Characteristic Roots of a Class of Fractional Oscillators

1 School of Information Science & Technology, East China Normal University No. 500, Dong-Chuan Road, Shanghai 200241, China 2Department of Computer and Information Science, University of Macau, Avenue Padre Tomas Pereira, Taipa 1356, Macau SAR, China 3 Faculty of Engineering, Multimedia University, Selangor Darul Ehsan, 63100 Cyberjaya, Malaysia 4Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy 5 Department of Mathematics, University of Rome, la Sapienza Piazzale Aldo Moro, 00185 Rome, Italy

This research is in the domain of fractional oscillators that attract increasing interests of physicists and engineers.More specifically, we aim at revealing specific properties of characteristic roots of a class of fractional oscillators.In doing so, we first consider an ordinary differential equation of order  given by where  is a natural number and   is any complex number.We always assume that at least one of the higher coefficients   ̸ = 0 for  > 1.The characteristic equation of ( 1) is given by The fundamental theorem of algebra says that the number of roots of (2) is  (G. A. Korn and T. M. Korn [45]).This theorem is stated in the domain of complex variables (Krantz [46]).
According to the fundamental theorem of algebra, there are only two characteristic roots with respect to the oscillator equation (5).They are One might be carelessly misled to consider that there exist only two characteristic roots regarding the fractional oscillator equation ( 8) because However, we shall show that the number of the roots in the above expression dramatically differs from what in the following expression: The contributions of this paper are twofold.One is to exhibit that the number of the characteristic roots of ( 8) is in general infinitely great.The other is to reveal the relationship between the range of  and the locations of the characteristic roots of (8) in a complex plane.In addition, if all   ( = 1, . . ., ) are simple complex pair of roots, the ordinary differential equation of order  (1) and its generalization given by may be taken as the product of oscillators of integer order and fractional order in series in the wide sense for  being even, respectively.
The rest of the paper is organized as follows.We shall give the results in Section 2, including the proof that there are infinite characteristic roots regarding (8), and the explanation that ( 1) and ( 12) may be taken as oscillators in series in the wide sense.Discussions are given in Section 3, which is followed by Conclusions.

Result 1.
The number of the characteristic roots of (8) may be infinitely great.
Denote by C the set of complex numbers.Let  ∈ C. Suppose that a power function is given by Then, the number of different values of  relies on the value of  for a given .More precisely, we express that by the following lemmas, which can be found in the literature, such as [45] or Yu [62].The general expression of  is in the form Therefore, from Lemma 2, we have the theorem below.
Theorem 3. The number of the characteristic roots of the fractional oscillator (8) is infinitely great if  ̸ = 1.
Proof.Let   = 0.Then, the characteristic roots  ,12 in (9) become the imaginary numbers expressed by According to Lemma 2, the number of the roots of either ( −  ,1 )  or ( −  ,2 )  is infinitely great.Thus, Theorem 3 results.

Result 2.
Equations ( 1) and ( 12) may be taken as oscillators in series.
Denote (   2  +     +   ) in ( 4) by   (): Without loss of generality,  is assumed to be even.In addition, we suppose that all   ( = 1, . . ., ) are simple complex pair of roots.Then, we have the theorem below.
Theorem 4. The ordinary differential equation (1) may be taken as an oscillator (i.e., product of oscillators) in the wide sense if  is even and all   ( = 1, . . ., ) are simple complex pair of roots.By wide sense, one means that it is a system consisting of the product of a series of conventional 2-order oscillators.
Proof.On the one hand,   () stands for the characteristic equation of the th oscillator of order 2 since  is even and all   ( = 1, . . ., ) are simple complex pair of roots.On the other hand, the characteristic equation of (1) can be expressed by Based on the theory of filter design (Mitra and Kaiser [63]), the system (1) in the case of  being even may be expressed by Figure 1.Therefore, the system (1) may be expressed by the product of a series of 2-order oscillators.
Denote by   () the characteristic equation of (12).Then, where Thus, the system of fractional order expressed by ( 12) may be the product of a series of fractional oscillators of (8).

