Research Article | Open Access
Reducibility for a Class of Almost-Periodic Differential Equations with Degenerate Equilibrium Point under Small Almost-Periodic Perturbations
This paper focuses on almost-periodic time-dependent perturbations of an almost-periodic differential equation near the degenerate equilibrium point. Using the KAM method, the perturbed equation can be reduced to a suitable normal form with zero as equilibrium point by an affine almost-periodic transformation. Hence, for the equation we can obtain a small almost-periodic solution.
1. Introduction and Main Result
Reducibility of nonautonomous finite-dimensional systems with quasiperiodic coefficients has basic importance in the analysis of dynamical systems; see [1, 2]. Unfortunately, we cannot guarantee in general such reducibility. In the last years, establishing the reducibility of finite-dimensional systems by means of the KAM tools is an active field of research, and many authors are devoted to the study of reducibility of such systems. In 1996, Jorba and Simó  considered reducibility of the following nonlinear quasiperiodic system near an elliptic equilibrium point: where is assumed to be elliptic. , and depend on time in a quasiperiodic way with basic frequencies , as . Under a nondegenerate condition and a nonresonant condition, using KAM iteration they proved that for most sufficiently small by an affine quasiperiodic transformation the system (1) is reducible to the following form: where is a constant matrix close to and is a high-order term close to . Therefore, the system (1) has a quasiperiodic solution near the zero equilibrium point. Some similar results were obtained in .
Recently, Xu and Jiang  considered the following nonlinear quasiperiodic differential equation: where is an integer, is a higher-order term, is a small perturbation term, and and are all real analytic in and , quasiperiodic in with frequency . Under the Diophantine condition, when is sufficiently small the differential equation (3) can be reduced to a suitable normal form with zero as equilibrium point by an affine quasiperiodic transformation, so it has a quasiperiodic solution near zero.
In 1996, Xu and You  considered the following linear almost-periodic differential equation: where is a constant matrix with different eigenvalues and is a almost-periodic matrix with the frequency . Under some small divisor condition, they proved that, for most sufficiently small the system (4) is reducible to the form by an affine almost-periodic transformation.
By the above inspired works [5, 6], we consider the following nonlinear almost-periodic differential equation: where is an integer, is a positive number, is a small parameter, is a higher-order term, and is a small perturbation term. Under some suitable conditions, we show that the differential equation (5) can be reduced to a suitable normal form with zero as equilibrium point by an affine almost-periodic transformation, so it has an almost-periodic solution near zero.
For our purpose, we first introduce some definitions and notations.
Definition 1. The function is called a quasiperiodic function of with frequencies , if there is a function , which is -periodic in all its arguments , such that .
If is analytic on a strip domain , we say that is analytic quasiperiodic on . Expand as a Fourier series , where and . Define .
Write and .
Definition 2. Let be real analytic in and t on , and let be quasiperiodic with respect to with the frequency . Then can be expanded as a Fourier series as follows: Define a norm by where and . It is easy to see that
Definition 3 (see ). A function is called an approximation function if it satisfies the following:(1), , and is a nondecreasing function;(2) is a decreasing function in ;(3).
Obviously, any positive power of an approximation function is again an approximation function, so is the product of two such functions.
Definition 4 (see ). Suppose that is the natural number set and is a set composed of the subset of . We say that is the finite spatial structure on if satisfies the following:(1)the empty set ;(2)if , then ;(3), and ( is called a weight function defined in ).
Definition 5. Assume , the set is called the support set of . Consider is called the weight of , and .
Definition 6. If with is quasiperiodic function with the frequency , then is said to be almost-periodic function with the finite spatial structure . If is the biggest subset of in the sense of integer modulus, then is called to be the frequency of .
If almost-periodic function has a rapidly converging Fourier series expansion where is the frequency and have only finitely many nonzero components, then is analytic in .
Definition 7. Let with the frequency . For , , is called the weight norm of in the finite spatial structure .
From Definition 7, we know that , for , .
Definition 8. Let with the frequency , for , , is called the weight norm of in the finite spatial structure .
Let be almost-periodic function; then is called the mean value of . The existence of the limit can be found in .
Throughout this paper, we assume that the following hypotheses hold the functions , , and are real analytic in all variables and almost-periodic in with common frequency vector ; they also have the the finite spatial structure ;, , where is a constant and is an approximation function; , where , and for fixed , , we have
Now we are ready to state the main result of this paper.
Theorem 9. Suppose that conditions hold. Then there exists sufficiently small , such that if then there exists an affine real analytic almost-periodic transformation of the form such that the differential equation (5) is changed to where as . Moreover, is a real analytic almost-periodic in with and is also an almost-periodic solution of (5).
