x)γ+

ω(x) –μI

x

has the eigenvalues±μ; moreover, since

c≤ | γ(s)| ω(γ(s))≤c

for two positive constants  <c<c, it follows thatγ(s) := γ

_(s)

ω(γ(s))e–

μs_{is a solution of}

x=

Fγ(s)–F(γ(s)) ω(γ(s))ω

_{γ(s)–μIx ()}

satisfying

c c

γ(s) ≤ γ(s) eμ(s–s)≤ c c

γ(s)

for all ≤s≤s. Then () satisfies the assumptions of [, Theorem .] and hence its conclusion withrankP+= , that is, the fundamental matrixX+(s) of () satisfies

_X+(s)P+X+–(s)≤ke–μ(s–s), ≤s≤s,

X+(s)(I–P+)X+–(s)≤keμ˜(s–s), ≤s≤s,

where ≤ ˜μ<μ. However, it is well known (see [–]) thatRP+is the space of initial conditions for the bounded solutions on [,∞[ of () that, then, tend to zero ass→ ∞ at the exponential rate e–μs_{. As a consequence a solutionu(s) of () is bounded on [,∞[} if and only if u(s)eμs_{is a bounded solution of (). Then we conclude that the space of} solutions of () that are bounded on [,∞[ is one dimensional.

Incidentally, since the fundamental matrix of () isX(s) =X+(s)eμs, we note that it sat-isfies

X(s)P+X–(s)≤k, ≤s≤s,

X(s)(I–P+)X–(s)≤ke(μ+μ˜)(s–s), ≤s≤s.

of solutions bounded inR. More precisely,μ˜ withμ<μ˜ < , and a projectionP–onR exists such that

X(s)(I–P–)X–(s)≤k, s≤s≤,

X(s)P–X–(s)≤ke(μ+μ˜)(s–s), s≤s≤,

anddimNP–= . Since γ

_(s)

ω(γ(s)) is a solution of () bounded onRwe deduce thatRP+=

NP–=span{ γ

_()

ω(γ())}and the result follows.

We conclude this section with a remark about condition (c) in [, Theorem .]. Con-sider a system inRnsuch as

x=D+A(s)x. ()

Then the following result holds.

Theorem . Suppose the following hold:

(i) Dhas two simple eigenvaluesμ∗<μ∗and all the other eigenvalues ofDhave either real part less thanμ∗or greater thanμ∗;

(ii) _∞A(s)ds<∞; (iii) A(s)→ass→ ∞.

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

k x(s) ≤ x(t) e–μ∗(t–s)≤k x(s) , for any≤s≤t, ()

as the dimension of the space of the generalized eigenvectors of the matrix D with real parts less than or equal toμ_∗;here k,k> are two suitable positive constants.Similarly there are as many solutions of ()such that

˜

k x(s) ≤ x(t) e–μ

∗_(t–s)

≤ ˜k x(s) , for any≤s≤t, ()

for suitable constantsk˜,k˜> ,as the dimension of the space of the generalized eigenvectors of the matrix D with real parts greater than or equal toμ∗.

Proof We prove the first statement concerning (). By a similar argument () is handled. Changing variables we may assume that

D=

⎛ ⎜ ⎝

μ_∗  

 D– 

  D+

⎞ ⎟ ⎠

and the eigenvalues ofD– have real parts less thanμ∗ and those ofD+have real parts greater than or equal toμ∗. So the system reads

x_=μ∗x+a(t)x+A(t)x+A(t)x,

x_=D–x+A(t)x+A(t)x+A(t)x,

x_=D+x+A(t)x+A(t)x+A(t)x,

where a(t)∈R andAij(t) are matrices (or vectors) of suitable orders. Setting yi(t) = e–μ∗t_x

i(t) we get

y_=a(t)y+A(t)y+A(t)y,

y_= (D––μ∗I)y+A(t)y+A(t)y+A(t)y,

y_= (D+–μ∗I)y+A(t)y+A(t)y+A(t)y.

()

Now we observe thaty(t) is a solution of () bounded at +∞if and only ifx(t) is a solution of () which is bounded onRwhen multiplied by e–μ∗t_{. Moreover, since|a}

(t)|,|A(t)|, and|A(t)|belong toL(R), the limitlimt→+∞y(t) exists for any solutiony(t) of () bounded onR+. So, let us fixt>  and taket≥t. Ify(t) is a solution of () bounded at +∞it must be, by the variation of constants formula,

y(t) =y∞ –

_∞

t

a(s)y(s) +A(s)y(s) +A(s)y(s)

ds,

y(t) = e(D––μ∗I)(t–t)y

+

t

t

e(D––μ∗I)(t–s)_A

(s)y(s) +A(s)y(s) +A(s)y(s)

ds,

y(t) = –

∞

t

e(D+–μ∗I)(t–s)_A

(s)y(s) +A(s)y(s) +A(s)y(s)

ds,

()

wherey

=y(t) andy∞ =limt→+∞y(t). Note that sinceσ(D––μ∗I)⊂ {λ∈C| λ< } andσ(D+–μ∗I)⊂ {λ∈C| λ> }anda(t),Aij(t) are bounded, we can interpret () as a fixed point theorem in the Banach space of bounded function on [t, +∞[:

