In this paper, we propose a method to calculate asymptotically period two sinks and define the range of stability of fixed points for a variety of discrete fractional systems of the order 0<α<2. The method is tested on various forms of fractional generalizations of the standard and logistic maps. Based on our analysis, we make a conjecture that chaos is impossible in the corresponding continuous fractional systems.

1.
J. A. T.
Machado
,
F. B.
Duarte
, and
G. M.
Duarte
,
Int. J. Bifurcation Chaos
22
,
1250249
(
2012
).
2.
J. A. T.
Machado
,
C. M. A.
Pinto
, and
A. M.
Lopes
,
Signal Process.
107
,
246
(
2015
).
3.
V. E.
Tarasov
and
V. V.
Tarasova
,
Int. J. Manage. Soc. Sci.
5
,
327
(
2016
).
4.
M. J.
Kahana
,
Foundations of Human Memory
(
Oxford University Press
,
New York
,
2012
).
5.
D. C.
Rubin
and
A. E.
Wenzel
,
Psychol. Rev.
103
,
743
(
1996
).
6.
J. T.
Wixted
,
J. Exp. Psychol.: Learn., Mem., Cognit.
16
,
927
(
1990
).
7.
J. T.
Wixted
and
E.
Ebbesen
,
Psychol. Sci.
2
,
409
(
1991
).
8.
J. T.
Wixted
and
E.
Ebbesen
,
Mem. Cognit.
25
,
731
(
1997
).
9.
C.
Donkin
and
R. M.
Nosofsky
,
Psychol. Sci.
23
,
625
(
2012
).
10.
J. R.
Anderson
,
Learning and Memory: An Integrated Approach
(
Wiley
,
New York
,
1995
).
11.
A. L.
Fairhall
,
G. D.
Lewen
,
W.
Bialek
, and
R. R.
de Ruyter van Steveninck
,
Nature
412
,
787
(
2001
).
12.
D. A.
Leopold
,
Y.
Murayama
, and
N. K.
Logothetis
,
Cereb. Cortex
13
,
422
(
2003
).
13.
A.
Toib
,
V.
Lyakhov
, and
S.
Marom
,
J. Neurosci.
18
,
1893
(
1998
).
14.
N.
Ulanovsky
,
L.
Las
,
D.
Farkas
, and
I.
Nelken
,
J. Neurosci.
24
,
10440
(
2004
).
15.
M. S.
Zilany
,
I. C.
Bruce
,
P. C.
Nelson
, and
L. H.
Carney
,
J. Acoust. Soc. Am.
126
,
2390
(
2009
).
16.
B. N.
Lundstrom
,
A. L.
Fairhall
, and
M.
Maravall
,
J. Neurosci.
30
,
5071
(
2010
).
17.
B. N.
Lundstrom
,
M. H.
Higgs
,
W. J.
Spain
, and
A. L.
Fairhall
,
Nat. Neurosci.
11
,
1335
(
2008
).
18.
C.
Pozzorini
,
R.
Naud
,
S.
Mensi
, and
W.
Gerstner
,
Nat. Neurosci.
16
,
942
(
2013
).
20.
M.
Edelman
,
Commun. Nonlinear Sci. Numer. Simul.
16
,
4573
(
2011
).
21.
A. S.
Deshpande
and
V.
Daftardar-Gejji
,
Chaos, Solitons Fractals
102
,
119
(
2017
).
22.
M.
Edelman
and
V. E.
Tarasov
,
Phys. Lett. A
374
,
279
(
2009
).
24.
M.
Edelman
and
L. A.
Taieb
, in
Advances in Harmonic Analysis and Operator Theory; Series: Operator Theory: Advances and Applications
, edited by
A.
Almeida
,
L.
Castro
, and
F.-O.
Speck
(
Springer
,
Basel
,
2013
), Vol.
229
, pp.
139
155
.
25.
J.
Jagan Mohan
,
Commun. Appl. Anal.
20
,
585
(
2016
).
26.
J.
Jagan Mohan
,
Fractional Differ. Calculus
7
,
339
(
2017
).