2. Normal Form for an Almost-Periodic Equation with Parameters
The proof of Theorem 9 is based on a norm form theorem for an almost-periodic equation with parameters. In this section, we first consider the following real almost-periodic differential equation with two parameters: where is the mean value of . , are parameters and .
Let and . Denote by the complex -neighborhood of in the two-dimensional complex space .
We will invoke the KAM iteration technique to prove the following normal form theorem.
Theorem 10. Let . Suppose that and are analytic on and , respectively. Let and let . Let such that . There exists a sufficiently small such that if
then there exists a real -smooth curve in ,
And for every , there exists an affine analytic almost-periodic transformation
which changes (19) to
with . Moreover,
Furthermore, for , is an analytic almost-periodic solution of the differential equation (19) with .
Lemma 11. Let , . Suppose that is real analytic on with Suppose that, on the domain , determines implicitly a real analytic curve , , such that , where , and are supposed to be well defined. Let , where with . Suppose that with . Then there exists a domain of , with , such that is real analytic on and satisfies . If then, on the domain , determines implicitly a real analytic curve , , such that and . Moreover, one has
Proof. Let . Since and , by it follows easily that and . Using Cauchy’s estimate we have By condition (29), the equation determines implicitly an analytic curve on , It follows that so Thus, . Let . For each , we have Noting that , for all , and , for all , we have Thus we prove Lemma 11.
Lemma 12. Assume that is an analytic almost-periodic function with respect to with the frequency ; it has the finite spatial structure , and
then there exists an almost-periodic function with the same spatial structure as , which satisfies
where is the direction derivative of along with . is the truncation of with order .
Moreover, for , ,
Proof. We assume
Insert the above formulas into the equation , and compare the coefficients on both sides, thus we can find
From Definition 7,
Thus, is convergent in the smaller domain with the norm .
Now we consider the following real analytic almost-periodic differential equation with parameters with .
Lemma 13. Consider the above equation, where . Let and . Let , and . Assume the following hold.(1) is real analytic on and satisfies
with , , and . (2). (3), , and are real analytic on and all the assumptions of Lemma 11 hold. Let be the domain defined in Lemma 11.
Then, for any , there exists an affine analytic almost-periodic transformation where is real analytic on , such that the above differential equation is transformed to where and will be get in the proof.
Moreover, one has
The new perturbation term satisfies
Proof. The proof is the standard KAM step and we divide it into several parts.
(A) Truncation. Let with and . It follows easily that Hence . Let By definition, we have
(B) Construnction of the Transformation. Define the transformation , where satisfies From Lemma 12, we have By the transformation , the equation becomes then
Define the transformation ; satisfies We assume
Then Similar to Lemma 12, we have
By the transformation , the equation becomes then
Thus, by the transformation , the equation is transformed to where With the estimates of and , we have
Let , , and . Then the transformation is analytic almost-periodic on with respect to and affine in .
(C) Estimates of Error Terms. Because , then . Thus, Let . Then, it follows that So where , , , and . Thus we have proved Lemma 13.
Iteration. Now we choose some suitable parameters so that the above KAM step can be iterated infinitely. At the initial step, let Let satisfy and . Inductively, we define And satisfies .
By , we have . Thus, if is sufficiently small, we have and . Moreover, by definition it follows that Thus , .
Now we prove that, for sufficiently small, hold for all .
Let ; from (21) we have . Moreover, we have for all . Thus, . So the inequalities in the assumption 2 of Lemma 13 hold for all .
Let , , , , , and . By Lemmas 11, 12, and 13, we have a sequence of closed domains with and a sequence of affine transformations We also have Let with . Then, after the transformation , (19) is changed to By the inductive assumptions of KAM iteration, we have .
The correction terms and satisfy By Lemma 11, we have . For Cauchy’s estimate we have
Noting that and , . Since and , it follows that and . For Cauchy’s estimate we have
Let ; then So . Obviously, we can choose sufficiently small so that , . Noting that and , we have So condition (29) holds for all .
From Lemma 11, defines implicitly a real analytic curve , , satisfying Furthermore, .
Convergence of KAM Iteration. Now we prove the convergence of KAM iteration. By the definition of , if is sufficiently small, it follows that Therefore, we have
Let We have
Thus So we have the convergence of to on .
From (86) it is easy to show that is convergent on . In fact, . For , it follows that Let , . For , we have
Moreover, by Cauchy’s estimate we have
Let ; then . Thus, it is easy to prove that is convergent uniformly on , and so is differentiable on . In fact, in the same way as in , we can prove that is -smooth on .
Since , for all , letting we have and . Obviously, , for . Let and let . By Cauchy’s estimate we have Thus and . Hence for .
Noting that and , we have The proof of Theorem 10 is complete.
3. Proof of Theorem 9
Let with and . Then (5) becomes Let