This study aims to investigate the harmonic resonance of third-order forced van der Pol oscillator with fractional-order derivative using the asymptotic method. The approximately analytical solution for the system is first determined, and the amplitude–frequency equation of the oscillator is established. The stability condition of the harmonic solution is then obtained by means of Lyapunov theory. A comparison between the traditional integer-order of forced van der Pol oscillator and the considered fractional-order one follows the numerical simulation. Finally, the numerical results are analyzed to show the influences of the parameters in the fractional-order derivative on the steady-state amplitude, the amplitude–frequency curves, and the system stability.

References

References
1.
Miller
,
K. S.
, and
Ross
,
B.
,
1993
,
An Introduction to the Fractional Calculus and Fractional Differential Equations
,
Wiley
,
New York
.
2.
Oldham
,
K. B.
, and
Spanier
,
J.
,
1974
,
The Fractional Calculus
,
Academic Press
,
Boston, MA
.
3.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
London
.
4.
Samko
,
S. G.
,
Kilbas
,
A. A.
, and
Marichev
,
O. I.
,
1993
,
Fractional Integrals and Derivatives: Theory and Applications
,
Gordon and Breach
,
Amsterdam, The Netherlands
.
5.
Baleanu
,
D.
,
Machado
,
J. A. T.
, and
Luo
,
A. C. J.
,
2012
,
Fractional Dynamics and Control
,
Springer
,
New York
.
6.
Baleanu
,
D.
,
Diethelm
,
K.
,
Scalas
,
E.
, and
Trujillo
,
J. J.
,
2012
,
Fractional Calculus Models and Numerical Methods
,
World Scientific
,
Singapore
.
7.
Bagley
,
R. L.
, and
Torvik
,
P. J.
,
1983
, “
A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity
,”
J. Rheol.
,
27
(
3
), pp.
201
210
.
8.
Bagley
,
R. L.
, and
Torvik
,
P. J.
,
1985
, “
Fractional Calculus in the Transient Analysis of Viscoelastically Damped Structures
,”
AIAA J.
,
23
(
6
), pp.
918
925
.
9.
Rossikin
,
Y. A.
, and
Shitikova
,
M. V.
,
1997
, “
Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids
,”
ASME Appl. Mech. Rev.
,
50
(
1
), pp.
15
57
.
10.
Zhang
,
W.
, and
Shimizu
,
N.
,
1998
, “
Numerical Algorithm for Dynamic Problems Involving Fractional Operator
,”
Int. J. JSME Ser. C
,
41
(
3
), pp.
364
370
.
11.
Shimizu
,
N.
, and
Zhang
,
W.
,
1999
, “
Fractional Calculus Approach to Dynamic Problems of Viscoelastic Materials
,”
Int. J. JSME Ser. C
,
42
(
4
), pp.
825
837
.
12.
Fukunaga
,
M.
,
Shimizu
,
N.
, and
Nasuno
,
H.
,
2009
, “
A Nonlinear Fractional Derivative Model of Impulse Motion for Viscoelastic Materials
,”
Phys. Scr.
,
136
, p.
01410
.
13.
Fukunaga
,
M.
, and
Shimizu
,
N.
,
2011
, “
Nonlinear Fractional Derivative Models of Viscoelastic Impact Dynamics Based on Entropy Elasticity and Generalized Maxwell Law
,”
ASME J. Comput. Nonlinear Dyn.
,
6
(
2
), p.
021005
.
14.
Fukunaga
,
M.
, and
Shimizu
,
N.
,
2013
, “
Comparison of Fractional Derivative Models for Finite Deformation With Experiments of Impulse Response
,”
J. Vib. Control
,
20
(
7
), pp.
1033
1041
.
15.
Khang
,
N. V.
, and
Chien
,
T. Q.
,
2016
, “
Subharmonic Resonance of Duffing Oscillator With Fractional-Order Derivative
,”
ASME J. Comput. Nonlinear Dyn.
,
11
(
5
), p.
051018
.
16.
Wahi
,
P.
, and
Chatterjee
,
A.
,
2004
, “
Averaging Oscillations With Small Fractional Damping and Delayed Terms
,”
Nonlinear Dyn.
,
38
, pp.
3
22
.
17.
Nishimoto
,
K.
,
1989
, “
Nishimoto's Fractional Calculus of Elementary Functions
,”
International Conference of Fractional Calculus and Its Applications
, Nihon University, Tokyo, Japan, pp.
112
122
.
18.
Tseng
,
C.-C.
,
Pei
,
S.-C.
, and
Hsia
,
S.-C.
,
2000
, “
Computation of Fractional Derivatives Using Fourier Transform and Digital FIR Differentiator
,”
Signal Process.
,
80
(
1
), pp.
151
159
.
19.
