We extend a typical system that possesses a transcritical bifurcation to a fractional-order version. The bifurcation and the resonance phenomenon in the considered system are investigated by both analytical and numerical methods. In the absence of external excitations or simply considering only one low-frequency excitation, the system parameter induces a continuous transcritical bifurcation. When both low- and high-frequency forces are acting, the high-frequency force has a biasing effect and it makes the continuous transcritical bifurcation transit to a discontinuous saddle-node bifurcation. For this case, the system parameter, the high-frequency force, and the fractional-order have effects on the saddle-node bifurcation. The system parameter induces twice a saddle-node bifurcation. The amplitude of the high-frequency force and the fractional-order induce only once a saddle-node bifurcation in the subcritical and the supercritical case, respectively. The system presents a nonlinear response to the low-frequency force. The system parameter and the low-frequency can induce a resonance-like behavior, though the high-frequency force and the fractional-order cannot induce it. We believe that the results of this paper might contribute to a better understanding of the bifurcation and resonance in the excited fractional-order system.

References

References
1.
Torvik
,
P. J.
, and
Bagley
,
R. L.
,
1984
, “
On the Appearance of the Fractional Derivative in the Behavior of Real Materials
,”
ASME J. Appl. Mech.
,
51
(
2
), pp.
294
298
.10.1115/1.3167615
2.
Diethelm
,
K.
, and
Freed
,
A. D.
,
1999
, “
On the Solution of Nonlinear Fractional-Order Differential Equations Used in the Modeling of Viscoplasticity
,”
Scientific Computing in Chemical Engineering II
,
Springer, Berlin Heidelberg
,
Germany
, pp.
217
224
10.1007/978-3-642-60185-9_24.
3.
Bagley
,
R. L.
, and
Calico
,
R. A.
,
1991
, “
Fractional Order State Equations for the Control of Viscoelastically Damped Structures
,”
J. Guid. Control Dyn.
,
14
(
2
), pp.
304
311
.10.2514/3.20641
4.
Adolfsson
,
K.
,
Enelund
,
M.
, and
Olsson
,
P.
,
2005
, “
On the Fractional Order Model of Viscoelasticity
,”
Mech. Time-Depend. Mater.
,
9
(
1
), pp.
15
34
.10.1007/s11043-005-3442-1
5.
West
,
B. J.
,
Bologna
,
M.
, and
Grigolini
,
P.
,
2003
,
Physics of Fractal Operators
,
Springer
,
New York
, pp.
235
270
10.1007/978-0-387-21746-8.
6.
Yang
,
F.
, and
Zhu
,
K.
,
2011
, “
A Note on the Definition of Fractional Derivatives Applied in Rheology
,”
Acta Mech. Sin.
,
27
(
6
), pp.
866
876
.10.1007/s10409-011-0526-9
7.
Wu
,
G. C.
,
Baleanu
,
D.
, and
Zeng
,
S. D.
,
2014
, “
Discrete Chaos in Fractional Sine and Standard Maps
,”
Phys. Lett. A
,
378
(
5–6
), pp.
484
487
.10.1016/j.physleta.2013.12.010
8.
Wu
,
G. C.
, and
Baleanu
,
D.
,
2014
, “
Discrete Fractional Logistic Map and Its Chaos
,”
Nonlinear Dyn.
,
75
(
1–2
), pp.
283
287
.10.1007/s11071-013-1065-7
9.
Wu
,
G. C.
, and
Baleanu
,
D.
, “
Discrete Chaos in Fractional Delayed Logistic Maps
,”
Nonlinear Dyn.
, (in press)10.1007/s11071-014-1250-3.
10.
Syta
,
A.
,
Litak
,
G.
,
Lenci
,
S.
, and
Scheffler
,
M.
,
2014
, “
Chaotic Vibrations of the Duffing System With Fractional Damping
,”
Chaos
,
24
(
1
), p.
013107
.10.1063/1.4861942
11.
Cao
,
J.
,
Ma
,
C.
,
Xie
,
H.
, and
Jiang
,
Z.
