Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 590421, 14 pages
http://dx.doi.org/10.1155/2013/590421
Research Article
A New Series of Three-Dimensional Chaotic Systems with
Cross-Product Nonlinearities and Their Switching
Xinquan Zhao,1 Feng Jiang,1 Zhigang Zhang,2 and Junhao Hu3
1
School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China
Department of Statistics and Applied Mathematics, Hubei University of Economics, Wuhan 430205, China
3
School of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China
2
Correspondence should be addressed to Xinquan Zhao; zhaoxq56@gmail.com
Received 31 October 2012; Accepted 25 December 2012
Academic Editor: Xinzhi Liu
Copyright © 2013 Xinquan Zhao et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper introduces a new series of three-dimensional chaotic systems with cross-product nonlinearities. Based on some
conditions, we analyze the globally exponentially or globally conditional exponentially attractive set and positive invariant set
of these chaotic systems. Moreover, we give some known examples to show our results, and the exponential estimation is explicitly
derived. Finally, we construct some three-dimensional chaotic systems with cross-product nonlinearities and study the switching
system between them.
where 𝑥𝑇 = (𝑥1 , 𝑥2 , 𝑥3 ) and
1. Introduction
Since Lorenz discovered the well-known Lorenz chaotic system, many other chaotic systems have been found, including
the well-known Rössler system and Chua’s circuit, which
serve as models of the study of chaos [1–12].
The Lorenz system plays an important role in the study
of nonlinear science and chaotic dynamics [13–18]. We know
that it is extremely difficult to obtain the information of
chaotic attractor directly from system. Most of the results
in the literature are based on computer simulations. When
calculating the Lyapunov exponents of the system, one needs
to assume that the system is bounded in order to conclude
chaos. Therefore, the study of the globally attractive set of the
Lorenz system is not only theoretically significant but also
practically important. Moreover, Liao et al. [19, 20] gave
globally exponentially attractive set and positive invariant set
for the classical Lorenz system and the generalized system by
constructive proofs. In addition, Yu et al. [21] studied the
problem of invariant set of systems, which was considered as
a more generalized Lorenz system.
In this paper, we consider the following three-dimensional autonomous systems with cross-product nonlinearities:
𝑥̇ = 𝐴𝑥 + 𝑓 (𝑥) + 𝐶,
(1)
𝑎11 𝑎12 𝑎13
𝐴 = [𝑎21 𝑎22 𝑎23 ] ;
[𝑎31 𝑎32 𝑎33 ]
𝑐1
𝐶 = [𝑐2 ] ;
[𝑐3 ]
𝑏𝑖11 𝑏𝑖12 𝑏𝑖13
𝐵𝑖 = [𝑏𝑖21 𝑏𝑖22 𝑏𝑖23 ] ,
[𝑏𝑖31 𝑏𝑖32 𝑏𝑖33 ]
𝑥𝑇 𝐵1 𝑥
𝑓 (𝑥) = [𝑥𝑇 𝐵2 𝑥] ;
𝑇
[𝑥 𝐵3 𝑥]
𝑖 = 1, 2, 3
(2)
with 𝑎𝑖𝑗 , 𝑏𝑖𝑗𝑘 , 𝑐𝑖 ∈ 𝑅, 𝑖, 𝑗, 𝑘 = 1, 2, 3. This second-order dynamic system may be regarded as the most general Lorenz system.
For such system, we can choose Lyapunov function:
𝑉 (𝑋 (𝑡)) =
1
2
2
[(𝜆 1 𝑥1 − 𝑑1 ) + (𝜆 2 𝑥2 − 𝑑2 )
2
2
(3)
+(𝜆 3 𝑥3 − 𝑑3 ) ] ,
which is obviously positive definite and radially unbounded,
where 𝑑𝑖 , 𝜆 𝑖 , 𝑖 = 1, 2, 3 are undetermined parameters. In this
paper, we will study this more general Lorenz system (1) than
the classical system and the generalized Lorenz system. The
result obtained contains earlier results as its special cases.
This paper is organized as follows. In Section 2, we define
the globally exponentially attractive set and positive invariant
2
set and the globally conditional exponentially attractive set
and positive invariant set of the three-dimensional chaotic
systems with cross-product nonlinearities. In Section 3, the
qualitative analysis of the exponentially attractive set and
positive invariant set of the chaotic systems has been done.
In Section 4, we also suggest an idea to construct chaotic
systems, and some new chaotic systems and switched chaotic
systems are illustrated.
2. Preliminaries
In this section, we present some basic definitions which
are needed for proving all theorems in the next section.
For convenience, denote 𝑋 := (𝑥1 , 𝑥2 , 𝑥3 ) and 𝑋(𝑡) :=
𝑋(𝑡, 𝑡0 , 𝑋0 ).
Definition 1. For the three-dimensional autonomous systems
with cross-product nonlinearities (1), if there exists compact
(bounded and closed) set Ω ∈ 𝑅3 such that for all 𝑋0 ∈ 𝑅3 ,
the following condition: 𝜌(𝑋(𝑡), Ω) := inf 𝑌∈Ω ‖𝑋(𝑡)−𝑌‖ → 0
as 𝑡 → +∞, holds, then the set Ω is said to be globally
attractive. That is, system (1) is ultimately bounded; namely,
system (1) is globally stable in the sense of Lagrange or
dissipative with ultimate bound.
Furthermore, if for all 𝑋0 ∈ Ω0 ⊆ Ω ⊂ 𝑅3 , 𝑋(𝑡, 𝑡0 , 𝑋0 ) ⊆
Ω0 , then Ω0 for 𝑡 ≥ 0 is called the positive invariant set of the
system (1).
Definition 2. For the three-dimensional autonomous systems
with cross-product nonlinearities (1), if there exist compact
set Ω ⊂ 𝑅3 such that for all 𝑋0 ∈ 𝑅3 and constants 𝑀 >
0, 𝛼 > 0 such that 𝜌(𝑋(𝑡), Ω) ≤ 𝑀𝑒−𝛼(𝑡−𝑡0 ) , then the threedimensional autonomous systems with cross-product nonlinearities system (1) are said to have globally exponentially
attractive set, or the system (1) is globally exponentially
stable in the sense of Lagrange, and Ω is called the globally
exponentially attractive set.
Definition 3. For the three-dimensional autonomous systems
with cross-product nonlinearities (1), if there exist compact
set Ω ⊂ 𝑅3 , a constant 𝛼 > 0, and a bounded function
𝑀(𝑥0 , 𝑡) > 0 on 𝑡, as 𝑡 ≥ 𝑡0 , such that 𝜌(𝑋(𝑡), Ω) ≤
𝑀(𝑥0 )𝑒−𝛼(𝑡−𝑡0 ) , where 𝑀(𝑥0 ) = sup 𝑀(𝑥0 , 𝑡), 𝑡 ≥ 𝑡0 , then the
system (1) is said to have globally conditional exponentially
attractive set, or the system (1) is globally conditional exponentially stable in the sense of Lagrange, and Ω is called the
globally conditional exponentially attractive set.