B:=y(t)∈C[t,∞[

|sup y(t) <∞

with the obvious norm. Sincea(t),Aij(t)∈L(R+) we see that the map () is a contrac-tion onB, providedtis sufficiently large, and then, for any given (y∞ ,y), it has a unique solutiony(t,y∞_ ,y

)∈B. Note thata priori y(t,y∞ ,y) is defined only on [t, +∞[ but of course we can extend it to [, +∞[ going backward with time. We now prove that positive constants  <c<cexist such thatc≤ |y(t,y)| ≤cfox anyt≥. Lett<t<t. We have

y(t) = e(D––μ∗I)(t–t)y

+

t

t

e(D––μ∗I)(t–s)_A

(s)y(s) +A(s)y(s) +A(s)y(s)

ds

and then

y(t) = e(D––μ∗I)(t–t)y+

t

t

e(D––μ∗I)(t–s)_A

(s)y(s) +A(s)y(s) +A(s)y(s)

ds

+

t

t

e(D––μ∗I)(t–s)_A

(s)y(s) +A(s)y(s) +A(s)y(s)

ds

= e(D––μ∗I)(t–t)_y

(t) +

t

t

e(D––μ∗I)(t–s)_A

(s)y(s) +A(s)y(s) +A(s)y(s)

So, for anyδ>  lettbe such thatsupt≥t|Aij(t)| ≤δand setsupt≥t|yj(t)|=y¯j. We have

¯

y=sup

t≥t

y(t) ≤keα(t–t)y¯+

t

t

keα(t–s)δ(y¯+y¯+y¯)ds

≤k

¯

y+ δ

α(y¯+y¯+y¯)

eα(t–t)₊ kδ

|α|(¯y+y¯+y¯)

withmax{μ|μ∈σ(D––μ∗I)}<α< . Taking the limit ast→+∞we get

¯

y≤ δ

|α|(¯y+¯y+y¯).

Sinceδ→ ast→+∞, from the above it follows thatlimt→∞|y(t)|= . Similarly we get

|y| ≤kδβ–(y¯+y¯+y¯),

where  <β<min{μ|μ∈σ(D+–μ∗I)}and thenlimt→∞|y(t)|= . As a consequence we obtainlim_t→∞|y(t)|–|y(t)|=  and then

lim

t→∞ y(t) = y ∞

 .

So, provided we takey∞_ =  we see that eventually (i.e.fort≥ ¯t, for somet¯> )

|y∞_ |

 ≤ y(t) ≤   y

∞



and the existence ofc,c>  such that

c≤ y(t) ≤c

for allt≥ follows from the fact that|y(t)|cannot vanish in any bounded interval. Finally since|x(t)|=|y(t)|eμ∗t_{we get, for ≤s≤t,}

|x(t)|

|x(s)|=

|y(t)|

|y(s)|e

μ∗(t–s) _⇒ c c

eμ∗(t–s)≤|x(t)|

|x(s)|≤ c c

eμ∗(t–s)

i.e.

c c

x(s) ≤ x(t) e–μ∗(t–s)_≤c c

x(s) .

The proof is complete.

Remark . (i) It follows from the proof of Theorem . that inequalities of () also hold replacing (i) with the weaker assumption thatμ∗ is a simple eigenvalue ofDand all the

others either have real parts less thanμ∗or≥μ∗(i.e.we do not need thatμ∗is simple). Similarly inequalities of () hold ifμ∗is a simple eigenvalue ofDand all the others either have real parts greater thanμ∗or≤μ∗(i.e.we do not need thatμ∗is simple).

3 Solutions asymptotic to the fixed point

It follows from ()-() thatγ(s) –x=O(eμ∓s) ass→ ±∞then, sinceγ(s) =F(γ(s)) = O(γ(s) –x) we obtainγ(s) =O(eμ∓s). Furthermore, from () we also get:

ωγ(s)=Oγ(s)=Oeμ∓s.

As a consequence

T±:= ±∞



ωγ(τ)dτ <∞.

Sinceω(γ(s)) >  it follows thatθ :R→]T–,T+[ is a strictly increasing diffeomorphism (see () for the definition of θ(s)). Thenxh(t) :=γ(θ–(t)) satisfies () on the interval ]T–,T+[ and

lim

t→T±xh(t) =x.

Moreover (see ())

lim

t→t±x

h(t) =_tlim_→t±

F(xh(t)) ω(xh(t))

= lim

s→±∞ F(γ(s)) ω(γ(s))=μ∓

γ±

ω(x)γ± = .

