A note on fractional-order derivatives of periodic functions

doi:10.1016/j.automatica.2010.02.023

Automatica

Volume 46, Issue 5, May 2010, Pages 945-948

https://doi.org/10.1016/j.automatica.2010.02.023 Get rights and content

Abstract

In this paper, it is shown that the fractional-order derivatives of a periodic function with a specific period cannot be a periodic function with the same period. The fractional-order derivative considered here can be obtained based on each of the well-known definitions Grunwald–Letnikov definition, Riemann–Liouville definition and Caputo definition. This concluded point confirms the result of a recently published work proving the non-existence of periodic solutions in a class of fractional-order models. Also, based on this point it can be easily proved the absence of periodic responses in a wider class of fractional-order models. Finally, some examples are presented to show the applicability of the paper achievements in the solution analysis of fractional-order systems.

Introduction

Nowadays, fractional-order dynamics, which are defined based on fractional-order differential equations (Podlubny, 1999), play a significant role in different control applications. For example, these dynamics have been extensively used in design and practical implementation of controllers (Feliu-Batlle et al., 2009, Monje et al., 2008, Tavazoei et al., 2008), in modeling of real-world phenomena (Ding and Ye, 2009, Fitt et al., 2009, Oldham, 2010, Reyes-Melo et al., 2008), and in identification of physical systems (Aoun et al., 2007, Hartley and Lorenzo, 2003, Poinot and Trigeassou, 2004). Due to the growing interest of fractional-order dynamics to be applied in different control applications, it seems analysis of this type of system is of great importance. This problem, i.e. analysis of fractional-order systems, is a motivation for some recent research works. In one of these works, the non-existence of periodic solutions in a class of fractional-order systems defined based on Caputo definition has been proved (Tavazoei & Haeri, 2009a). By using the proof given in Tavazoei and Haeri (2009a), a remarkable property for fractional-order derivatives of periodic functions is presented in this paper. Based on this property, the absence of periodic solutions in a wider class of fractional-order models can be proved. More precisely, in Tavazoei and Haeri (2009a) it has been proved that autonomous fractional-order systems described by state-space like forms containing Caputo derivatives cannot have non-constant periodic solutions. In the present work, the absence of periodic solutions is shown for autonomous fractional-order systems whose differential equations contain only a fractional-order derivative defined based on Grunwald–Letnikov definition, Riemann–Liouville definition, or Caputo definition. Also, the validity of the main result of Tavazoei and Haeri (2009a) is shown for autonomous fractional-order systems described by state-space like forms containing Grunwald–Letnikov or Riemann–Liouville derivatives.

The paper is organized as follows. Section 2 devotes to presenting a proof for the following claim: “The fractional-order derivatives of a periodic function with a specific period cannot be a periodic function with the same period”. Based on the achievements of Section 2, some results are concluded in Section 3 which may be useful in the solution analysis of fractional-order systems. In Section 4, some examples are presented to show the applicability of the paper outcomes. Finally, the paper is concluded in Section 5.

Section snippets

Fractional-order derivatives of periodic functions

The fractional-order derivative has been defined by extending the concept of integer-order derivative. Three commonly used definitions for fractional-order derivative are Grunwald–Letnikov definition, Riemann–Liouville definition and Caputo definition (Podlubny, 1999). When $0 < α \notin N$ , the $α$ th order derivative of function $g (t)$ with respect to $t$ and the terminal value 0 is given as follows.

Grunwald–Letnikov definition: $_{0}^{G L} D_{t}^{α} g (t) = lim_{\binom{h \to 0}{n h = t}} h^{- α} \sum_{r = 0}^{n} {(- 1)}^{r} (\begin{matrix} α \\ r \end{matrix}) g (t - r h)$ where $(\begin{matrix} α \\ r \end{matrix}) = \frac{Γ (α + 1)}{r! Γ (α - r + 1)}$ .

