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: 34A09, 34C23, 37G99.
Keywords:
implicit ode; perturbation; Melnikov method; RLC circuits1 Introduction
In [1], motivated by [2,3], the equation modeling nonlinear RLC circuits
has been studied. It is assumed that and are smooth functions with at least quadratic at the origin and satisfying suitable assumptions. Setting the equation reads
It is assumed that, for some , we have and . So for (2) has the Hamiltonian
passing through . Clearly and the Hessian of ℋ at is
so that the condition means that is a saddle for ℋ. Multiplying the second equation by we get the system
Note that (2) falls in the class of implicit differential equations (IODE) like
with . Obviously, vanishes on the line and the condition implies that the line consists of noncritical 0singularities for (3) (see [[4], 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 Ipoint (see [[4], pp.163166]). Quasilinear implicit differential equations, such as (4), find applications in a large number of physical sciences and have been studied by several authors [412]. On the other hand, there are many other works on implicit differential equations [1318] 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 :
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 [1] are , and , for some (that is, is an Ipoint for (4)) and the existence of a solution in a bounded interval J tending to as t tends to the endpoints of J.
It is well known [4,8] that and imply
and then is equivalent to the fact that .
Let . It has been proved in [19] that (5) implies that is at most 2. So, if , with then cannot be hyperbolic for the map .
In this paper we study coupled IODEs such as
with , , 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 satisfies , and . Thus is not a Ipoint. Multiplying the first equation by and the second by we obtain the system
We assume that , and satisfy the following assumptions:
(C1) , and the unperturbed equation
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 1periodic in t with for any , and ε sufficiently small.
(C3) Let be the eigenvectors of with the eigenvalues , resp. Then (or else ).
From (C2) we see that is a bounded solution of the equation
and that 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 , the coupled equations (7) has a solution in a neighborhood of the orbit and reaching 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 . 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
By following [1,19] we note that since as then is a bounded solution of the linear equation . Hence for some . We get then, for ,
So
From [[22], Theorem 4.3, p.335 and Theorem 4.5, p.338] it follows that
and there exist a constant and a solution of such that
Note that since otherwise , contradicting (10). Hence is an eigenvector of the eigenvalue of . We have then
for a suitable constant . As a consequence,
Next
Taking logarithms, dividing by s and letting we get
that is, in (10) can be replaced with . Similarly, changing s with −s:
Next, set
and
From (C2), we know for any , so . Hence condition (C3) means that tends transversally to the singular manifold at .
As in [19] it is easily seen that
So is a bounded solution of (14). Next, setting as in [19]
and , it is easily seen that satisfies whose linearization along is
i.e.
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 [19] 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 2.1Assume (C2) and (C3) hold. Then, up to a multiplicative constant, 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 and . Let , then the linear map
has the eigenvalues ±μ; moreover, since
for two positive constants , it follows that is a solution of
satisfying
for all . Then (17) satisfies the assumptions of [[19], Theorem 5.3] and hence its conclusion with , that is, the fundamental matrix of (17) satisfies
where . However, it is well known (see [2325]) 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.
Incidentally, since the fundamental matrix of (14) is , we note that it satisfies
Using a similar argument in with , and [[19], Theorem 5.4] instead of [[19], Theorem 5.3] with we see that (14) has at most a one dimensional space of solutions bounded in ℝ. More precisely, with , and a projection on exists such that
and . Since is a solution of (14) bounded on ℝ we deduce that and the result follows. □
We conclude this section with a remark about condition (c) in [[19], Theorem 5.3]. Consider a system in such as
Then the following result holds.
Theorem 2.2Suppose the following hold:
(i) Dhas two simple eigenvaluesand all the other eigenvalues ofDhave either real part less thanor greater than;
Then there are as many solutionsof (18) satisfying
as the dimension of the space of the generalized eigenvectors of the matrixDwith real parts less than or equal to; hereare two suitable positive constants. Similarly there are as many solutions of (18) such that
for suitable constants, as the dimension of the space of the generalized eigenvectors of the matrixDwith real parts greater than or equal to.