Hencexis not anI-point of (). In this paper we want to look for solutions of the coupled equation () that belong to a neighborhood of{(xh(t),xh(t))|T–<t<T+}, they are defined in the interval ]T–+α,T++α[, for someα=α(ε), and tend to (x,x) at the same rate as (xh(t),xh(t)). To this end we first perform a change of the time variable as follows. Set

t=α+θ(s)∈]T–+α,T++α[

and plugzj(s) =xj(α+θ(s)) in (). We get

ω(zj)zj=ω

γ(s)F(zj) +εGj

z,z,α+θ(s),ε,κ

, j= , . ()

Since we are looking for solutions of () tending to (x,x) at the same rate asγ(s), in () we make the change of variables

zj(s) =γ(s) +ϕ(s)yj(s) =x+ϕ(s)

η(s) +yj(s)

, j= , , ()

whereη(s) is the bounded function γ(s)–x

ϕ(s) . Since

ωx+ϕ(s)

η(s) +y

≥∇ω(x),ϕ(s)

η(s) +y–K ϕ(s)

η(s) +y 

=ϕ(s)∇ω(x),η(s) +y

–Kϕ(s) η(s) +y 

()

for a suitable constantK>  and anys∈R,|y| ≤ we get, using (C), ():

ωx+ϕ(s)

η(s) +y≥ ϕ(s)

for|s|>  large and|y|<δsufficiently small. Then () andω(γ(t)) >  imply the existence ofM>  andδ>  so that

ωx+ϕ(s)

η(s) +y≥Mϕ(s)

for anys∈Rand|y| ≤δ. Now plugging () into () we derive the equations

y_j= ω(γ(s)) ϕ(s)ω(γ(s) +ϕ(s)yj)

Fγ(s) +ϕ(s)yj

–F(γ(s)) ϕ(s) –

ϕ(s) ϕ(s)yj

+ε ω(γ(s))

ϕ(s)ω(γ(s) +ϕ(s)yj) Gj

γ(s) +ϕ(s)y,γ(s) +ϕ(s)y,θ(s) +α,ε,κ

,

j= , . ()

From () it follows that

ϕ(s) ϕ(s) =

μ–e–μ–(s)+μ+e–μ+(s)

e–μ–(s)_{+ e}–μ+(s) →μ∓ ass→ ±∞.

Next we note that fromGj(x,x,t,ε,κ) =  it follows that the quantities

Gj(γ(s) +ϕ(s)y,γ(s) +ϕ(s)y,α+θ(s),ε,κ) ϕ(s)

=Gj(x+ϕ(s)(η(s) +y),x+ϕ(s)(η(s) +y),α+θ(s),ε,κ)

ϕ(s) , j= , 

are bounded uniformly ins∈Randκ∈Rm,y,y,εbounded. The linearization of () aty= ,ε=  is

y_j=

Fγ(s)–F(γ(s))ω

_(γ(s))

ω(γ(s)) – ϕ(s) ϕ(s)I

yj, j= , . ()

Taking the limit ass→+∞we get the systems

y_j=

F(x) – μ–γ+ ω(x)γ+

ω(x) –μ–I

yj, j= , . ()

Similarly taking the limit ass→–∞we get the systems

y_j=

F(x) – μ+γ– ω(x)γ–

ω(x) –μ+I

yj, j= , . ()

From the proof of Lemma . (see also [, Lemma .]) we know that () has the positive simple eigenvaluesμ+–μ–and –μ–, and () has the negative simple eigenvaluesμ––μ+ and –μ+. From the roughness of exponential dichotomies it follows that both equations in () have an exponential dichotomy on bothR+andR–with projections, resp.P+=  andP–=I. Hence (see also []) all solutions of the system

y= –

Fγ(s)∗–ω

_(γ(s))∗_F(γ(s))∗

ω(γ(s)) – ϕ(s) ϕ(s)I

adjoint to (), are bounded as|s| → ∞. We letψ(s) andψ(s) be any two linearly inde-pendent solutions of ().

4 Melnikov function and the original equation

In this section we will give a condition for solving () for y(t),y(t) near the solution y(t) =y(t) =  of the same equation withε= . Writing

F(y) :=y– ω(γ(s))

ϕ(s)ω(γ(s) +ϕ(s)y)F

γ(s) +ϕ(s)y+F(γ(s)) ϕ(s) +

ϕ(s)

ϕ(s)y ()

and

Hj(y,y,η,ε) := –

ω(γ(s)) ϕ(s)ω(γ(s) +ϕ(s)yj)

×Gj

γ(s) +ϕ(s)y,γ(s) +ϕ(s)y,θ(s) +α,ε,κ

, j= , , ()

we look for solutionsy(t),y(t) :R→Rof

F(y) +εH(y,y,η,ε) = ,

F(y) +εH(y,y,η,ε) = 

()

in the Banach space ofC-functions onR, bounded together with their derivatives and with small norms. We observe thatF() =  and equationF()y=  reads

y=

Fγ(s)–F(γ(s))ω

_(γ(s))

ω(γ(s)) – ϕ(s) ϕ(s)

y. ()

In Section  (see also [, ]) we have seen that () has an exponential dichotomy on both

R+andR+with projectionsP+=I–P–= . So the only bounded solutiony(t) ofF()y=  isy(t) = . In other wordsN F() ={}. So we are lead to prove the following.