In general, from the definition we see that a globally exponential attractive set is not necessarily a positive invariant set.
But our results obtained in the next section indeed show that
a globally exponentially attractive set is a positive invariant
set.
Note that it is difficult to verify the existence of Ω in
Definition 2. Since the Lyapunov direct method is still a powerful tool in the study of asymptotic behaviour of nonlinear
dynamical systems, the following definition is more useful in
applications.
Journal of Applied Mathematics
Definition 4. For three-dimensional autonomous systems
with cross-product nonlinearities (1), if there exist a positive
definite and radially unbounded Lyapunov function 𝑉(𝑋(𝑡))
and positive numbers 𝐿 > 0, 𝛼 > 0 such that the following
inequality
𝑉 (𝑋 (𝑡)) − 𝐿 ≤ (𝑉 (𝑋0 ) − 𝐿) 𝑒−𝛼(𝑡−𝑡0 )
(4)
is valid for 𝑉(𝑋(𝑡)) > 𝐿(𝑡 ≥ 𝑡0 ), then the system (1) is said to
be globally exponentially attractive or globally exponentially
stable in the sense of Lagrange, and Ω := {𝑋 | 𝑉(𝑋(𝑡)) ≤
𝐿, 𝑡 ≥ 𝑡0 } is called the globally exponentially attractive set.
Definition 5. For the three-dimensional autonomous systems
with cross-product nonlinearities (1), if there exist a positive
definite and radially unbounded Lyapunov function 𝑉(𝑋(𝑡))
and a bounded function 𝐿(𝑥0 , 𝑡) > 0, on 𝑡, as 𝑡 ≥ 𝑡0 , and 𝛼 > 0
such that the following inequality
𝑉 (𝑋 (𝑡)) − 𝐿 (𝑥0 ) ≤ (𝑉 (𝑋0 ) − 𝐿 (𝑥0 )) 𝑒−𝛼(𝑡−𝑡0 )
(5)
is valid for 𝑉(𝑋(𝑡)) > 𝐿(𝑥0 ), 𝑡 ≥ 𝑡0 , where 𝐿(𝑥0 ) =
sup 𝐿(𝑥0 , 𝑡), 𝑡 ≥ 𝑡0 , then the system (1) is said to be globally
conditional exponentially attractive or globally conditional
exponentially stable in the sense of Lagrange, and Ω := {𝑋 |
𝑉(𝑋(𝑡)) ≤ 𝐿(𝑥0 ), 𝑡 ≥ 𝑡0 } is called the globally conditional
exponentially attractive set.
3. Qualitative Analysis
We call the dynamic system (1) the first class three-dimensional chaotic system with cross-product nonlinearities (1), if
there are some nonzero numbers {𝜆 1 , 𝜆 2 , 𝜆 3 } so as to satisfy
conditions
𝜆21 𝑏111 = 0,
𝜆21 (𝑏112 + 𝑏121 ) + 𝜆22 𝑏211 = 0,
𝜆22 𝑏222 = 0,
𝜆22 (𝑏221 + 𝑏212 ) + 𝜆21 𝑏122 = 0,
𝜆23 𝑏333 = 0,
𝜆23 (𝑏313 + 𝑏331 ) + 𝜆21 𝑏133 = 0,
𝜆21 (𝑏113 + 𝑏131 ) + 𝜆23 𝑏311 = 0,
(6)
𝜆22 (𝑏223 + 𝑏232 ) + 𝜆23 𝑏322 = 0,
𝜆23 (𝑏323 + 𝑏332 ) + 𝜆22 𝑏233 = 0,
𝜆21 (𝑏123 + 𝑏132 ) + 𝜆22 (𝑏213 + 𝑏231 ) + 𝜆23 (𝑏321 + 𝑏312 ) = 0.
Condition (6) is satisfied by some known three-dimensional quadratic autonomous chaotic systems, the wellknown Lorenz system [1–3], the Rössler system [5], the
Rucklidge system [6], and the Chen system [7, 8]. Lorenz
systems are widely studied and the references therein [9–
12, 19–21]. For example, consider the classical Lorenz system
𝑥̇ = 𝜎 (𝑦 − 𝑥) ,
𝑦̇ = 𝜌𝑥 − 𝛾𝑦 − 𝑥𝑧,
𝑧̇ = 𝑥𝑦 − 𝛽𝑧,
(7)
Journal of Applied Mathematics
3
Lemma 6. Suppose 𝜆 𝑖 > 0, 𝑖 = 1, 2, 3,
and the general Lorenz systems
𝜆21 𝑎12 + 𝜆22 𝑎21 + 𝜆 3 𝑑3 (𝑏312 + 𝑏321 ) = 0,
𝑥̇ = −𝑎𝑥 + 𝑏𝑦 + 𝑦𝑧,
𝑦̇ = 𝑐𝑥 − 𝑦 − 𝑥𝑧,
𝜆21 𝑎13 + 𝜆23 𝑎31 + 𝜆 2 𝑑2 (𝑏213 + 𝑏231 ) = 0,
(8)
𝜆22 𝑎23 + 𝜆23 𝑎32 + 𝜆 1 𝑑1 (𝑏123 + 𝑏132 ) = 0,
𝑧̇ = 𝑑𝑦 − 𝑧 + 𝑥𝑦,
(13)
𝑎11 + 𝜇1 < 0,
𝑥̇ = −𝑧,
𝑦̇ = −𝑦 − 𝑥2 ,
𝑎22 + 𝜇2 < 0,
(9)
𝑎33 + 𝜇3 < 0.
𝑧̇ = 1.7𝑥 + 𝑦 + 1.7.
The function (12) has maximum.
Thus it can be seen that condition (6) is very important
in qualitative analysis of the exponentially attractive set and
positive invariant set of Lorenz systems.
We will research this dynamic system in two cases.
