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Characteristic Roots of a Class of Fractional Oscillators
The fundamental theorem of algebra determines the number of characteristic roots of an ordinary differential equation of integer order. This may cease to be true for a differential equation of fractional order. The results given in this paper suggest that the number of the characteristic roots of a class of oscillators of fractional order may in general be infinitely great. Further, we infer that it may also be the case for the characteristic roots of a differential equation of fractional order greater than 1. The relationship between the range of the fractional order and the locations of characteristic roots of oscillators in the complex plane is considered.
Oscillators are an essential component in devices in electron positron collider systems (see, e.g., Zhao et al. , Ma et al. , Zang et al. , Ding et al. , Marder et al. , Barroso , Miller et al. , and Lemke , just citing a few). As a matter of fact, oscillations are phenomena widely observed in sciences and engineering relating to high energy physics (see, e.g., Akhmediev et al. , Bachas , Winter et al. , Dodonov , Tan , Diamandis et al. , Greenwald et al. , Mathews et al. , Faiman , Cocho et al. , Baldiotti et al. , Kyu Shin , Kirson , Clement , Sikström et al. , Asghari et al. , Um et al. , Bahar and Yasuk , Hassanabadi et al. , Bhattacharya and Roy , and Saad et al. , simply mentioning a few).
There are various structures of oscillators, such as Mathieu oscillator (Floris ), Liénard type oscillator (Yaşar ), relativistic oscillator (Osborne ), Schrödinger equation type oscillator (Cornwall and Tiktopoulos ), and Duffing oscillator (Baltanás et al.  and Erturk and Inman ). In fact, oscillators play a role in various fields, ranging from experimental physics to electronics engineering (see, e.g., Riley et al. , Soong and Grigoriu , Harris , Papoulis , Bendat and Piersol , Devasahayam , Karrenberg , Edson , and Balaban et al. ).
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 for . The characteristic equation of (1) is given by
Suppose that the roots of are . For each root of multiplicity of , either real or complex, we always consider the roots in what follows unless otherwise stated. Using the partial fraction expansion, can be expressed by
Now, we rewrite (3) by the following expression: where , , , and are constants. Without loss of generality, we can suppose that the only simple zero of is if is odd.
The factor in (4) corresponds to the oscillator equation in the form
The characteristic equation of (5) is in the form
There are two classes of fractional oscillators. One is in the form (Ryabov and Puzenko [47, Eq. (1)], Ahmad et al. , Radwan et al. , Drozdov [50, Eq. (9)], Tofighi and Pour , Tofighi [52, Eq. (2)], Blaszczyk et al. [53, Eq. (10)], and Narahari Achar et al. [54, 55])
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 () 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.
2.1. Result 1
The number of the characteristic roots of (8) may be infinitely great.
Denote by C the set of complex numbers. Let . Suppose that a power function is given by
Lemma 1. If is a rational number expressed by the irreducible fraction , where , the number of values of is .
Lemma 2. If is an irrational number or imaginary number, the number of values of is infinitely great.
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 .
Proof. Let = 0. Then, the characteristic roots in (9) become the imaginary numbers expressed by
2.2. Result 2
Denote in (4) by :
Without loss of generality, is assumed to be even. In addition, we suppose that all () 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 () 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 () are simple complex pair of roots. On the other hand, the characteristic equation of (1) can be expressed by
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
The previous section says that there are infinite roots in if . In the case of , reduces to the characteristic equation of the conventional oscillator (5) with two roots only. Thus, the fraction 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 the power source. According to the Kirchhoff voltage law, one has
Let and . Denote by . Then, (21) becomes the form
Generalizing (22) to the fractional order yields
Denote by the impulse response function of (24). Then, using the techniques in fractional calculus and differential equations [64–81], we have (see  for details) where is the Bessel function of the first kind of order .
The following theorems reflect the particularity of roots of .
Theorem 5. If , 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
Denote by the Laplace transform of . Then, according to the final-value theorem, we have
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 , 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 implies , we immediately see that both the right side and the left one on the above expression are respectively unbounded when . Thus, for , the fractional oscillator (24) is nonstable according to the theory of systems (Gabel and Roberts , Dorf and Bishop ). 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 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 . 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 ?(ii)Might there be interesting oscillation behavior of (12) if all in (18) and if is even?
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 and at least some of characteristic roots of (24) are in the right side of the complex plane if . In the case of , (24) reduce to an ordinary damping-free oscillator.
This work was partly supported by the National Natural Science Foundation of China (NSFC) under the project Grant nos. 61272402, 61070214, 60873264, and the 973 plan under the project no. 2011CB302800.
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