,
2010
, “
Nonlinear Dynamics of Duffing System With Fractional Order Damping
,”
ASME J. Comput. Nonlinear Dyn.
,
5
(
4
), p.
041012
.10.1115/1.4002092
12.
Yang
,
J. H.
,
Sanjuán
,
M. A. F.
,
Xiang
,
W.
, and
Zhu
,
H.
,
2013
, “
Pitchfork Bifurcation and Vibrational Resonance in a Fractional-Order Duffing Oscillator
,”
Pramana
,
81
(
6
), pp.
943
957
.10.1007/s12043-013-0621-5
13.
Yang
,
J. H.
,
Liu
,
H. G.
, and
Cheng
,
G.
,
2013
, “
The Pitchfork Bifurcation and Vibrational Resonance in a Quintic Oscillator
,”
Acta Phys. Sin.
,
62
(
18
), p.
180503
10.7498/aps.62.180503.
14.
El-Saka
,
H. A.
,
Ahmed
,
E.
,
Shehata
,
M. I.
, and
El-Sayed
,
A. M. A.
,
2009
, “
On Stability, Persistence and Hopf Bifurcation of Fractional Order Dynamical System
,”
Nonlinear Dyn.
,
56
(
1
), pp.
121
126
.10.1007/s11071-008-9383-x
15.
Babakhani
,
A.
,
Baleanu
,
D.
, and
Khanbabaie
,
R.
,
2012
, “
Hopf Bifurcation for a Class of Fractional Differential Equations With Delay
,”
Nonlinear Dyn.
,
69
(
3
), pp.
721
729
.10.1007/s11071-011-0299-5
16.
Abdelouahab
,
M.
,
Hamri
,
N.
, and
Wang
,
J.
,
2012
, “
Hopf Bifurcation and Chaos in Fractional-Order Modified Hybrid Optical System
,”
Nonlinear Dyn.
,
69
(
1–2
), pp.
275
284
.10.1007/s11071-011-0263-4
17.
Sun
,
K.
,
Wang
,
X.
, and
Sprott
,
J. C.
,
2010
, “
Bifurcations and Chaos in Fractional-Order Simplified Lorenz System
,”
Int. J. Bifurcation Chaos
,
20
(
4
), pp.
1209
1219
.10.1142/S0218127410026411
18.
Guckenheimer
,
J.
, and
Holmes
,
P.
,
2002
,
Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields
,
Springer-Verlag
,
New York
, pp.
117
165
10.1007/978-1-4612-1140-2.
19.
Thomsen
,
J. J.
,
2003
,
Vibrations and Stability: Advanced Theory, Analysis, and Tools
,
Springer-Verlag, Berlin
,
Heidelberg, Germany
.
20.
Medio
,
A.
, and
Lines
,
M.
,
2001
,
Nonlinear Dynamics: A Primer
,
Cambridge University Press
,
Cambridge
, pp.
133
148
10.1017/CBO9780511754050.
21.
Blekhman
,
I. I.
,
2000
,
Vibrational Mechanics: Nonlinear Dynamic Effects, General Approach, Applications
,
World Scientific
,
Singapore
.
22.
Blekhman
,
I. I.
,
2003
,
Selected Topics in Vibrational Mechanics
,
World Scientific
,
Singapore
10.1142/9789812794529.
23.
Monje
,
C. A.
,
Chen
,
Y.
,
Vinagre
,
B. M.
,
Xue
,
D.
, and
Feliu-Batlle
,
V.
,
2010
,
Fractional-Order Systems and Controls: Fundamentals and Applications
,
Springer
,
London, UK
, pp.
10
12
.
24.
Thomsen
,
J. J.
,
2002
, “
Some General Effects of Strong High-Frequency Excitation: Stiffening, Biasing, and Smoothening
,”
J. Sound Vib.
,
253
(
4
), pp.
807
831
.10.1006/jsvi.2001.4036
25.
Jeyakumari
,
S.
,
Chinnathambi
,
V.
,
Rajasekar
,
S.
, and
Sanjuán
,
M. A. F.
,
2011
, “
Vibrational Resonance in an Asymmetric Duffing oscillator
,”
Int. J. Bifurcation Chaos
,
21
(
1
), pp.