First, supposing 𝑏111 = 𝑏122 = 𝑏133 = 𝑏211 = 𝑏222 = 𝑏233 =
𝑏311 = 𝑏322 = 𝑏333 = 0, 𝑏𝑖𝑖𝑗 = −𝑏𝑖𝑗𝑖 , 𝑖, 𝑗 = 1, 2, 3, ∃𝑏𝑖𝑗𝑘 ≠ −
𝑏𝑖𝑘𝑗 , 𝑖 ≠ 𝑗 ≠ 𝑘, the dynamic system (1) can be rewritten as
Proof. Consider
𝑓𝑥󸀠1 (𝜇, 𝑋) = 2 (𝜆21 𝑎11 + 𝜇1 𝜆21 ) 𝑥1
+ (𝜆21 𝑐1 + 𝜇1 𝑑1 𝜆 1 − 𝜇1 𝜆21 𝑑1 − 𝜆 1 𝑑1 𝑎11
−𝜆 2 𝑑2 𝑎21 − 𝜆 3 𝑑3 𝑎31 ) ,
𝑓𝑥󸀠2 (𝜇, 𝑋) = 2 (𝜆22 𝑎22 + 𝜇2 𝜆22 ) 𝑥2
3
𝑥1̇ = ∑𝑎1𝑗 𝑥𝑗 + (𝑏123 + 𝑏132 ) 𝑥2 𝑥3 + 𝑐1 ,
+ (𝜆22 𝑐2 + 𝜇2 𝑑2 𝜆 2 − 𝜇2 𝜆22 𝑑2 − 𝜆 1 𝑑1 𝑎12
𝑗=1
3
𝑥2̇ = ∑𝑎2𝑗 𝑥𝑗 + (𝑏213 + 𝑏231 ) 𝑥1 𝑥3 + 𝑐2 ,
−𝜆 2 𝑑2 𝑎22 − 𝜆 3 𝑑3 𝑎32 ) ,
(10)
𝑗=1
𝑓𝑥󸀠3 (𝜇, 𝑋) = 2 (𝜆23 𝑎33 + 𝜇3 𝜆23 ) 𝑥3
3
𝑥3̇ = ∑𝑎3𝑗 𝑥𝑗 + (𝑏312 + 𝑏321 ) 𝑥1 𝑥2 + 𝑐3 .
+ (𝜆23 𝑐3 + 𝜇3 𝑑3 𝜆 3 − 𝜇3 𝜆23 𝑑3 − 𝜆 1 𝑑1 𝑎13
𝑗=1
(14)
−𝜆 2 𝑑2 𝑎23 − 𝜆 3 𝑑3 𝑎33 ) ,
The construction techniques of this kind of Lorenz
systems are to pay attention to satisfing formula
𝑓𝑥󸀠󸀠2 (𝜇, 𝑋) = 2𝜆21 (𝑎11 + 𝜇1 ) ,
1
𝜆21 (𝑏123 + 𝑏132 ) + 𝜆22 (𝑏213 + 𝑏231 ) + 𝜆23 (𝑏321 + 𝑏312 ) = 0,
𝑓𝑥󸀠󸀠2
2
(11)
(𝜇, 𝑋) = 2𝜆22 (𝑎22 + 𝜇2 ) ,
𝑓𝑥󸀠󸀠2 (𝜇, 𝑋) = 2𝜆23 (𝑎33 + 𝜇3 ) ,
3
𝑓𝑥󸀠󸀠1 𝑥2
where 𝜆 𝑖 , 𝑖 = 1, 2, 3 are parameters and
(𝜇, 𝑋) = 𝑓𝑥󸀠󸀠2 𝑥1 (𝜇, 𝑋) = 𝑓𝑥󸀠󸀠1 𝑥3 (𝜇, 𝑋)
= 𝑓𝑥󸀠󸀠2 𝑥3 (𝜇, 𝑋) = 𝑓𝑥󸀠󸀠3 𝑥2 (𝜇, 𝑋) = 0.
3
𝑓 (𝜇, 𝑋) = ∑𝜆2𝑖 (𝑎𝑖𝑖 + 𝜇𝑖 ) 𝑥𝑖2
𝑖=1
3
3
𝑖=1
𝑗=1
+ ∑ (𝜆2𝑖 𝑐𝑖 + 𝜇𝑖 𝑑𝑖 𝜆 𝑖 − 𝜇𝑖 𝜆2𝑖 𝑑𝑖 − ∑𝜆 𝑗 𝑑𝑗 𝑎𝑗𝑖 ) 𝑥𝑖
3
+ ∑ (𝜆 𝑖 𝑑𝑖 𝑐𝑖 −
𝑖=1
Thus the Hesse matrix 𝐻𝑓 of the 𝑓(𝜇, 𝑋) is a negative
definite matrix; furthermore max𝑋∈𝑅3 𝑓(𝜇, 𝑋) exists.
These parameters 𝑑𝑖 , 𝜇𝑖 , 𝜆 𝑖 , 𝑖 = 1, 2, 3 will be determined
by solving the maximum of 𝑓1 (𝜇, 𝑋) and formula (12), and let
𝑀+ = {
𝜆 𝑖 𝑑𝑖2 ) ,
(12)
where 𝜇𝑖 , 𝑖 = 1, 2, 3 are undetermined parameters. And we
always assume that the supremum 𝑓(𝜇, 𝑋) < +∞ in the
paper.
𝑀, 𝑀 > 0,
0, 𝑀 ≤ 0.
(15)
Theorem 7. If condition (6) exists, 𝜂 = min{𝜇1 , 𝜇2 , 𝜇3 }, 𝑀 =
max𝑋∈𝑅3 𝑓(𝜇, 𝑋), 𝜆 𝑖 > 0, 𝑖 = 1, 2, 3, then the estimation
[𝑉 (𝑋 (𝑡)) −
1
1 +
𝑀 ] ≤ [𝑉 (𝑋 (𝑡0 )) − 𝑀+ ] 𝑒−2𝜂(𝑡−𝑡0 )
2𝜂
2𝜂
(16)
4
Journal of Applied Mathematics
3
holds, and the set
𝑗=1
1
Ω = {𝑋 | 𝑉 (𝑋 (𝑡)) ≤ 𝑀+ }
2𝜂
≤ −2𝜂 [𝑉 (𝑋 (𝑡)) −
2
2
1 +
𝑀 ] ≤ 0,
2𝜂
2
= {𝑋 | [(𝜆 1 𝑥1 − 𝑑1 ) + (𝜆 2 𝑥2 − 𝑑2 ) + (𝜆 3 𝑥3 − 𝑑3 ) ]
≤
2
≤ −𝜂 ∑ (𝜆 𝑗 𝑥𝑗 − 𝑑𝑗 ) + 𝑀+
when 𝑉 (𝑋 (𝑡)) >
1 +
𝑀 }
𝜂
1 +
𝑀 ,
2𝜂
(19)
(17)
is the globally exponentially attractive set and positive invariant
set of system (10); that is,
lim 𝑉 (𝑋 (𝑡)) ≤
𝑡→∞
1 +
𝑀 .