27.
I.
Area
,
J.
Losada
, and
J. J.
Nieto
,
Abstr. Appl. Anal.
2014
,
392598
.
28.
E.
Kaslik
and
S.
Sivasundaram
,
Nonlinear Anal. Real World Appl.
13
,
1489
(
2012
).
29.
M. S.
Tavazoei
and
M.
Haeri
,
Automatica
45
,
1886
(
2009
).
30.
J.
Wang
,
M.
Feckan
, and
Y.
Zhou
,
Commun Nonlinear Sci. Numer. Simul.
18
,
246
(
2013
).
31.
M.
Yazdani
and
H.
Salarieh
,
Automatica
47
,
1834
(
2011
).
32.
G. M.
Zaslavsky
,
A. A.
Stanislavsky
, and
M.
Edelman
,
Chaos
16
,
013102
(
2006
).
33.
M.
Edelman
, in
Nonlinear Dynamics and Complexity
, Nonlinear Systems and Complexity, edited by
A.
Afraimovich
,
A. C. J.
Luo
, and
X.
Fu
(
Springer
,
New York
,
2014
), pp.
79
120
.
34.
M.
Edelman
,
Discontinuity, Nonlinearity, Complexity
1
,
305
(
2013
).
35.
A. A.
Stanislavsky
,
Eur. Phys. J. B
49
,
93
(
2006
).
36.
J.
Cermak
and
L.
Nechvatal
,
Nonlinear Dyn.
87
,
939
(
2017
).
37.
D.
Matignon
, in
Proceedings of the International Meeting on Automated Compliance Systems and the International Conference on Systems, Man, and Cybernetics (IMACS-SMC 96)
(Lille, France,
1996
), pp.
963
968
.
38.
I.
Grigorenko
and
E.
Grigorenko
,
Phys. Rev. Lett.
91
,
034101
(
2003
).
39.
E.
Ahmed
,
A. M. A.
El-Sayed
, and
H. A. A.
El-Saka
,
Phys. Lett. A
358
,
1
(
2006
).
40.
H. A.
El-Saka
,
E.
Ahmed
,
M. I.
Shehata
, and
A. M. A.
El-Sayed
,
Nonlinear Dyn.
56
,
121
(
2009
).
41.
Y.
Li
,
Y. Q.
Chen
, and
I.
Podlubny
,
Comput. Math. Appl.
59
,
1810
(
2010
).
42.
N.
Aguila-Camacho
,
M. A.
Duarte-Mermoud
, and
J. A.
Gallegos
,
Commun. Nonlinear Sci. Numer. Simul.
19
,
2951
(
2014
).
43.
T.
Li
and
Y.
Wang
,
Discrete Dyn. Nat. Soc.
2014
,
724270
.
44.
B. K.
Lenka
and
S.
Banerjee
,
Nonlinear Dyn.
85
,
167
(
2016
).
45.
A. A.
Stanislavsky
,
Chaos
16
,
043105
(
2006
).
46.
F.
Chen
and
Z.
Liu
,
J. Appl. Math.
2012
,
879657
.
47.
F.
Jarad
,
T.
Abdeljawad
,
D.
Baleanu
, and
K.
Bicen
,
Abstr. Appl. Anal.
2012
,
476581
.
48.
J. J.
Mohan
,
N.
Shobanadevi
, and
G. V. S. R.
Deekshitulu
,
Ital. J. Pure Appl. Math.
32
,
165
(
2014
).
49.
M.
Wyrwas
,
E.
Pawluszewicz
, and
E.
Girejko
,
Kybernetika
15
,
112
136
(
2015
).
50.
D.
Baleanu
,
G.-C.
Wu
,
Y.-R.
Bai
, and
F.-L.
Chen
,
Commun. Nonlinear Sci. Numer. Simul.
48
,
520
(
2017
).
51.
I.
Petras
,
Fractional Calculus Appl. Anal.
12
,
269
(
2009
).
52.
C. P.
Li
and
F. R.
Zhang
,
Eur. Phys. J.: Spec. Top.
193
,
27
(
2011
).