Theorem . Let Y,X be Banach spaces,ε∈Rbe a small parameter andη∈Rd.Let

F:Y→X,H,:Y×Y×Rd+→X, (y,y,η,ε)→H,(y,y,η,ε)be C-functions such that

(a) F() = ; (b) N F() ={};

(c) there existψ, . . . ,ψd∈X∗such thatRF() ={ψ, . . . ,ψd}◦. SetM:Rd_→Rd_by

M(η) :=

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

ψH(, , ,η, ) .. .

ψdH(, , ,η, ) ψH(, , ,η, )

.. .

ψdH(, , ,η, )

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

and suppose there existsη¯∈Rd_{such thatM(η) = ¯ and the derivativeM(η)¯ is invertible.} Then there exist r> and unique C_{-functionη=η(ε),defined in a neighborhood ofε=} ∈Rsuch that

lim

ε→η(ε) =η¯ ()

and forη=η(ε),ε= , ()has a unique solution(y(ε),y(ε))∈Y×Y satisfying

y(ε),y(ε)≤r.

Moreover,yj(ε) =εyj(ε)for C-functionsyj(ε)∈Y and we have

F_()y

j() +Hj(, , ,η, ) = ,¯ j= , . ()

Proof We look for solutions (y,y,η) of () that are close to (y,y,η) = (, ,η). Let¯ P:X→Xbe the projection such thatRP=RF(). NotecodimRF() =d. From the implicit function theorem, we solve the projected equations

PF(y) +εPH(y,y,η,ε) = ,

PF(y) +εPH(y,y,η,ε) = 

for uniquey,=Y,(η,ε)∈Ysuch that

Y,(η, ) = ,

provided|ε| ≤εis sufficiently small andηin a fixed closed ball⊂Rdwithη¯∈

◦

. Note thatY,areC-smooth. SettingQ=I–P, we need to solve the bifurcation equations:

QFYj(η,ε)

+εQHj

Y(η,ε),Y(η,ε),η,ε

= , j= , . ()

Observe that

Qx=  ⇔ x∈RP=RF() ⇔ ψix=  for alli= , . . . ,d. ()

ThenQF() =  and so

QFYj(η,ε)

=Oj

ε, j= , ,

uniformly with respect toη. We conclude that () can be written as

εQHj(, ,η, ) = –QRj(η,ε), j= , , ()

where

Rj(η,ε) :=F

Yj(η,ε)

+εHj

Y(η,ε),Y(η,ε),η,ε

–Hj(, ,η, )

Note thatRj(η,ε) areC-functions of (η,ε) and thatε–Rj(η,ε) =O(ε) uniformly with re-spect toη, so

Rj(η,ε) :=

–ε–Rj(η,ε), ifε= ,

, ifε= 

isC_{in (η,ε). By (), system () is equivalent to}

M(η) =

ψiR(η,ε) ψiR(η,ε)

i=,...,d

=O(ε). ()

Because of the assumptions we can apply the implicit function theorem to () to obtain aC_{-functionη(ε) defined in a neighborhood ofε=  satisfying () and such that ()} holds. Setting

yj(ε) :=Yj

η(ε),ε, j= , 

we see thaty(ε),y(ε) are boundedC-solutions of () withη=η(ε) such thaty() = , y() = . Then we can writey(ε) =εy(ε),y(ε) =εy(ε)) for continuousy(ε),y(ε)∈Y where

Fεyj(ε)

+εHj

εy(ε),εy(ε),η(ε),ε

= , j= , .

Clearly () follows differentiating the above equality atε= . The proof is complete.

Remark . Note that, because ofM(η) = , () is equivalent to¯

PF()yj() +Hj(, , ,η, ) = ,¯ j= , ,

which has the unique solution

yj() = –

PF()–Hj(, , ,η, ),¯ j= , . ()

Now we apply Theorem . to () withF(y),H(y,y,η,ε),H(y,y,η,ε) as in (), () and

Y=C_bR,R, X=C_bR,R, η= (α,κ)∈R×Rm,

whereCk_b(R,R_{) is the Banach space ofC}k_{-functions bounded together with their} deriva-tives with the usualsup-norm.

