Synchronization of Discrete Fractional Odd Logistic System
Based on Parametric Adaptive Control Algorithm
Shengdan Hu and Yihong Wang
Department of Computer Science, Shanghai Normal University Tianhua College, Shanghai, China
hushengdan@163.com, dhhlwyh@163.com
Keywords: Fractional odd logistic map, synchronization, difference scheme, chaos.
Abstract: In this study, a simple but efficient method for chaos synchronization of fractional difference system is
proposed, which is based upon the parametric adaptive control algorithm. Using this new method, chaos
synchronization for discrete fractional odd logistic system is implemented.
1 INTRODUCTION
Since Pecora and Carroll introduced an idea of
achieving synchronization between the drive and
response systems, chaos synchronization has been
widely explored and studied due to its potential
applications in secure communication, ecological
systems and system identification. Recent studies
show that chaos of fractional differential systems
can be synchronized, see and references cited
therein. Compared with the fruitful results in the
chaos synchronization of continuous fractional
differential equations, the fractional difference
equation is a particularly new topic. The dynamical
behaviors of the fractional one and two dimensional
maps and the results show that the chaos does exist
there. The DFC (Pyragas K., 1992 and Pyragas K.,
1992) is proved to be an efficient tool to discrete the
chaotic systems with a memory effect. Naturally, a
question maybe put forth: how to achieve the
fractional synchronization of such maps? In this
paper, we investigate the chaos synchronization of
the discrete fractional odd logistic map in the design
the synchronized systems based upon the parametric
adaptive control algorithm. The remainder of this
paper is organized as follows. In section 2,
introduces the definitions and the properties of the
discrete fractional calculus. Section 3 presents
fractional odd logistic map on time scales and shows
the discrete chaotic solutions while the difference
orders and the coefficients are changing. Section 4 is
the conclusion.
2 FRACTIONAL ODD LOGISTIC
MAP
Considering the discrete fractional calculus, we start
with some necessary definitions from discrete
fractional calculus theory and preliminary results so
that this paper is self-contained.
Definition 1. (F.M. Atici, P.W. Eloe, 2009) Let
th fractional sum of
is defined by
∆
≔
1
,
Where a is the starting point,
= + 1 and
is defined for = mod (1) and ∆
-
is defined
for = ( + ) mod (1). In particular ∆
-
maps a
function defined on
to functions defined on
, where
,1,2⋯. In addition,
Γ1
Γ1
Definition 2. ( T. Abdeljawad, 2011) For 0,
:
→ and α be given, the Caputo-like delta
difference is defined by
∆
≔∆
∆
1
Γ
t
,
1
where is the difference order, and
580
Hu, S. and Wang, Y.
Synchronization of Discrete Fractional Odd Logistic System Based on Parametric Adaptive Control Algorithm.
In 3rd International Conference on Electromechanical Control Technology and Transportation (ICECTT 2018), pages 580-583
ISBN: 978-989-758-312-4
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
Theorem 1. ( F.L. Chen, X.N. Luo, Y. Zhou,
2011) For the delta fractional difference equation
∆
1,1
∆
,
1,0,1,…,1.
the equivalent discrete integral equation can be
obtained as
0
1
Γ
1,
1
1
where the initial iteration reads
0
!
∆
.
The complex difference equation with long-term
memory is obtained here. It can reduce to the
classical one when the difference order = 1 but the
integer one does not hold the discrete memory.
Through a discrete fractional odd logistic map it
reveals that the dynamical behavior holds discrete
memory even the difference order is very small. For
the famous odd logistic map
1
,0,1,2⋯
The odd logistic map is based to maps of the
plane with dihedral symmetry.
We can redefined it as
∆
1
From the discrete fractional calculus, the
fractional one can be given as
∆
t
1
1
,0
1,∈
Where 1. From [1], we can obtain the
following discrete integral form from 01.