275
286
.10.1142/S0218127411028416
26.
Thomsen
,
J. J.
,
1999
, “
Using Fast Vibrations to Quench Friction-Induced Oscillations
,”
J. Sound Vib.
,
228
(
5
), pp.
1079
1102
.10.1006/jsvi.1999.2460
27.
Jeevarathinam
,
C.
,
Rajasekar
,
S.
, and
Sanjuán
,
M. A. F.
,
2011
, “
Theory and Numerics of Vibrational Resonance in Duffing Oscillators With Time-Eelayed Feedback
,”
Phys. Rev. E
,
83
(
6
), p.
066205
.10.1103/PhysRevE.83.066205
28.
Yang
,
J. H.
, and
Zhu
,
H.
,
2013
, “
Bifurcation and Resonance Induced by Fractional-Order Damping and Time Delay Feedback in a Duffing System
,”
Commun. Nonlinear Sci. Num. Simul.
,
18
(
5
), pp.
1316
1326
.10.1016/j.cnsns.2012.09.023
29.
Yang
,
J. H.
, and
Zhu
,
H.
,
2012
, “
Vibrational Resonance in Duffing Systems With Fractional-Order Damping
,”
Chaos
,
22
(
1
), p.
013112
.10.1063/1.3678788
30.
Borromeo
,
M.
, and
Marchesoni
,
F.
,
2006
, “
Vibrational Ratchets
,”
Phys. Rev. E
,
73
(
1
), p.
016142
.10.1103/PhysRevE.73.016142
31.
Rajasekar
,
S.
,
Abirami
,
K.
, and
Sanjuán
,
M. A. F.
,
2011
, “
Novel Vibrational Resonance in Multistable Systems
,”
Chaos
,
21
(
3
), p.
033106
.10.1063/1.3610213
32.
Blekhman
,
I. I.
, and
Landa
,
P. S.
,
2004
, “
Effect of Conjugate Resonances and Bifurcations Under the Biharmonic Excitation of a Pendulum With a Vibrating Suspension Axis
,”
Dokl. Phys.
,
49
(
3
), pp.
187
190
.10.1134/1.1710687
33.
Tcherniak
,
D
.,
1999
, “
The Influence of Fast Excitation on a Continuous System
,”
J. Sound Vib.
,
227
(
2
), pp.
343
360
.10.1006/jsvi.1999.2349
34.
Hansen
,
M. H.
,
2000
, “
Effect of High-Frequency Excitation on Natural Frequencies of Spinning Discs
,”
J. Sound Vib.
,
234
(
4
), pp.
577
589
.10.1006/jsvi.1999.2796
35.
Jensen
,
J. S.
,
1997
, “
Fluid Transport due to Nonlinear Fluid–Structure Interaction
,”
J. Fluid Struct.
,
11
(
3
), pp.
327
344
.10.1006/jfls.1996.0080
36.
Landa
,
P. S.
, and
McClintock
,
P. V. E.
,
2000
, “
Vibrational Resonance
,”
J. Phys. A: Math. Gen.
,
33
(
45
), pp.
L433
L438
.10.1088/0305-4470/33/45/103
37.
Blekhman
,
I. I.
, and
Landa
,
P. S.
,
2004
, “
Conjugate Resonances and Bifurcations in Nonlinear Systems Under Biharmonical Excitation
,”
Int. J. Nonlinear Mech.
,
39
(
3
), pp.
421
426
.10.1016/S0020-7462(02)00201-9
38.
Chizhevsky
,
V. N.
, and
Giacomelli
,
G.
,
2006
, “
Experimental and Theoretical Study of Vibrational Resonance in a Bistable System With Asymmetry
,”
Phys. Rev. E
,
73
(
2
), p.
022103
.10.1103/PhysRevE.73.022103
39.
Chizhevsky
,
V. N.
,
2008
, “
Analytical Study of Vibrational Resonance in an Overdamped Bistable Oscillator
,”
Int. J. Bifurcation Chaos
,
18
(
6
), pp.
1767
1773
.10.1142/S021812740802135X
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