2𝜂
(18)
Proof. Differentiating the Lyapunov function 𝑉(𝑋(𝑡)) in (3)
with respect to time 𝑡 along the trajectory of system (10) yields
󵄨
𝑉̇ (𝑋 (𝑡))󵄨󵄨󵄨󵄨(10) = 𝜆 1 (𝜆 1 𝑥1 − 𝑑1 ) 𝑥1̇ + 𝜆 2 (𝜆 2 𝑥2 − 𝑑2 ) 𝑥2̇
+ 𝜆 3 (𝜆 3 𝑥3 − 𝑑3 ) 𝑥3̇
3
where 𝜇𝑗 > 0, 𝑗 = 1, 2, 3. Integrating both sides of (19) yields
(16) and (17). By the definition, taking into account limit on
both sides of the above inequality (16) as 𝑡 → +∞ results in
inequality (18).
Now, the characters of some of the chaotic systems known
are analysed by condition (6). When 𝑎11 = −𝜎, 𝑎12 = 𝜎, 𝑎21 =
𝜌, 𝑎22 = −𝛾, 𝑎33 = −𝛽, 𝑏213 = −1, 𝑏312 = 1, else 𝑎𝑖𝑗 = 0, 𝑏𝑖𝑗𝑘 =
0, 𝑐1 = 𝑐2 = 𝑐3 = 0, and 𝜆 1 = √𝜆, 𝜆 2 = 𝜆 3 = 1, 𝑑1 = 𝑑2 =
0, 𝑑3 = 𝜆𝜎 + 𝜌, 𝜇1 = 𝜎, 𝜇2 = 𝛾, 𝜇3 = min{𝜎, 𝛾}, 𝜂 = 𝜂1 = 𝜇3 ,
𝛽 > 𝜂1 , system (10) can be rewritten as system (7):
𝑉 (𝑋 (𝑡)) =
1
2
[𝜆𝑥12 + 𝑥22 + (𝑥3 − 𝜆𝜎 − 𝜌) ] ,
2
𝑓 (𝜇, 𝑋) = − (𝛽 − 𝜂1 ) 𝑥32 + (𝛽 − 2𝜂1 ) (𝜆𝜎 + 𝜌) 𝑥3
(20)
2
+ 𝜂1 (𝜆𝜎 + 𝜌) .
2
= − ∑𝜇𝑗 (𝜆 𝑗 𝑥𝑗 − 𝑑𝑗 ) + 𝜆 1 (𝜆 1 𝑥1 − 𝑑1 )
𝑗=1
We have 𝑀 = 𝛽2 (𝜆𝜎 + 𝜌)2 /4(𝛽 − 𝜂1 ). Thus
3
× ( ∑𝑎1𝑗 𝑥𝑗 + 𝜇1 𝜆−1
1 (𝜆 1 𝑥1 − 𝑑1 )
𝑗=1
2
[𝑉 (𝑋 (𝑡)) −
𝛽2 (𝜆𝜎 + 𝜌)
]
8 (𝛽 − 𝜂1 ) 𝜂1
2
+ (𝑏123 + 𝑏132 ) 𝑥2 𝑥3 + 𝑐1 )
≤ [𝑉 (𝑋 (𝑡0 )) −
𝛽2 (𝜆𝜎 + 𝜌)
] 𝑒−2𝜂1 (𝑡−𝑡0 ) ,
8 (𝛽 − 𝜂1 ) 𝜂1
2
+ 𝜆 2 (𝜆 2 𝑥2 − 𝑑2 )
Ω1 = {𝑋 | 𝑉 (𝑋) ≤
3
× ( ∑𝑎2𝑗 𝑥𝑗 + 𝜇2 𝜆−1
2 (𝜆 2 𝑥2 − 𝑑2 )
𝑗=1
𝛽2 (𝜆𝜎 + 𝜌)
}
8 (𝛽 − 𝜂1 ) 𝜂1
2
= {𝑋 | 𝜆𝑥12 + 𝑥22 + (𝑥3 − 𝜆𝜎 − 𝜌) ≤
+ (𝑏213 + 𝑏231 ) 𝑥1 𝑥3 + 𝑐2 )
2
𝛽2 (𝜆𝜎 + 𝜌)
}
4 (𝛽 − 𝜂1 ) 𝜂1
(21)
is the globally exponentially attractive set and positive invariant set of system (7).
+ 𝜆 3 (𝜆 3 𝑥3 − 𝑑3 )
3
× ( ∑𝑎3𝑗 𝑥𝑗 + 𝜇3 𝜆−1
3 (𝜆 3 𝑥3 − 𝑑3 )
𝑗=1
Example 8. Further, taking ito accout 𝜇1 = 𝜎, 𝜇2 = 𝛾, 𝜇3 =
𝛽/2, 𝜂 = 𝜂2 = min{𝜎, 𝛾, 𝛽/2}, the estimate
2
+ (𝑏312 + 𝑏321 ) 𝑥1 𝑥2 + 𝑐3 )
3
2
≤ −𝜂∑ (𝜆 𝑗 𝑥𝑗 − 𝑑𝑗 ) + 𝑓 (𝜇, 𝑋)
𝑗=1
[𝑉 (𝑋 (𝑡)) −
𝛽(𝜆𝜎 + 𝜌)
]
4𝜂2
2
𝛽(𝜆𝜎 + 𝜌)
≤ [𝑉 (𝑋 (𝑡0 )) −
] 𝑒−2𝜂2 (𝑡−𝑡0 )
4𝜂2
(22)
Journal of Applied Mathematics
5
𝑑2 = 0, 𝑑3 = 𝑏 + 2𝑐, system (6), 𝑉(𝑋(𝑡)), and 𝑓(𝑢, 𝑋) can be
rewritten as system (8):
holds and that
2
Ω2 = {𝑋 | 𝑉 (𝑋) ≤
𝛽(𝜆𝜎 + 𝜌)
}
4𝜂2
2
= {𝑋 | 𝜆𝑥12 + 𝑥22 + (𝑥3 − 𝜆𝜎 − 𝜌) ≤
2
𝛽(𝜆𝜎 + 𝜌)
}
2𝜂2
𝑉 (𝑋 (𝑡)) =
(23)
𝑓 (𝜇, 𝑋) = − (𝑎 − 𝜇1 ) 𝑥12 + (𝑎 − 2𝜇1 ) 𝑑𝑥1 − 2 (1 − 𝜇2 ) 𝑥22
− 2 (𝑏 + 𝑐) 𝑑𝑥2 − (1 − 𝜇3 ) 𝑥32
is the globally uniform exponentially attractive set and positive invariant set of system (7).
Proof. Again applying Lyapunov function given in (19) and
evaluating the derivative of 𝑑𝑉1 /𝑑𝑡 along the trajectory of
system (16) lead to
2
𝑓 (𝜇, 𝑋) = −
𝑥32 𝛽 (𝜆𝜎 + 𝜌) 𝛽
+
,
2
2
+ (𝑏 + 2𝑐) (1 − 2𝜇3 ) 𝑥3 + 𝜇1 𝑑2 + 𝜇3 (𝑏 + 2𝑐)2 .