Munkhammar
,
J. D.
,
2004
, “
Rieman–Liouville Fractional Derivatives and Taylor–Rieman Series
,” Project Report, Uppsala University, Uppsala, Sweden, Report No. 2004:7.
20.
Attari
,
M.
,
Haeri
,
M.
, and
Tavazoei
,
M. S.
,
2010
, “
Analysis of a Fractional Order van der Pol-Like Oscilattor Via Describing Function Method
,”
Nonlinear Dyn.
,
61
, pp.
265
274
.
21.
Barbosa
,
R. S.
,
Tenreiro Machado
,
J. A.
,
Vinagre
,
B. M.
, and
Calderón
,
A. J.
,
2007
, “
Analysis of the van der Pol Oscillator Containing Derivatives of Fractional Order
,”
J. Vib. Control
,
13
(
9–10
), pp.
1291
1301
.
22.
Chen
,
J.-H.
, and
Chen
,
W.-C.
,
2008
, “
Chaotic Dynamics of the Fractionally Damped van der Pol Equation
,”
Chaos Solitons Fractals
,
35
(
1
), pp.
188
198
.
23.
Ge
,
Z.-M.
, and
Hsu
,
M.-Y.
,
2007
, “
Chaos in a Generalized van der Pol System and in Its Fractional Order System
,”
Chaos Solitons Fractals
,
33
(
5
), pp.
1711
1745
.
24.
Ge
,
Z.-M.
, and
Hsu
,
M.-Y.
,
2008
, “
Chaos Excited Chaos Synchronizations of Integral and Fractional Order Generalized van der Pol Systems
,”
Chaos Solitons Fractals
,
36
(
3
), pp.
592
604
.
25.
Ge
,
Z.-M.
, and
Zhang
,
A.-R.
,
2007
, “
Chaos in a Modified van der Pol System and in Its Fractional Order Systems
,”
Chaos Solitons Fractals
,
32
(
5
), pp.
1791
1822
.
26.
Tavazoei
,
M. S.
,
Heari
,
M.
,
Attari
,
M.
,
Bolouki
,
S.
, and
Siami
,
M.
,
2009
, “
More Details on Analysis of Fractional-Order van der Pol Oscillator
,”
J. Vib. Control
,
15
(
6
), pp.
803
819
.
27.
Carla
,
M. A. P.
, and
Tenreira Machado
,
J. A.
,
2011
, “
Complex-Order van der Pol Oscillator
,”
Nonlinear Dyn.
,
65
(
3
), pp.
247
254
.
28.
Shen
,
Y.
,
Wei
,
P.
,
Sui
,
C.
, and
Yang
,
S.
,
2014
, “
Subharmonic Resonance of van der Pol Oscillator With Fractional—Order Derivative
,”
Math. Probl. Eng.
,
2014
, p.
738087
.
29.
Wei
,
P.
,
Shen
,
Y.-J.
, and
Yang
,
S.-P.
,
2014
, “
Super-Harmonic Resonance of Fractional-Order van der Pol Oscillator (in Chinese)
,”
Acta Phys. Sin.
,
63
(
1
), p.
010503
.
30.
Mitropolskii
,
I. A.
, and
Dao
,
N. V.
,
1997
,
Applied Asymptotic Methods in Nonlinear Oscillations
,
Kluwer Academic Publisher
,
Dordrecht, The Netherlands
.
31.
Dao
,
N. V.
,
1998
,
Stability of Dynamic Systems With Examples and Solved Problems
,
VNU Publishing House
,
Hanoi, Vietnam
.
32.
Dao
,
N. V.
,
1979
,
Nonlinear Oscillations of Higher Order Systems
,
NCSR Vietnam
,
Hanoi, Vietnam
.
33.
Dao
,
N. V.
,
1979
, “
Nonlinear Oscillation of Third Order Systems—Part 1: Autonomous Systems
,”
J. Tech. Phys.
,
20
(
4
), pp.
511
519
.
34.
Dao
,
N. V.
,
1980
, “
Nonlinear Oscillation of Third Order Systems—Part 2: Non-Autonomous Systems
,”
J. Tech. Phys.
,
21
(
1
), pp.
125
134
.
35.
Dao
,
N. V.
,
1980
, “
Nonlinear Oscillation of Third Order Systems—Part 3: Parametric Systems
,”
J. Tech. Phys.
,
21
(
2
), pp.
253
265
.
36.
Golmankhaneh
,
K. A.
,
Arefi
,
R.
, and
Baleanu
,
D.
,
2015
, “
Synchronization in a Nonidential Fractional Order of a Proposed Modified System
,”
J. Vib. Control
,
21
(
6
), pp.
1154
1161
.
37.
Sanders
,
J. A.
, and
Verhulst
,
F.
,
1985
,
Averaging Methods in Nonlinear Dynamical Systems
,
Springer
,
New York
.
You do not currently have access to this content.