1
Γ
1,
1
∈
2
where
is a discrete kernel function and
. As a result, the
numerical formula can be presented explicitly
1
Γ
Γ
Γ
1
1,
1
.3
For the fractional odd logistic map, an explicit
numerical formula ban be given as
1
Γ
Γ
Γ
1
1
1
.4
3 CHAOS SYNCHRONIZATION
OF THE FRACTIONAL ODD
LOGISTIC MAP
3.1 Sufficient conditions of system
synchronization
Making use of equation (4), we obtain the iteration
equations of chaotic system as follows:
1
Γ
Γ
Γ
1
1
1
.5
1
Γ
Γ
Γ
1
1
1
.6
We iterate the following three equations to
synchronize the two systems by controlling the
initial states and parameters of the response system.
1
Γ
Γ
Γ
1
,
1
Γ
Γ
Γ
1
1
,
7
1
KΓ
1
Γ
1
.
Where,
is the master system,
is the
response system,
is the adaptive control
parameter, and K is the control stiffness and it is
adjustable.
Lemma 1.( T. Abdeljawad, 2011)
Γ
Γ
1
Γ
Γ
1
Γ
1
Γ
1
.
Theorem 2. Iteration system (7) is convergent
when 6Γ
1
0.
Proof. Let ∆
,
, where
is the synchronization error
system, then we can get the following equation from
system (7):
∆
1
Γ
Γ
Γ
1
.8
Subtracting
from the both sides of equation
1
,
yields
Synchronization of Discrete Fractional Odd Logistic System Based on Parametric Adaptive Control Algorithm
581
∆
∆
1
1
KΓ
1
Γ
1
Γ
Γ
1
Γ
1
0
9
It is obvious that for any 1,2,⋯.∈
0,1 because of 0∈0,1. Then apply lemma 1
to equation (9), the following inequality holds
∆
1
KΓ
1
Γ
1
Γ
Γ
Γ
1
∆
0
.10
When
1
KΓ
1
Γ
1
Γ
Γ
Γ
1
1,
also if 6Γ
1
0, we can get
lim
→
∆
0,
or lim
→
0, the iteration system (7) is
convergent.
3.2 Numerical simulations
In this section, three cases for different fractional
order and different K in system (7) will be given
to verify the synchronization of the odd logistic
system. We assume the initial condition associated
with master system and response system as
0
0.4 and
0
0.4.
Case1:1.
We choose
0
0.4 and K=-1 as the
parameters. By numerical simulation, we can see the
dynamical behaviors of master system and response
system as show in figure 1(a). The error system
showed in figure 1(b) is stable at zero, so the
synchronization can reach.
(a)
(b)
Figure 1 α=1, u(0)=0.4, K=-1
Case2:0.3.
The parameters are chosen as
0
0.4 and
K=-4, and the dynamical behaviors of master
system, response system and error system are
showed in figure 2(a), figure 2(b) respectively.
(a)
(b)
Figure 2 α=0.3, u(0)=0.4, K=-4
Case3:0.1.
ICECTT 2018 - 3rd International Conference on Electromechanical Control Technology and Transportation
582
In this case, we also choose
0
0.4 and K=-
4 the same as the parameters in case 2. Figure 3
shows the history of
,
and
.
(a)
(b)
Figure 3 α=0.1, u(0)=0.4, K=-4
4 CONCLUSION
In this paper, fractional odd logistic system is
investigated, and parametric adaptive control
algorithm is applied to synchronize two chaotic
systems. We proved that the sufficient conditions of
system synchronization is -6Γ(α+1)<K<0.
Moreover, numerical simulations are given and the
results show that the algorithm can work efficiently
for synchronization. Future works regarding this
topic include varying parameters of the control
system or applying the adaptive control algorithm to
other systems. Also, the studies of this paper may
have some referenced value for secure
communication.
ACKNOWLEDGEMENTS
This work is financially supported by the Zhejiang
Natural Science Foundation (Grant no.
LQ12A01010), the special fund for outstanding
young teachers in Shanghai universities research,
and the foundation of Shanghai Normal University
Tianhua College.
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Synchronization of Discrete Fractional Odd Logistic System Based on Parametric Adaptive Control Algorithm
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