We already observed thatF() =  andN F() = . Moreover,

RF_{() =x= (x}

,x)∈X

∞

–∞ψ ∗

i(s)xi(s)ds= ,i= , 

,

whereψi(s) have been defined in the previous Section . Sod=  andm= . We recall, from [], thatψj(s) =ϕ(s)vj(θ(s)) wherevj(t) are solutions of the adjoint equation of ():

ωxh(t)

v=ω

_(x

h(t))∗ ω(xh(t))

Fxh(t)

∗

v–Fxh(t)

∗

andvj() =RP∗–∩NP+∗. Hence () reads

M(α,κ) =

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

_∞

–∞

ψ_∗(s)

ϕ(s) G(γ(s),γ(s),α+θ(s), ,κ)ds

∞

–∞

ψ_∗(s)

ϕ(s) G(γ(s),γ(s),α+θ(s), ,κ)ds

∞

–∞

ψ_∗(s)

ϕ(s)G(γ(s),γ(s),α+θ(s), ,κ)ds

∞

–∞

ψ_∗(s)

ϕ(s)G(γ(s),γ(s),α+θ(s), ,κ)ds

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

or passing to timet=θ(s):

M(α,κ) =

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

T+

T– v ∗

(t)

G(xh(t),xh(t),t+α,,κ)

ω(xh(t)) dt T+

T– v ∗

(t)

G(xh(t),xh(t),t+α,,κ)

ω(xh(t)) dt T+

T– v ∗

(t)

G(xh(t),xh(t),t+α,,κ)

ω(xh(t)) dt T+

T– v ∗

(t)

G(xh(t),xh(t),t+α,,κ)

ω(xh(t)) dt ⎞ ⎟ ⎟ ⎟ ⎟

⎠. ()

A direct application of Theorem . gives the following.

Theorem . Let m= andM(α,κ)be given as in()where v(t),v(t)are two inde-pendent bounded solutions(onR)of the adjoint equation().Suppose thatα¯andκ¯exist so that

M(α,¯ κ¯) =  and ∂M

∂(α,κ)(α,¯ κ)¯ ∈GL(,R). ()

Then there existε¯> ,ρ> ,unique C-functionsα(ε)andκ(ε)withα() =α¯andκ() =κ¯, defined for|ε|<ε,¯ and a unique solution(z(s,ε),z(s,ε))of ()withα=α(ε),κ=κ(ε),  <|ε|<ε,¯ such that

sup

s∈R

zj(s,ε) –γ(s) ϕ(s)–<ρ, j= , . ()

Moreover,

sup

s∈R

zj(s,ε) –γ(s) ϕ(s)–=O(ε), j= , .

Remark . (i) Equation () implies

zj(s,ε) =γ(s) +εyj(s,ε)ϕ(s), j= , ,

forC_-functionsy

j(s,ε) withsup|ε|≤¯εsups∈R|yj(s,ε)|<∞. Then we have

zj(s,ε) =γ(s) +εyj(s,ε)ϕ(s) =γ(s) +εyj(s, )ϕ(s) +εwj(s,ε)ϕ(s)

withwj(s,ε) =yj(s,ε) –yj(s, ), solimε→sups∈R|wj(s,ε)|= . Hence

lim

ε→sup_s∈R

which gives a first order approximation ofzj(s,ε). Next,yj(s, ) can be computed using () adapted to this case. Henceyj(s, ) are bounded solutions of

y_j=

Fγ(s)–F(γ(s))ω

_(γ(s))

ω(γ(s)) – ϕ(s) ϕ(s)

yj+  ϕ(s)Gj

γ(s),γ(s),θ(s) +α,¯ κ¯.

Since () has exponential dichotomies on bothR+(with projectionP+= ) andR–(with projectionP–=I) it follows that

yj(s, ) =

–_s∞X(s)X–_(z) 

ϕ(z)Gj(γ(z),γ(z),θ(z) +α,¯ κ¯)dz fors≥,

s

–∞X(s)X–(z)ϕ(z)Gj(γ(z),γ(z),θ(z) +α,¯ κ)¯ dz fors≤.

()

X(s) is the fundamental solution of (). Note formulas () are well defined ats= ,i.e.,

yj(–, ) =yj(+, ), due to the first assumption of (). Next, passing to timet=θ(s) and takingzj(t) :=ϕ(θ–(t))yj(θ–(t), ), we get

zj(t) =

θ–(t)

–∞

ϕθ–(t)Xθ–(t)X–(z)ϕ(z)–Gj

γ(z),γ(z),θ(z) +α,¯ κ¯dz

=

t

T–

ϕθ–(t)Xθ–(t)X–θ–(u)ϕθ–(u)–Gj(xh(u),xh(u),u+α,¯ κ)¯ ω(xh(u))

du

forT–<t≤ and

zj(t) = –

T+

t

ϕθ–(t)Xθ–(t)X–θ–(u)ϕθ–(u)–Gj(xh(u),xh(u),u+α,¯ κ¯) ω(xh(u))

du

for ≤t<T+. Note thatzj(t) solves

ωxh(t)

z=Fxh(t)

z–Fxh(t)

ω(xh(t))z ω(xh(t))

+Gj

xh(t),xh(t),t+α,¯ κ¯

withsup_{t∈]T–,T+[}|zj(t)|ϕ(θ–(t))–<∞, andϕ(θ–(t))X(θ–(t)) is a fundamental solution of ().