Absence of periodic solutions in a class of fractional-order systems

The results of Theorem 1, Theorem 2, which are effective in solution analysis of fractional-order systems, are given in this section. From Theorem 1, Theorem 2, the following corollaries are resulted. In these corollaries, the operator $_{0} D_{t}^{α}$ can be each of the three operators $_{0}^{G L} D_{t}^{α}$ , $_{0}^{R L} D_{t}^{α}$ , and $_{0}^{C} D_{t}^{α}$ .

Corollary 1

A differential equation of fractional-order in the form $_{0} D_{t}^{α} f (t) + Ψ (f (t), f^{(1)} (t), \dots, f^{(n)} (t)) = 0,$ where $0 < α \notin N$ , cannot have any non-constant smooth periodic solution.

Proof

Suppose that $\tilde{f} (t)$ is a solution for

Some examples

In this section, some examples are provided to show the applicability of the analytical achievements of the paper. These examples are brought in the rest.

Example 1

The Laplace transform of fractional-order derivative of Sine function is given as $L (_{0}^{R L} D_{t}^{α} sin (t)) = \frac{s^{α}}{s^{2} + 1}$ where $0 < α < 1$ . The inverse Laplace transform of right hand side of Eq. (29) is obtained as $_{0}^{R L} D_{t}^{α} sin (t) = t^{1 - α} E_{2, 2 - α} (- t^{2}),$ where $E_{α, β} (z)$ denotes two-parameter function of Mittag–Leffler (Li et al., 2009, Podlubny, 1999). This function is

Conclusion

In this paper, it was proved that the fractional-order derivatives (obtained based on the Grunwald–Letnikov definition, Riemann–Liouville definition, or Caputo definition) of a periodic function with a specific period cannot be a periodic function with the same period. Based on this proved statement, it was concluded that the existence of periodic solutions in autonomous fractional-order systems defined in the form (26) or form (28) is impossible. This point specifies one of the basic

References (20)

M. Aoun et al.
Synthesis of fractional Laguerre basis for system approximation
Automatica
(2007)
K. Diethelm et al.
Multi-order fractional differential equations and their numerical solution
Applied Mathematics and Computation
(2004)
Y. Ding et al.
A fractional-order differential equation model of HIV infection of CD4⁺ T-cells
Mathematical and Computer Modelling
(2009)
J.T. Edwards et al.
The numerical solution of linear multi-term fractional differential equations: systems of equations
Journal of Computational and Applied Mathematics
(2002)
V. Feliu-Batlle et al.
Smith predictor based robust fractional-order control: application to water distribution in a main irrigation canal pool
Journal of Process Control
(2009)
A.D. Fitt et al.
A fractional differential equation for a MEMS viscometer used in the oil industry
Journal of Computational and Applied Mathematics
(2009)
T.T. Hartley et al.
Fractional-order system identification based on continuous order-distributions
Signal Processing
(2003)
Y. Li et al.
Mittag–Leffler stability of fractional order nonlinear dynamic systems
Automatica
(2009)
C. Li et al.
Remarks on fractional derivatives
Applied Mathematics and Computation
(2007)
C.A. Monje et al.
Tuning and auto-tuning of fractional order controllers for industry applications
Control Engineering Practice
(2008)

There are more references available in the full text version of this article.

Cited by (116)