(28)
Thus
2
(24)
2
𝛽(𝜆𝜎 + 𝜌)
.
𝑀=
2
(𝑎 − 2𝜇1 ) 𝑑2 (𝑏 + 𝑐)2 𝑑2 (𝑏 + 2𝑐)2 (1 − 2𝜇3 )
𝑀=
+
+
4 (𝑎 − 𝜇1 )
2 (1 − 𝜇2 )
4 (1 − 𝜇3 )
(29)
+ 𝜇1 𝑑2 + 𝜇3 (𝑏 + 2𝑐)2 .
The conclusion of Example 9 is obtained.
We have
Example 9. Furthermore, choose 𝜇1 = 𝜎, 𝜇2 = 𝛾, 𝜇3 = 𝛽, 0 <
𝜉1 < 𝛽, 𝜂 = 𝜂3 = min{𝜎, 𝛾, 𝜉1 }. Get
𝑓 (𝜇, 𝑋) = − (𝛽 − 𝜂3 ) 𝑥32 + (𝛽 − 2𝜂3 ) (𝜆𝜎 + 𝜌) 𝑥3
+ 𝜂2 (𝜆𝜎 + 𝜌) ,
(25)
2
𝛽 (𝜆𝜎 + 𝜌)
𝑀=
.
4 (𝛽 − 𝜂3 )
[𝑉 (𝑋 (𝑡)) −
2
2
𝛽 (𝜆𝜎 + 𝜌)
]
8 (𝛽 − 𝜂3 ) 𝜂3
(26)
2
𝛽2 (𝜆𝜎 + 𝜌)
≤ [𝑉 (𝑋 (𝑡0 )) −
] 𝑒−2𝜂3 (𝑡−𝑡0 )
8 (𝛽 − 𝜂3 ) 𝜂3
(30)
2
(31)
is the estimation of the globally exponentially attractive and
positive invariant sets of system (8).
If 𝑏111 = 𝑏222 = 𝑏333 = 0, ∃𝑏𝑖𝑗𝑗 ≠ 0, 𝑖, 𝑗 = 1, 2, 3, the
dynamic system (1) is shown as
3
3
𝑥1̇ = ∑ 𝑎1𝑗 𝑥𝑗 + ∑ 𝑏1𝑖𝑖 𝑥𝑖2 + ∑ (𝑏1𝑖𝑗 + 𝑏1𝑗𝑖 ) 𝑥𝑖 𝑥𝑗 + 𝑐1 ,
𝑗=1
holds and that
𝑖=2,3
3
𝑖 ≠ 𝑗=1
3
𝑐𝑥2̇ = ∑𝑎2𝑗 𝑥𝑗 + ∑ 𝑏2𝑖𝑖 𝑥𝑖2 + ∑ (𝑏2𝑖𝑗 + 𝑏2𝑗𝑖 ) 𝑥𝑖 𝑥𝑗 + 𝑐2 ,
2
Ω3 = {𝑋 | 𝑉 (𝑋) ≤
Ω4 = {𝑋 | 𝑉 (𝑥 (𝑡)) − 𝑀}
= {𝑋 | (𝑥1 − 𝑑) + 2𝑥22 + (𝑥3 − 𝑏 − 2𝑐) ≤ 2𝑀}
Then, the estimate
2
𝑉 (𝑥 (𝑡)) − 𝑀 ≤ [𝑉 (𝑥 (𝑡0 )) − 𝑀] 𝑒−2𝜂(𝑡−𝑡0 ) ,
then
2
2
1
2
2
[(𝑥1 − 𝑑) + 2𝑥22 + (𝑥3 − 𝑏 − 2𝑐) ] ,
2
𝛽2 (𝜆𝜎 + 𝜌)
}
8 (𝛽 − 𝜂3 ) 𝜂3
𝑗=1
𝑖=1,3
3
2
= {𝑋 | 𝜆𝑥12 + 𝑥22 + (𝑥3 − 𝜆𝜎 − 𝜌) ≤
2
𝛽2 (𝜆𝜎 + 𝜌)
}
4 (𝛽 − 𝜂3 ) 𝜂3
(27)
𝑖 ≠ 𝑗=1
3
𝑥3̇ = ∑ 𝑎3𝑗 𝑥𝑗 + ∑ 𝑏3𝑖𝑖 𝑥𝑖2 + ∑ (𝑏3𝑖𝑗 + 𝑏3𝑗𝑖 ) 𝑥𝑖 𝑥𝑗 + 𝑐3 .
𝑗=1
𝑖=1,2
𝑖 ≠ 𝑗=1
(32)
In this case, we can take into account
is the globally exponentially attractive set and positive invariant set of system (7).
Example 10. Taking 𝑎11 = −𝑎, 𝑎12 = 𝑏, 𝑎21 = 𝑐, 𝑎22 = −1,
𝑎32 = 𝑑, 𝑎33 = −1, 𝑏123 = 𝑏312 = 1, 𝑏213 = −1 else 𝑎𝑖𝑗 = 0,
𝑏𝑖𝑗𝑘 = 0, 𝑐1 = 𝑐2 = 𝑐3 = 0, and 𝜆 1 = 𝜆 3 = 1, 𝜆 2 = √2, 𝑑1 = 𝑑,
3
𝑓 (𝜇, 𝑋) = ∑ [𝜆2𝑖 (𝑎𝑖𝑖 + 𝜇𝑖 ) + 𝜆 [𝑖+1]3 𝑑[𝑖+1]3 𝑏[𝑖+1]3 ,𝑖𝑖
𝑖=1
+𝜆 [𝑖+2]3 𝑑[𝑖+2]3 𝑏[𝑖+2]3 ,𝑖𝑖 ] 𝑥𝑖2
6
Journal of Applied Mathematics
3
80
𝑖<𝑗,1
60
+ ∑ (𝜆2𝑖 𝑎𝑖𝑗 + 𝜆2𝑗 𝑎𝑗𝑖 + 𝜆 1 𝑑1 (𝑏1𝑖𝑗 + 𝑏1𝑗𝑖 )
+𝜆 2 𝑑2 (𝑏2𝑖𝑗 +𝑏2𝑗𝑖 )+𝜆 3 𝑑3 (𝑏3𝑖𝑗 +𝑏3𝑗𝑖 )) 𝑥𝑖 𝑥𝑗
3
40
20
+ ∑ (𝜆2𝑖 (𝑐𝑖 − 𝜇𝑖 𝑑𝑖 ) + 𝜇𝑖 𝜆 𝑖 𝑑𝑖 − 𝑎1𝑖 𝜆 1 𝑑1
𝑖=1
0
20
−𝑎2𝑖 𝜆 2 𝑑2 − 𝑎3𝑖 𝜆 3 𝑑3 ) 𝑥1
10
0
3
+ ∑𝜆 𝑖 𝑑𝑖 (𝑐𝑖 − 𝑑𝑖 ) ,
−10
𝑖=1
1
Theorem 11. Suppose that 𝐺0 = (𝑥10 , 𝑥20 , . . . , 𝑥𝑛0 ) is the stable
point of the 𝑓(𝜇, 𝑋) defined by (33). If the Hesse matrix of
the 𝑓(𝜇, 𝑋) is a negative definite matrix, the 𝑓(𝜇, 𝑋) has
maximum 𝑀and the estimation
1
[𝑉 (𝑋 (𝑡)) − 𝑀+ ]
2𝜂
≤ [𝑉 (𝑋 (𝑡0 )) −
0
−5
15
Figure 1: Simulation of system (40).