(ii) Using (), the functionsxj(t,ε) :=zj(θ–(t–α(ε)),ε) are bounded solutions of () in the interval ]T–+α(ε),T++α(ε)[ such that

lim

ε→_t∈]sup_T

–,T+[ xj

t+α(ε),ε–xh(t) –εzj(t) ϕ

θ–(t)–ε–= .

Summarizing, we obtain the following corollary of Theorem ..

Corollary . Let m= andM(α,κ)be given as in()where v(t),v(t)are two inde-pendent bounded solutions(onR)of the adjoint equation().Suppose thatα¯andκ¯exist so that

M(α,¯ κ¯) =  and ∂M

∂(α,κ)(α,¯ κ)¯ ∈GL(,R).

 <|ε|<ε,¯ such that

sup

T–<t<T+ xj

t+α(ε),ε–xh(t) ϕ

θ–(t)–<ρ, j= , .

Moreover,

sup

T–<t<T+ xj

t+α(ε),ε–xh(t) ϕ

θ–(t)–=O(ε), j= , .

5 Applications to RLC circuits

In this section we study the coupled equations

u+u

+εγ

u+u

+u–ελ

u+u

+εsint= ,

u+u

+εγ

u+u

+u–ελ

u+u

+εχsin(t+) = ,

()

which is motivated by [, ]. Note that () is obtained by coupling two equations modeling nonlinear RLC circuits as in (). Hereγ,γ,λ,χand are positive parameters. Setting wj= (uj+uj),j= , , () reads

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

(u+ )u=w,

w_+εγw+u–ελw+εsint= , (u+ )u=w,

w_+εγw+u–ελw+εχsin(t+) = .

()

By solving the second and fourth equations of () forw_andw_, we get:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

(u+ )u=w,

w_= –u+εsint+λu+γw+ελ_ε(χ_λsin_–(t+)+λu+γw),

(u+ )u=w,

w= –u+εχsin(t+)+λu+γ_εw_λ+_–ελ(sint+λu+γw),

()

provided|ελ| = . Sinceω(u,w) =ω(u) = u+ , to write the system () in the form () with parameterκ= (γ,γ,λ) and (χ,) fixed, we have to multiply the second and fourth equation by u,+ , respectively, and we obtain the system

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

(u+ )u=w,

(u+ )w= –(u+ )u+ε(u+ )sint+λu+γw+ελ_ε(χ_λsin_–(t+)+λu+γw),

(u+ )u=w,

(u+ )w= –(u+ )u+ε(u+ )χsin(t+)+λu+γ_εw_λ+_–ελ(sint+λu+γw),

()

with (uncoupled) unperturbed equation forε=  (see ()):

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

(u+ )u=w,

(u+ )w= –(u+ )u, (u+ )u=w,

(u+ )w= –(u+ )u.

Neglecting left multiplicators u+  (u=u,u) in (), we obtain the following system (see condition (C)):

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

u_=w,

w_= –(u+ )u, u_=w,

w_= –(u+ )u.

()

Clearly, condition (C) is satisfied withx= (–_, ) and

F(x) =w, –(u+ )u∗, ω(x) = u+ , x= (u,w)∗.

The equationu+ (u+ )u=  has the prime integralu₊

u+u. A solutionu(s) satisfyinglim_s→∞u(s) = –_has to satisfyu+_u+u–_ = ⇔u+ (u– )(u+_)=  with the solution

u(s) =  

 – tanh s



bounded onR. Hence

γ(s) =

 

 – tanh s



, –cschssinh s



∗

.

Noteω(γ(s)) = u(s) +  =_sech_s> . From

F(x) =

 

 

,

we getμ_±=± andγ_±= (,±)∗. Since∇ω(x) = (, )∗, we derive∇ω(x),γ±=  > ,

and condition (C) holds as well. Next, we have

G(x,x,t,ε,κ) =

u+  ε_λ_{– }



sint+λu+γw+ελ(χsin(t+) +λu+γw)

,

G(x,x,t,ε,κ) =

u+  ε_λ_{– }



χsin(t+) +λu+γw+ελ(sint+λu+γw)

;

henceGi(x,x,t,ε,κ) =  and assumption (C) is also verified. Herexi= (ui,wi)∗,i= , . Furthermore by ()

θ(s) =

s



ωγ(τ)dτ=

s

  sech

τ

dτ= tanh s ,

soT±=± and

xh(t) =γ

θ–(t)= 

 –t 

,t

t  – 

∗

Thus

ωxh(t)

= 



 –t, ∇ωxh(t)

= (, )∗,

F(xh(t)) ω(xh(t))

= 



–t,t– ∗, Fxh(t)

=

 

t

 –  

.