Collective behaviors of fractional-order FithzHugh–Nagumo network
2024, Physica A: Statistical Mechanics and its Applications
Brain connection is the mechanism of determining the brain function and cognition, and the small world network is widely investigated among these connections. In this paper, the small world complex brain network is constructed by the fractional-order neurons, and the fractional-order memristor is used to connect these neurons. Compared with the integer-order counterpart, the fractional-order memristor describes precisely the distortion of hysteresis curves, and it exhibits more abundant dynamical behaviors to function as the synapse. In addition, the neurons with different fractional orders represent the individual differences during the cell differentiation. For the two coupled neurons, the phase lock is achieved in the bursting and spiking neurons, and it denotes a time delay between the two firing process, whereas the chaotic neurons cannot synchronize. In small world network, the collective behaviors of neurons tend to synchronize with the increasing of the coupling intensity. These results promote the understanding of information coding and transformation in complex brain network.
Steady-state solutions of the Whittaker–Hill equation of fractional order
2024, Communications in Nonlinear Science and Numerical Simulation
We investigate the steady-state solutions of the Whittaker–Hill equation, including a fractional derivative term. Using appropriate Fourier series, we characterize the behavior of the eigenvalue surfaces as a function of the differential equation parameters and describe the corresponding eigensolutions. We also examine the situation where the fractional derivative is zero to serve as a comparison for the fractional solutions. For some special combinations of the parameters, the fractional derivative term leads to the appearance of degenerate complex eigenvalues.
On fractional order computational solutions of low-pass electrical transmission line model with the sense of conformable derivative
2023, Alexandria Engineering Journal
In this study, we solve fractional order PDEs with free parameters that regulate wave motion in a low-pass electrical transmission line equation (LPET) using the unified technique. To convert the governing equation into an ordinary differential equation (ODE), we have used a simple linear fractional transformation. The unified technique can therefore be used to find various single and extracting wave solutions to the equation of attention. Using Maple-18 software, all of the established solutions are validated and verified. By selecting appropriate values for the fractional order and free parameters, the two dimensional (2D) and three dimensional (3D) profiles of some of the assimilation solutions demonstrating kink, singular kink, cross kink, bright-dark kink, lump type kink, anti-kink, dark-anti kink, singular periodic, periodic, singular soliton solutions are displayed. The effect of free parameters and fractionality on the dynamical properties of the precise and exploited solutions are graphically illustrated and discussed in detail. We have also compared our results to those found in the literature review. Wherever possible, the judgment demonstrates no significant differences between our realized and published solutions. The results of this investigation show that the adopting technique is both productive and powerful in extracting wave solutions to nonlinear evolution problems that arise in the areas of applied mathematics, mathematical physics, and engineering.
Hidden attractors in a new fractional-order Chua system with arctan nonlinearity and its DSP implementation
2023, Results in Physics
This paper reports a new fractional-order Chua’s system with arctan function and an algorithm for determining the initial value of fractional-order hidden attractors. Firstly, dynamics of the system is investigated by employing the stability analysis, dissipativeness, phase diagrams, 0-1 test, bifurcation diagram and SampEn complexity measure. It shows rich dynamics in the proposed system. Secondly, using the proposed algorithm, the initial values of the new fractional-order Chua’s system are calculated. And according to two sets of initial conditions, hidden attractors are found and verified by the numerical simulation. Meanwhile, the basin attraction is analyzed to confirm that the hidden attractor is indeed not near the fixed point. Thirdly, the new fractional-order Chua’s system is implemented on the DSP platform, and the arctan function’s processing algorithm is proposed. The DSP experimental results show that the system can produce a pair of coexisting fractional-order hidden attractors, and the experimental results are consistent with the numerical simulation results. It verifies the effectiveness of the presented methods and shows the potential engineering application value of the proposed Chua’s system.
Memory Fusion Controller of Fractional-Order Systems with Sliding Memory Window under Intermittent Sampled-Data Transmission
2024, IEEE Transactions on Cybernetics
An improved Hénon map based on G-L fractional-order discrete memristor and its FPGA implementation
2024, European Physical Journal Plus

View all citing articles on Scopus

^☆: The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Mayuresh V. Kothare under the direction of Editor André L. Tits.

View full text

Technical communiqueA note on fractional-order derivatives of periodic functions☆

Abstract

Introduction

Section snippets

Fractional-order derivatives of periodic functions

Absence of periodic solutions in a class of fractional-order systems

Some examples

Conclusion

Automatica

Applied Mathematics and Computation

Mathematical and Computer Modelling

Journal of Computational and Applied Mathematics

Journal of Process Control

Journal of Computational and Applied Mathematics

Signal Processing

Automatica

Applied Mathematics and Computation

Control Engineering Practice

Technical communique
A note on fractional-order derivatives of periodic functions☆