(33)
where [⋅]3 denotes modulo-3.
−10
10
5
(34)
holds; that is,
lim 𝑉 (𝑋 (𝑡)) ≤
0
−0.5
−1
−1.5
1 + −2𝜂(𝑡−𝑡0 )
𝑀 ]𝑒
2𝜂
𝑡→∞
0.5
1 +
𝑀 ,
2𝜂
(35)
−2
1.5
1
0.5
0
−0.5
10
5
0
15
Figure 2: Simulation of system (43).
and the set
Ω = {𝑋 | 𝑉 (𝑋 (𝑡)) ≤
The 𝑓(𝜇, 𝑋) has the maximum 𝑀. Differentiating the Lyapunov function 𝑉(𝑋(𝑡)) in (3) with respect to time 𝑡 along
the trajectory of system (32) yields
1 +
𝑀 }
2𝜂
2
2
2
= {𝑋 | [(𝜆 1 𝑥1 − 𝑑1 ) + (𝜆 2 𝑥2 − 𝑑2 ) + (𝜆 3 𝑥3 − 𝑑3 ) ]
≤
1 +
𝑀 }
𝜂
+ 𝜆 3 (𝜆 3 𝑥3 − 𝑑3 ) 𝑥3̇
(36)
is the globally exponentially attractive set and positive invariant
set of system (32).
3
2
≤ −𝜂∑ (𝜆 𝑗 𝑥𝑗 − 𝑑𝑗 ) + 𝑓 (𝜇, 𝑋)
𝑗=1
3
2
≤ −𝜂∑ (𝜆 𝑗 𝑥𝑗 − 𝑑𝑗 ) + 𝑀+
Proof. If 𝐺0 is the stable point of the 𝑓(𝜇, 𝑋), that is,
∇(𝑓)𝐺0 = (𝑓𝑥󸀠1 , 𝑓𝑥󸀠2 , 𝑓𝑥󸀠3 ) = 0,
󵄨
𝑉̇ (𝑋 (𝑡))󵄨󵄨󵄨󵄨(32) = 𝜆 1 (𝜆 1 𝑥1 − 𝑑1 ) 𝑥1̇ + 𝜆 2 (𝜆 2 𝑥2 − 𝑑2 ) 𝑥2̇
(39)
𝑗=1
(37)
and the Hesse matrix 𝐻𝑓 of the 𝑓(𝜇, 𝑋) is a negative definite
matrix, namely,
󵄨󵄨𝑓󸀠󸀠 𝑓󸀠󸀠 󵄨󵄨
󵄨󵄨 𝑥1 𝑥1 𝑥1 𝑥2 󵄨󵄨
󸀠󸀠
󵄨
󵄨󵄨
𝑓𝑥1 𝑥1 < 0,
󵄨󵄨𝑓󸀠󸀠 𝑓󸀠󸀠 󵄨󵄨󵄨 > 0,
󵄨󵄨 𝑥2 𝑥1 𝑥2 𝑥2 󵄨󵄨
󵄨󵄨𝑓󸀠󸀠 𝑓󸀠󸀠 𝑓󸀠󸀠 󵄨󵄨
(38)
󵄨󵄨 𝑥1 𝑥1 𝑥1 𝑥2 𝑥1 𝑥3 󵄨󵄨
󵄨󵄨
󵄨
󵄨󵄨𝑓󸀠󸀠 𝑓󸀠󸀠 𝑓󸀠󸀠 󵄨󵄨󵄨
󵄨󵄨 𝑥2 𝑥1 𝑥2 𝑥2 𝑥2 𝑥3 󵄨󵄨 < 0.
󵄨󵄨󵄨 󸀠󸀠
󵄨󵄨
󸀠󸀠
󸀠󸀠 󵄨󵄨
󵄨󵄨𝑓
󵄨 𝑥3 𝑥1 𝑓𝑥3 𝑥2 𝑓𝑥3 𝑥3 󵄨󵄨
≤ −2𝜂 [𝑉 (𝑋 (𝑡)) −
1 +
𝑀 ] ≤ 0,
2𝜂
when 𝑉 (𝑋 (𝑡)) >
1 +
𝑀 .
2𝜂
The proof is complete.
4. Switched Chaotic Systems
Condition (6) has helpfully provided us with instructions on
how to find the new chaotic systems. We construct a series
Journal of Applied Mathematics
7
150
200
100
150
50
100
0
50
−50
100
0
20
50
0
−50
−50
−100 −100
50
0
100
0
−20
−40 −40
Figure 3: Simulation of system (44).
−20
20
0
40
Figure 5: Simulation of system (46).
200
150
150
100
100
50
50
0
200
0
100
50
0
−50 −60
−20
−40
20
0
40
100
0
−100
+ (−
of new chaotic systems that the condition (6) is fulfilled and
study the switching system between them.
+(
Example 12. Consider a Lorenz system shown in Figure 1:
𝑥2̇ = 28𝑥1 − 𝑥2 − 𝑥1 𝑥3 ,
(40)
𝑓2󸀠 𝑥1 (𝜇, 𝑋) = 0,
𝑥3̇ = −3𝑥2 − 𝑥3 + 10𝑥12 + 𝑥1 𝑥2 .
48 √
112 √
12.5 −
13.5) 𝑥1
25
45
168 √
2643
39293196661
12.5 +
) 𝑥2 +
,
25
2
405000
𝑥1 = −
48 √
112 √
12.5 +
13.5) ,
25
45
1
112 √
48
13.5)
( √12.5 +
1250 5
9
≈ 0.064,
Solution. Here
󸀠
(𝜇, 𝑋) = −13.5𝑥2 + (
𝑓2𝑥
2
𝑎33 = −1,
els 𝑎𝑖𝑗 = 0,
𝑏123 = 𝑏132 = 0.5,
𝑏312 = 𝑏321 = 0.5,
𝜆 2 = √13.5,
𝑑3 = 440.5,
𝑎21 = 28,
𝑎22 = −1,
𝑎32 = −3,
𝑏113 = 𝑏131 = −0.4,
𝑏213 = 𝑏231 = −0.5,
els 𝑏𝑖𝑗𝑘 = 0,
𝑏311 = 10,
𝜆 1 = √12.5,
6
,
25
𝜆 3 = 1,
𝑑1 = −
𝜇1 = 2,
1
𝜇2 = 𝜇3 = ,
2
−200
−100
𝑓2󸀠 𝑥1 (𝜇, 𝑋) = −250𝑥1 − (
𝑥1̇ = −12𝑥1 + 5𝑥2 − 0.8𝑥1 𝑥3 + 𝑥2 𝑥3 ,
𝑎12 = 5,
100
Figure 6: Simulation of system (47).