So now () has the form

z_= t  –tz+

  –tz,

z_= –z,

()

which has the solutionx_h(t) = (–_t,_(t_{– )). In other words, we deal with}

z_= t  –tz

– 

 –tz, ()

possessing the solution _(t– ). Following [, p.], the second solution of () is given by

 

t– arctanht

– t

.

Consequently a fundamental matrix solution of () has the form

Z(t) =

–_t _((_tt_––)–tarctanh_t)

 (t– )



(t– )arctanh

t

–

t



.

NotedetZ(t) =_–_t. The adjoint system of () is (see ())

ζ_= t

t_{– }ζ+ζ,

ζ_=  t_{– }ζ

()

with the fundamental matrix solution

Z–∗(t) =

 ( –t

_)((t_{– )arctanh}t

– t)  (t

_{– )(t}_{– )}

 (t

_{– ) –} t(t

_{– )arctanh}t

  t(t

_{– )}

.

Notelimt→±Z–∗(t) =

_{ }

 

. Thus we take

v(t) =

 (t

_{– )(t– (t}_{– )arctanh}t

) 

(t– ) – 

t(t– )arctanh

t



and

v(t) =



(t– )(t– ) 

t(t– )

We now computeM(α,κ). We have (see the appendix)

T+

T–

v∗_(t)G(xh(t),xh(t),t+α, ,κ) ω(xh(t))

dt

= 

λ+ sinα(cos – sin) +cos(– – Ci +log – Si)

+sin( + Ci – log – Si)=aλ+asinα, ()

whereSi(t) =_tsin_ssdsis the sine integral function,Ci(t) = –_t∞cos_ssdsis the cosine inte-gral function [, p.] andis the Euler constant. Similarly

T+

T–

v∗_(t)G(xh(t),xh(t),t+α, ,κ) ω(xh(t))

dt

= –γ

 – cosα(cos + sin) =aγ+acosα ()

and

T+

T–

v∗_(t)G(xh(t),xh(t),t+α, ,κ) ω(xh(t))

dt=aλ+aχsin(α+),

T+

T–

v∗_(t)G(xh(t),xh(t),t+α, ,κ) ω(xh(t))

dt=aγ+aχcos(α+).

()

Note

a= , a= .,. a= –



, a

.

= ..

Consequently, the Melnikov function is now

M(α,κ) =

⎛ ⎜ ⎜ ⎜ ⎝

aλ+asinα aγ+acosα aλ+aχsin(α+) aγ+aχcos(α+)

⎞ ⎟ ⎟ ⎟

⎠. ()

The equationM(α,κ) =  is equivalent to

λ= –a a

sinα= –a a

χsin(α+),

γ= – a a

cosα, γ= – a a

χcos(α+).

()

So havingχ>  and>  such that the equation

sinα–χsin(α+) =  ()

applied to (). Ifχcos= , then () is equivalent to

tanα= χsin

 –χcos. ()

Hence assuming ∈]π, π[ andχcos< , the right hand side of () is negative, and then

α=arctan

χsin

 –χcos ∈

–π , 

()

satisfiessinα<  andcosα> . Sinceαsatisfies () andsinα< , conditioncos(α+ ) >  is equivalent totan(α+) < . Then using (), we derive

 >tan(α+) =

sin

cos–χ, = π

 . ()

Whencos < , then () is not satisfied, sincesin < . So we take ∈]_π, π[ and () gives alsoχ<cos. Clearlyχ<cos implies  >χcos.

Summarizing we see that for any fixedχand satisfying

 <χ<cos, ∈

π  , π

, ()

the Melnikov function () has a simple zero (α,γ,,γ,,λ) given by () and (), and γ,> ,γ,> ,λ> . Hence in the region given by () we apply Corollary . to () with parametersγ,γ,λnearγ,,γ,,λdetermined by () and (),i.e.,

λ= –

aχsin a

"

 +χ_{– χcos},

γ,= –

a( –χcos) a

"

 +χ_{– χcos},

γ,= –χ

a(cos–χ) a

"

 +χ_{– χcos}

()

for any fixedandχsatisfying (). Summarizing, we get the following.

Theorem . For any fixed,χsatisfying()and thenα,γ,,γ,,λgiven by() and(),there is anε> and smooth functionsα,γ,γ,λ: ]–ε,ε[→Rwithα() =α, γ() =γ,,γ() =γ,,λ() =λ,such that for anyε∈]–ε,ε[\{},system()withγ=

γ(ε),γ=γ(ε),λ=λ(ε),possesses a unique solution(u(ε,t),u(ε,t))on]– +α(ε),  + α(ε)[such that

lim

ε→_t∈sup_]–,[

uj

ε,t+α(ε)–  

 –t 



 –t–= ,

lim

ε→_t∈sup_]–,[

u_jε,t+α(ε)+ t 

= .