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 have real parts less than and those of have real parts greater than or equal to . So the system reads
where and are matrices (or vectors) of suitable orders. Setting we get
Now we observe that is a solution of (22) bounded at +∞ if and only if is a solution of (21) which is bounded on ℝ when multiplied by . Moreover, since , , and belong to , the limit exists for any solution of (22) bounded on . So, let us fix and take . If is a solution of (22) bounded at +∞ it must be, by the variation of constants formula,
where and . Note that since and and , are bounded, we can interpret (22) as a fixed point theorem in the Banach space of bounded function on :
with the obvious norm. Since we see that the map (23) is a contraction on B, provided is sufficiently large, and then, for any given , it has a unique solution . Note that a priori is defined only on but of course we can extend it to going backward with time. We now prove that positive constants exist such that fox any . Let . We have
and then
So, for any let be such that and set . We have
with . Taking the limit as we get
Since as , from the above it follows that . Similarly we get
where and then . As a consequence we obtain and then
So, provided we take we see that eventually (i.e. for , for some )
and the existence of such that
for all follows from the fact that cannot vanish in any bounded interval. Finally since we get, for ,
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 is a simple eigenvalue of D and all the others either have real parts less than or (i.e. we do not need that is simple). Similarly inequalities of (20) hold if is a simple eigenvalue of D and all the others either have real parts greater than or (i.e. we do not need that is simple).
(ii) Note that a result related to Theorem 2.2 has been proved in [26].
3 Solutions asymptotic to the fixed point
It follows from (11)(12) that as then, since we obtain . Furthermore, from (13) we also get:
As a consequence
Since it follows that is a strictly increasing diffeomorphism (see (15) for the definition of ). Then satisfies (8) on the interval and
Moreover (see (13))
Hence is not an Ipoint 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 , for some , and tend to at the same rate as . To this end we first perform a change of the time variable as follows. Set
Since we are looking for solutions of (7) tending to at the same rate as , in (24) we make the change of variables
where is the bounded function . Since
for a suitable constant and any , we get, using (C3), (26):
for large and sufficiently small. Then (27) and imply the existence of and so that
for any and . Now plugging (25) into (24) we derive the equations
From (11) it follows that
Next we note that from it follows that the quantities
are bounded uniformly in and , , , ε bounded.
The linearization of (28) at , is
Taking the limit as we get the systems
Similarly taking the limit as 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 and , and (31) has the negative simple eigenvalues and . From the roughness of exponential dichotomies it follows that both equations in (29) have an exponential dichotomy on both and with projections, resp. and . Hence (see also [19]) all solutions of the system
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 this section we will give a condition for solving (28) for near the solution of the same equation with . Writing
and
in the Banach space of functions on ℝ, bounded together with their derivatives and with small norms. We observe that and equation reads
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.
Theorem 4.1LetY, Xbe Banach spaces, be a small parameter and. Let, , befunctions such that
and suppose there existssuch thatand the derivativeis invertible. Then there existand uniquefunction, defined in a neighborhood ofsuch that
and for, , (35) has a unique solutionsatisfying
Moreover, forfunctionsand we have
Proof We look for solutions of (35) that are close to . Let be the projection such that . Note . From the implicit function theorem, we solve the projected equations
provided is sufficiently small and η in a fixed closed ball with . Note that are smooth. Setting , we need to solve the bifurcation equations:
Observe that
uniformly with respect to η. We conclude that (40) can be written as
where
Note that are functions of and that uniformly with respect to η, so
is in . By (41), system (42) is equivalent to
Because of the assumptions we can apply the implicit function theorem to (43) to obtain a function defined in a neighborhood of satisfying (43) and such that (38) holds. Setting
we see that , are bounded solutions of (35) with such that , . Then we can write , ) for continuous where
Clearly (39) follows differentiating the above equality at . The proof is complete. □
Remark 4.2 Note that, because of , (39) is equivalent to
which has the unique solution
Now we apply Theorem 4.1 to (28) with , , as in (33), (34) and
where is the Banach space of functions bounded together with their derivatives with the usual supnorm.
We already observed that and . Moreover,
where have been defined in the previous Section 3. So and . We recall, from [19], that where are solutions of the adjoint equation of (16):
A direct application of Theorem 4.1 gives the following.
Theorem 4.3Letandbe given as in (46) where, are two independent bounded solutions (on ℝ) of the adjoint equation (45). Suppose thatandexist so that
Then there exist, , uniquefunctionsandwithand, defined for, and a unique solutionof (24) with, , , such that
Moreover,
Remark 4.4 (i) Equation (48) implies
for functions with . Then we have
which gives a first order approximation of . Next, can be computed using (44) adapted to this case. Hence are bounded solutions of
Since (36) has exponential dichotomies on both (with projection ) and (with projection ) it follows that
is the fundamental solution of (36). Note formulas (50) are well defined at , i.e., , due to the first assumption of (47). Next, passing to time and taking , we get
with , and is a fundamental solution of (16).