Discussions
The previous section says that there are infinite roots in In the case of  = 1,    () reduces to the characteristic equation of the conventional oscillator (5) with two roots only.Thus, the fraction  ̸ = 1 dramatically alters the behavior of characteristic roots of oscillators.For facilitating our discussions, we omit the subscript  in what follows if not confused.More precisely, we specifically consider the fractional oscillator in the form Figure 2 shows an RLC resonance circuit in series, where , , and  represent resistor, inductor, and capacity, respectively.In Figure 2, () is the electronic current and V() the power source.According to the Kirchhoff voltage law, one has Let  = √1/ and / = 2.Denote (1/)(V()/) by ().Then, (21) becomes the form Generalizing (22) to the fractional order  yields Below, we specifically study the circuit in Figure 2 with  = 0, as indicated in Figure 3.
The following theorems reflect the particularity of roots of   ().Theorem 5.If 0 <  < 1, all roots of   () are located in the left side of the complex plane.Proof.Note that From the above, we have the asymptotic expression in the form Applying (27) to (25) produces Denote by () the Laplace transform of ℎ().Then, according to the final-value theorem, we have lim The above implies that all poles of () except the origin are strictly in the left side of  plane.In the right of the  plane, () is analytic.This completes the proof.Theorem 6.If  > 1, at least, parts of roots of   () are located in the right side of the complex plane.
Proof.Note that From the above, we have the following: Since  > 1 implies V > 1/2, we immediately see that both the right side and the left one on the above expression are respectively unbounded when  → ∞.Thus, for  > 1, the fractional oscillator ( 24) is nonstable according to the theory of systems (Gabel and Roberts [77], Dorf and Bishop [78]).Consequently, at least, some of poles of () are in the right of the  plane.Therefore, at least, parts of roots of   () are located in the right side of the complex plane.
Most of previous discussions take oscillators of fractional order (24) as a specific object.Note that the number of the characteristic roots of differential equation in general in the form of (12) may also be infinitely great.Hence, comes the following theorem in passing.
Theorem 7. Fractional-order differential equation expressed by (12) has infinite characteristic roots if  > 1 and if there is at least a pair of roots that are simple complex.
Proof.The characteristic equation of ( 12) may be decomposed in the form of (18) due to  > 1.Because there is at least a pair of roots that are simple complex, the number of the characteristic roots of ( 19) is infinitely great.Thus, the number of characteristic roots of ( 12) is infinitely great.This completes the proof.
The previous discussions exhibit interesting phenomena of the characteristic roots of the oscillators of the fractional type of (24).In the future, we will work on exploring the answers of the questions described below.
(i) Are all poles of () with respect to (24) in the right of the  plane when  > 1?
(ii) Might there be interesting oscillation behavior of ( 12) if all   = 0 in (18) and if  is even?

Conclusions
We have explained that the number of the characteristic roots of fractional-order oscillators of ( 24) is usually infinitely great.This conclusion has been further inferred to the case of fractional-order differential equation of (12).We have exhibited that all characteristic roots of ( 24) are strictly located in the left side of the complex plane if 0 <  < 1 and at least some of characteristic roots of (24) are in the right side of the complex plane if  > 1.In the case of  = 1, (24) reduce to an ordinary damping-free oscillator.

Figure 2 :Figure 3 :
Figure 2: Illustration of RLC resonance circuit in series.
2 , . . .,   .For each root  of multiplicity of , either real or complex, we Advances in High Energy Physics always consider  the  roots in what follows unless otherwise stated.Using the partial fraction expansion, () can be expressed by ,   , and  are constants.Without loss of generality, we can suppose that the only simple zero of () is   if  is odd.The factor (   2  +     +   ) in (4) corresponds to the oscillator equation in the form 1.If  is a rational number expressed by the irreducible fraction /, where  ≥ 1, the number of values of   is .If  is an irrational number or imaginary number, the number of values of   is infinitely great.