Figure 4: Simulation of system (45).
𝑎11 = −12,
0
𝑑2 =
4
,
45
1
𝜂= ,
2
𝑓2 (𝜇, 𝑋) = −125𝑥12 − 6.75𝑥22 − 0.5𝑥32
󸀠
(𝜇, 𝑋) = 0,
𝑓2𝑥
2
𝑥2 =
30 56 √
881
12.5 +
(
) ≈ 99.65,
135 25
2
󸀠
(𝜇, 𝑋) = −𝑥3 ,
𝑓2𝑥
3
󸀠
𝑓2𝑥
(𝜇, 𝑋) = 0,
3
󸀠󸀠
𝑓2𝑥
2 (𝜇, 𝑋) = −250,
1
󸀠󸀠
𝑓2𝑥
2
3
168 √
2643
12.5 +
),
25
2
𝑥3 = 0,
󸀠󸀠
𝑓2𝑥
2 (𝜇, 𝑋) = −13.5,
2
(𝜇, 𝑋) = −1,
󸀠󸀠
󸀠󸀠
󸀠󸀠
𝑓2𝑥
(𝜇, 𝑋) = 𝑓2𝑥
(𝜇, 𝑋) = 𝑓2𝑥
(𝜇, 𝑋) = 0,
1 𝑥2
2 𝑥1
1 𝑥3
󸀠󸀠
󸀠󸀠
󸀠󸀠
(𝜇, 𝑋) = 𝑓2𝑥
(𝜇, 𝑋) = 𝑓2𝑥
(𝜇, 𝑋) = 0.
𝑓2𝑥
3 𝑥1
2 𝑥3
3 𝑥2
(41)
8
Journal of Applied Mathematics
×1011
15
150
10
100
5
50
0
0
−5
20
−50
100
10
0
−10
−20
−2
2
0
4
6
8
×1012
50
0
−50
(a) System (40)–(43)
−100 −100
−50
0
50
100
(b) System (40)–(44)
250
200
200
150
150
100
100
50
50
0
100
0
20
50
0
−50
−100 −15
0
−5
−10
5
×1012
0
−20
−40 −20
(c) System (40)–(45)
−40
0
20
40
(d) System (40)–(46)
150
100
50
0
100
50
0
−50
−100 −150
−100
−50
0
50
(e) System (40)–(47)
Figure 7: Switched system between system (40) and others.
The Hesse matrix of the 𝑓(𝜇, 𝑋) is a negative definite matrix,
max 𝑓(𝜇, 𝑋) ≈ 164045.42. The set
Ω = {𝑋 | (√12.5𝑥1 +
5 2
4 2
) + (√13.5𝑥2 − )
26
45
(42)
2
+(𝑥3 − 440.5) ≤ 328090.84}
is the globally exponentially attractive set and positive invariant set of system (40).
Note. (a) If the Hesse matrix of the 𝑓(𝜇, 𝑋) is not a negative
definite matrix, the 𝑓(𝜇, 𝑋) has no maximum 𝑀.
(b) If ∃𝑎𝑖0 𝑖0 ≥ 0, lim𝑥𝑖 → ∞ 𝑓(𝜇, 𝑋) = +∞, this type of
0
chaotic system needs further research.
(c) We call the dynamic system (1) the second class threedimensional chaotic system with cross-product nonlinearities, if it does not satisfy condition (6). For this class of
chaotic systems, 𝑓(𝜇, 𝑋) is a cubic polynomial and there is
not maximum if we choose energy function (3) differentiating
this Lyapunov function with respect to 𝑡 along the trajectory
of system (1). It is very useful to research these problems.
Journal of Applied Mathematics
9
×1011
15
×1012
4
10
2
5
0
0
−2
−5
20
10
0
2
0
−10 −2
4
8
×1012
6
−4
4
×1012
2
0
−2
−4
(a) System (43)–(40)
×1011
15
10
10
5
5
0
0
10
5
0
−10
−5 −15
−6
−4
2 12
×10
0
(b) System (43)-(44)
×1011
15
−5
15
×1011
−8
−2
5
×1012
0
−5
−5
20
10
0
−10
(c) System (43)–(45)
−20 −2
0
2
4
6
8
×1012
(d) System (43)–(46)
×1015
2
1.5
1
0.5
0
15
10
5
0
−5
0
2
4
6
8
10
×1015
(e) System (43)–(47)
Figure 8: Switched system between system (43) and others.
Example 13. The new chaotic system shown in Figure 2 is
𝑥1̇ = −2𝑥1 + 5𝑥3 + 5.7𝑥1 𝑥3 + 4.7𝑥2 𝑥3 + 4,
𝑥2̇ = −𝑥1 − 2𝑥2 + 3𝑥3 + 5,
𝑥3̇ = −6𝑥1 − 2𝑥2 − 4𝑥3 + 0.2𝑥12 + 3.4𝑥1 𝑥2 + 7.
Example 14. The chaotic system shown in Figure 3 is
𝑥1̇ = −11𝑥1 + 0.15𝑥1 𝑥2 + 1.38𝑥2 𝑥3 + 1,
(43)
𝑥2̇ = 30𝑥1 − 𝑥2 − 𝑥1 𝑥3 + 0.1𝑥2 𝑥3 ,
𝑥3̇ = 15𝑥2 − 2.5𝑥3 + 𝑥1 𝑥2 .
(44)
10
Journal of Applied Mathematics
×1012
3
100
2
50
1
0
−50
100
0
−1
100
50
0
−50
−100 −100
50
0
−50
50
100
0
−50
(a) System (44)–(40)
−15
−10
5
×1012
0
(b) System (44)-(43)
1000
150
800
100
600
400
50
200
0
−50
100
−100 −20
−5
0
−200
100
50
0
−50
−100 −3
0
−1
−2
1
×1013
50
0
−50
−100 −150
(c) System (44)-(45)
−100
−50
0
50
100
(d) System (44)–(46)
150
100
50
0
−50
100
50
0
−50
−100 −200
−100
0
100
200
(e) System (44)–(47)
Figure 9: Switched system between Example (44) and others.