(ii) Using (49), the functions are bounded solutions of (7) in the interval such that
Summarizing, we obtain the following corollary of Theorem 4.3.
Corollary 4.5Letandbe given as in (46) where, are two independent bounded solutions (on ℝ) of the adjoint equation (45). Suppose thatandexist so that
Then there exist, , uniquefunctionsandwithand, defined for, and a unique solutionof (7) with, , , such that
Moreover,
5 Applications to RLC circuits
In this section we study the coupled equations
which is motivated by [2,3]. Note that (51) is obtained by coupling two equations modeling nonlinear RLC circuits as in (1). Here , , λ, χ and ϖ are positive parameters. Setting , , (51) reads
By solving the second and fourth equations of (52) for and , we get:
provided . Since , to write the system (52) in the form (7) with parameter and fixed, we have to multiply the second and fourth equation by , respectively, and we obtain the system
with (uncoupled) unperturbed equation for (see (8)):
Neglecting left multiplicators () in (55), we obtain the following system (see condition (C2)):
Clearly, condition (C1) is satisfied with and
The equation has the prime integral . A solution satisfying has to satisfy with the solution
bounded on ℝ. Hence
we get and . Since , we derive , and condition (C3) holds as well. Next, we have
hence and assumption (C2) is also verified. Here , . Furthermore by (15)
Thus
So now (16) has the form
which has the solution . In other words, we deal with
possessing the solution . Following [[27], p.327], the second solution of (58) is given by
Consequently a fundamental matrix solution of (57) has the form
Note . The adjoint system of (57) is (see (45))
with the fundamental matrix solution
and
We now compute . We have (see the appendix)
where is the sine integral function, is the cosine integral function [[28], p.886] and Γ is the Euler constant. Similarly
and
Note
Consequently, the Melnikov function is now
So having and such that the equation
has a simple zero with , and , formulas (64) give a simple zero of (63) with positive , , , and Corollary 4.5 can be applied to (51). If , then (65) is equivalent to
Hence assuming and , the right hand side of (66) is negative, and then
satisfies and . Since satisfies (65) and , condition is equivalent to . Then using (66), we derive
When , then (68) is not satisfied, since . So we take and (68) gives also . Clearly implies .
Summarizing we see that for any fixed χ and ϖ satisfying
the Melnikov function (63) has a simple zero given by (64) and (67), and , , . Hence in the region given by (69) we apply Corollary 4.5 to (51) with parameters , , λ near , , determined by (64) and (67), i.e.,
for any fixed ϖ and χ satisfying (69). Summarizing, we get the following.
Theorem 5.1For any fixedϖ, χsatisfying (69) and then, , , given by (67) and (70), there is anand smooth functionswith, , , , such that for any, system (51) with, , , possesses a unique solutiononsuch that
Proof We apply Corollary 4.5 to (53). Since, in this case, , according to Corollary 4.5 (53) has a solution , , such that
Now, and , where . So, taking ,
and then
or, equivalently,
Since we see that, for ε sufficiently small,
and hence, using (74) and (72),
Vice versa, if (75) and the first of (72) hold, then (73) gives
Hence (71) and (72) are equivalent. The proof is complete. □
Of course, solutions given by Theorem 5.1 vary smoothly with respect satisfying (69).
Remark 5.2 Missed in the above analysis is the second possibility when . Then and (66) is negative if
so we get and . Then inequality of (68) is satisfied since . So we conclude that the result of Theorem 5.1 is valid also for
Appendix
Let be the second component of . Note that is an even function and then
Similarly,
Now, using and integration by parts of the second integral,
Furthermore, since
we compute
and
Next
Furthermore,
and
Hence
Clearly . Next, using , as , we derive
Setting we see , so , and using , we obtain
Summarizing we arrive at
So inserting the above computations into (77) and (78), we get (60) and the first formula of (62). Next, since the second component of is odd, we get
But
and
So we obtain (61). Similarly,
which implies the second formula of (62).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The work presented here was carried out in collaboration between the authors. The authors contributed to every part of this study equally and read and approved the final version of the manuscript.
Acknowledgements
BF is partially supported by PRINMURST ‘Equazioni Differenziali Ordinarie e Applicazioni’ (Italy). MF is partially supported by the Grants VEGAMS 1/0071/14, VEGASAV 2/0029/13 and Marche Polytechnic University, Ancona (Italy).
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