Proof We apply Corollary . to (). Since, in this case,ϕ(θ–_{(t)) =}  

–t

+t, according to

Corollary . () has a solution (uj(t),wj(t)),j= , , such that

lim

ε→_t∈sup_]–,[

uj

ε,t+α(ε)– 

 –t 



 –t–= ,

lim

ε→_t∈sup_]–,[

wj

ε,t+α(ε)– t 

t  – 

 –t–= .

()

Now,wj(ε,t) = ( + uj(ε,t))uj(ε,t) andt(

t

 – ) = ( + uh(t))uh(t), whereuh(t) =_( –t



). So, takingt:=t+α(ε),

wj

ε,t–t 

t  – 

= + uj

ε,tu_jε,t– + uh(t)

u_h(t)

= + uj

ε,tu_jε,t–u_h(t)+ uj

ε,t–uh(t)

u_h(t)

= + uh(t)

u_jε,t–u_h(t)+ uj

ε,t–uh(t)

u_jε,t–u_h(t)

+ uj

ε,t–uh(t)

u_h(t)

=

 –t

 + 

uj

ε,t–uh(t)

u_jε,t–u_h(t)

– t 

uj

ε,t–uh(t)

and then

wj

ε,t–t 

t  – 

 –t–

=

 + 

uj(ε,t) –uh(t)  –t

u_jε,t–u_h(t)– t 

uj(ε,t) –uh(t)

 –t , ()

or, equivalently,

 + 

uj(ε,t) –uh(t)  –t

u_jε,t–u_h(t)

=

wj

ε,t– t 

t  – 

 –t–+ t 

uj(ε,t) –uh(t)

 –t . ()

Sincesup_–<t<|uj(ε,t) –uh(t)|( –t)–=O(ε) we see that, forεsufficiently small,

 <

_ + uj(ε,t

_{) –u}

h(t)  –t

< 

,

and hence, using () and (),

lim

ε→_–<sup_t< u

j

Vice versa, if () and the first of () hold, then () gives

lim

ε→_–<sup_t<

wj

ε,t+α(ε)– t 

t  – 

 –t–= .

Hence () and () are equivalent. The proof is complete.

Of course, solutions given by Theorem . vary smoothly with respect (,χ) satisfying ().

Remark . Missed in the above analysis is the second possibility when∈],π[. Then

sin >  and () is negative if

κcos> , ()

so we get∈],π_[ andκ> . Then inequality of () is satisfied sinceκ>  >cos. So we conclude that the result of Theorem . is valid also for

κcos> , ∈

,π 

,

λ=

aχsin a

"

 +χ_{– χcos},

γ,=

a( –χcos) a

"

 +χ_{– χcos},

γ,=χ

a(cos–χ) a

"

 +χ_{– χcos}.

Appendix

Letv(t) = _(t– ) –_t(t– )arctanh_t be the second component ofv(t). Note that v(t) is an even function and then

T+

T–

v∗_(t)G(xh(t),xh(t),t+α, ,κ) ω(xh(t))

dt

= –



– v(t)

γ t 

t  – 

+λ



 –t 



+sintcosα+sinαcost

dt

= –



– v(t)

λ 

 –t 



+sinαcost

dt

= λ





– v(t)

t– dt–



–

v(t)cost dtsinα. ()

Similarly,

T+

T–

v∗_(t)G(xh(t),xh(t),t+α, ,κ) ω(xh(t))

dt

= λ





– v(t)

t– dt–χ



–

Now, usinglim_t→±∓(t_{– )arctanh(t) =  and integration by parts of the second integral,}



– v(t)

t– dt

=  –  

t– – t

t– arctanht



t– dt

=  –    –

tt– t– arctanht

dt =  –  

t_(t_{– )}  arctanh

t

–

t_(t_{– )} 

  –tdt

 – =  –    –

tt– dt=

 +   = . Furthermore, since  –

v(t)cost dt=

 –  

t– – t

t– arctanht



cost dt,

we compute



–

t– cost dt= (sin + cos),

and

I(t) :=

tt– costarctanht

dt

=– +tcost+t– +tsintarctanht



+

((– +t)cost+t(– +t)sint)

– +t dt.

Next

(– +t)cost+t(– +t)sint

– +t dt

=

(cost+tsint)dt– 

–cost+tsint

– +t dt

= –tcost+ sint– 

–cost+tsint

– +t dt.

Furthermore,

_cos

t – +tdt=

 

_cos

t t– dt–

 

_cos

t t+ dt

= 

cos( –z) z dz

z=–t–  

cos(v– ) v dv

v=t+

= 

_cos

cosz+sinsinz

z dz