Example 16. The chaotic system shown in Figure 5 is
Example 15. The chaotic system shown in Figure 4 is
𝑥1̇ = 30 (𝑥2 − 𝑥1 ) − 0.48𝑥12 ,
𝑥2̇ = 80𝑥1 − 6𝑥2 − 𝑥1 𝑥3 ,
𝑥3̇ = 𝑥1 𝑥2 − 5𝑥3 .
𝑥1̇ = −12𝑥1 + 5𝑥2 + 𝑥2 𝑥3 ,
(45)
𝑥2̇ = 28𝑥1 − 𝑥2 − 𝑥1 𝑥3 ,
𝑥3̇ = −3𝑥2 − 𝑥3 + 4𝑥12 + 𝑥1 𝑥2 .
(46)
Journal of Applied Mathematics
11
150
×1012
4
100
3
2
50
0
100
1
50
0
−50
−100
−10
−15
−5
0
5
×1012
0
100
50
0
−50 −20
(a) System (45)–(40)
−15
−10
0
−5
5
×1012
(b) System (45)–(43)
150
150
100
100
50
50
0
−50
100
50
0
−50
−100 −3
0
−1
−2
1
×1012
0
100
50
0
−50
(c) System (45)-(44)
−100 −15
−10
−5
0
5
×1012
(d) System (45)-(46)
150
100
50
0
100
50
0
−50
−100
−15
−10
−5
0
5
×1012
(e) System (45)–(47)
Figure 10: Switched system between Example (45) and others.
Example 17. The chaotic system shown in Figure 6 is
𝑥̇ = (
20
) 𝑥 − 𝑦𝑧 + 9,
7
𝑦̇ = −10𝑦 + 𝑥𝑧 + 0.5𝑧2 ,
(47)
𝑧̇ = −4𝑧 + 𝑥𝑦 + 𝑦𝑧.
Note. When we analyse Examples 13 to 17 by the previous
means, for sup𝑋∈𝑅3 𝑓(𝜇, 𝑋) = +∞, the globally exponentially
attractive set and positive invariant set of them have not been
obtained. The globally exponentially attractive set and positive invariant set really exist by their trajectories. Particularly,
by Lü et al. chaotic system [11] and Example 17 we conjecture
that they have globally conditional exponentially attractive
set and positive invariant set, according to preliminary study.
These are waiting for us to do further research. Meanwhile,
we can compute that the maximum Lyapunov exponents
of Examples 12–17 are 1.06, 0.02, 1.84, 0.01, 0.92, and 0.95,
respectively.
12
Journal of Applied Mathematics
200
×1012
4
150
3
100
2
50
1
0
20
0
20
0
−20
0
−20
−40 −40
40
20
0
−20
−40
(a) System (46)–(40)
200
400
150
200
100
0
50
50
0
5
15
20
×1012
0
1
×1013
(b) System (46)–(40)
600
−200
100
−5
10
0
50
0
−50
−100 −100
−50
100
50
0
0
−50
−100
(c) System (46)–(44)
−3
−2
−1
(d) System (46)-(45)
200
150
100
50
0
100
50
0
−50
−100
−150
−100
−50
0
50
(e) System (46)-(47)
Figure 11: Switched system between Example (46) and others.
5. Simulation of Switched System
In this section, we will show some simulation results of the
following switching system
𝑥̇ = 𝐴 𝜎 𝑥 + 𝑓𝜎 (𝑥) + 𝐶𝜎 ,
(48)
where 𝑥𝑇 = (𝑥1 , 𝑥2 , 𝑥3 ), 𝜎 is the switching law, and
𝜎
𝜎
𝜎
𝑎12
𝑎13
𝑎11
[𝑎𝜎 𝑎𝜎 𝑎𝜎 ]
𝐴 𝜎 = [ 21 22 23 ] ,
𝜎
𝜎
𝜎
[𝑎31 𝑎32 𝑎33 ]
𝑐1𝜎
[𝑐𝜎 ]
𝐶𝜎 = [ 2 ] ,
𝜎
[𝑐3 ]
𝑥𝑇 𝐵1𝜎 𝑥
[
]
𝑓𝜎 (𝑥) = [𝑥𝑇 𝐵2𝜎 𝑥]
𝑇 𝜎
[𝑥 𝐵3 𝑥]
(49)
𝜎
, 𝑐𝑖𝜎 ∈ 𝑅, 𝑖, 𝑗, 𝑘 = 1, 2, 3. Each pair of (𝐴 𝜎 , 𝐶𝜎 , 𝐵1𝜎 ,
with 𝑎𝑖𝑗𝜎 , 𝑏𝑖𝑗𝑘
𝐵2𝜎 , 𝐵3𝜎 ) takes the form from Example 8 to Example 14. The
switching law is that the system will stay in each subsystem
for a constant time. In the following, we assume that (a, b)
denotes a switched system which switches between system
(a) and system (b). It can be seen from Figures 7 to 12 that
the switched systems (18,20), (18,22), (18,23), (20,18), (20,22),
(20,23), (22,18), (22,20), (22,23), (23,18), (23,20), and (23,22)
can also yield chaotic systems.
6. Conclusion
In this paper, the methods in [19–21] have been extended
to study the globally exponentially or globally conditional
Journal of Applied Mathematics
13
×1012
4
150
3
100
2
50
0
100
1
0
100
50
0
−50
−100
−150
−100
50
0
−50
50
0
−50
(a) System (47)–(40)
−100 −20
−15
−10
−5
0
5
×1012
(b) System (47)–(43)
150
150
100
100
50
50
0
−50
100
50
0
−50
−100 −150
−50
−100
0
50
100
0
100
50
0
−50
(c) System (47)–(44)
−100
−3
−2
−1
0
1
×1013
(d) System (47)–(45)
200
150
100
50
0
100
50
0
−50
−100
−150
−100
−50
0
50
(e) System (47)-(46)
Figure 12: Switched system between Example (47) and others.
exponentially attractive set and positive invariant set of the
three-dimensional chaotic system family with cross-product
nonlinearities. We have given two theorems for studying this
question and given some examples to show that such system
indeed has the globally exponentially or globally conditional
exponentially attractive set and positive invariant set, and
the exponential estimation is explicitly derived. We have
also suggested an idea to construct the chaotic systems, and
some new chaotic systems have been illustrated. The simulation results are given for switched system between these new
chaotic systems. It is very interesting to further research that
the Hesse matrix of the 𝑓(𝜇, 𝑋) is not a negative definite
matrix, and the dynamic system (1) is a second class threedimensional chaotic system with cross-product nonlinearities.
Acknowledgments
This work is supported by the Fundamental Research
Funds for the Central Universities, China Postdoctoral Science Foundation funded project under Grant 2012M511615,
National Natural Science Foundation of China under Grants
60474011 and 60904005, and the Hubei Provincial Natural
Science Foundation of China under Grant 2009CDB026.
14
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