53.
M.
Rivero
,
S. V.
Rogozin
,
J. A. T.
Machado
, and
J. J.
Trujilo
,
Math. Probl. Eng.
2013
,
356215
.
54.
I.
Petras
,
Fractional-Order Nonlinear Systems
(
Springer
,
Berlin
,
2011
).
55.
Y.
Zhou
,
Basic Theory of Fractional Differential Equations
(
World Scientific
,
Singapore
,
2014
).
57.
M.
Edelman
, in
International Conference on Fractional Differentiation and Its Applications (ICFDA)
(
2014
), pp.
1
6
.
58.
M.
Edelman
,
Discontinuity, Nonlinearity, Complexity
4
,
391
(
2015
).
59.
M.
Edelman
, in
Chaotic, Fractional, and Complex Dynamics: New Insights and Perspectives
, Understanding Complex Systems, edited by
M.
Edelman
,
E.
Macau
, and
M. A. F.
Sanjuan
(
eBook, Springer
,
2018
), pp.
147
171
.
60.
V. E.
Tarasov
,
J. Phys. A: Math. Theor.
42
,
465102
(
2009
).
61.
V. E.
Tarasov
,
J. Math. Phys.
50
,
122703
(
2009
).
62.
V. E.
Tarasov
,
Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media
(
HEP, Springer
,
Beijing, Berlin, Heidelberg
,
2011
).
63.
V. E.
Tarasov
and
G. M.
Zaslavsky
,
J. Phys. A: Math. Theor.
41
,
435101
(
2008
).
64.
G.-C.
Wu
,
D.
Baleanu
, and
S.-D.
Zeng
,
Phys. Lett. A
378
,
484
(
2014
).
65.
S. G.
Samko
,
A. A.
Kilbas
, and
O. I.
Marichev
,
Fractional Integrals and Derivatives Theory and Applications
(
Gordon and Breach
,
New York
,
1993
).
66.
A. A.
Kilbas
,
H. M.
Srivastava
, and
J. J.
Trujillo
,
Theory and Application of Fractional Differential Equations
(
Elsevier
,
Amsterdam
,
2006
).
67.
I.
Podlubny
,
Fractional Differential Equations
(
Academic Press
,
San Diego
,
1999
).
68.
K. S.
Miller
and
B.
Ross
, in
Univalent Functions, Fractional Calculus, and Their Applications
, edited by
H. M.
Srivastava
, and
S.
Owa
(
Ellis Howard
,
Chichester
,
1989
), pp.
139
151
.
69.
H. L.
Gray
and
N. F.
Zhang
,
Math. Comput.
50
,
513
(
1988
).
70.
F.
Atici
and
P.
Eloe
,
Proc. Am. Math. Soc.
137
,
981
(
2009
).
71.
F.
Atici
and
P.
Eloe
,
Electron. J. Qual. Theory Differ. Equ. Spec. Ed.
I3
,
1
(
2009
).
72.
G. A.
Anastassiou
, see http://arxiv.org/abs/0911.3370 for the definition of the Caputo-like difference operator (
2009
).
73.
N. R. O.
Bastos
,
R. A. C.
Ferreira
, and
D. F. M.
Torres
, “
Discrete-time fractional variational problems
,”
Signal Process.
91
,
513
(
2011
).
74.
R. A. C.
Ferreira
and
D. F. M.
Torres
,
Appl. Anal. Discrete Math.
5
,
110
(
2011
).
75.
A. A.
Kilbas
,
B.
Bonilla
, and
J. J.
Trujillo
,
Dokl. Math.
62
,
222
(
2000
).
76.
A. A.
Kilbas
,
B.
Bonilla
, and
J. J.
Trujillo
,
Demonstratio Math.
33
,
583
(
2000
).
77.
G. M.
Zaslavsky
,
Hamiltonian Chaos and Fractional Dynamics
(
Oxford University Press
,
Oxford
,
2008
).
78.
F.
Chen
,
X.
Luo
, and
Y.
Zhou
,
Adv. Differ. Equations
2011
,
713201
.
You do not currently have access to this content.