# Analysis of a Lorenz-Like Chaotic System by Lyapunov Functions

Authors: Fuchen Zhang
Journal: Complexity
 (2019)

---

## Abstract

In this paper, we investigate the ultimate bound set and positively invariant set of a 3D Lorenz-like chaotic system, which is different from the well-known Lorenz system, Rössler system, Chen system, Lü system, and even Lorenz system family. Furthermore, we investigate the global exponential attractive set of this system via the Lyapunov function method. The rate of the trajectories going from the exterior of the globally exponential attractive set to the interior of the globally exponential attractive set is also obtained for all the positive parameters valuesa,b,c. The innovation of this paper is that our approach to construct the ultimate bounded and globally exponential attractivity sets assumes that the corresponding sets depend on some artificial parameters (λ and m); that is, for the fixed parameters of the system, we have a series of sets depending on λ and m. The results contain the known result as a special case for the fixed λ and m. The efficiency of the scheme is shown numerically. The theoretical results may find wide applications in chaos control and chaos synchronization.


---

## Body

## 1. Introduction

In 1963, Edward Lorenz discovered a chaotic attractor numerically when he studied the Rayleigh–Bénard convection [1]. This discovery stimulated rapid development of chaos theory and a large number of chaotic systems were reported. Since then, chaos phenomenon and chaotic systems have been intensively studied due to great potential applications of chaotic systems in some engineering and technology fields [2–34]. A lot of new chaotic systems can be found, such as Rössler system [2], Chen system [3], Lü system [4], and Lorenz–Stenflo system [12, 23–25]. How to get the bounds of chaotic systems is one of very central problems in the theory of dynamical systems. Bounds for a new chaotic system are very important for the study of the qualitative behavior of a new chaotic system and chaos control. The bounds of the famous Lorenz system have been studied by Leonov et al. in [26–28]. Zhang et al. [17] give the new results of the bounds of the famous Lorenz system and their new results contain the existing results [26–28] as special cases. The bounds of the Lorenz–Stenflo system have been investigated in [23]. How to get the bounds of the Chen system and the Lü system is an important yet nontrivial open problem due to the important potential applications of the Chen system and the Lü system [29, 30]. Zhang et al. [16, 21, 30] investigate the open problems of the bounds of the Chen system and the Lü system and get many important results. However, it is very difficult to get the bounds of chaotic systems [16, 21, 28] and some results have been derived only for some chaotic systems [16, 17, 21, 28]. The bounds of a large number of chaotic systems are still unknown. This paper is devoted to computing the bounds of a compact domain, which contains all compact invariant sets of a Lorenz-like chaotic system.In this paper, we consider the Lorenz-like system [31]:(1)dxdt=-σx-y-ayz,dydt=rx-y-xz,dzdt=xy-bz,where r,σ,b>0 and parameter a is real. System (1) is when a=0 coincides with the classical Lorenz system [1]; when σ>ar and b=1, it could be transformed to the Glukhovsky-Dolzhansky system describing fluid convection in the rotating ellipsoidal cavity [32] (see also [8, 15] within this paper), and when σ=-ar, it could be transformed to the Rabinovich system describing interaction between waves in plasma [33] (see also [8, 9] within this paper). When a=-1, system (1) coincides with the Qi system [22]:(2)dxdt=ay-x+yz,dydt=cx-y-xz,dzdt=xy-bz,where x, y, and z are real variables; a, b, and c are positive parameters of system (2). When a=35, b=7, and c=25, system (2) is chaotic [22], as shown in Figure 1. System (2) is different from the well-known Lorenz system, Rössler system, Chen system, Lü system, and even Lorenz system family [22]. So, many dynamic behaviors of chaotic system (2) are still unknown, motivating the work to be presented in this paper.Figure 1
Chaotic attractor of system (2) in xOyz space for a=35, b=7, and c=25.Further in this paper we are going to study the dynamic behavior of system (1) in the special case when a=-1, and we compare our results with the results that were obtained previously [31] for the general case a∈R.


## 2. Dynamic Behavior of System (2)

Theorem 1.
Assume thatλ>0, m>0, a>0, b>0, and c>0, with (3)Ωλ,m=x,y,z∣λx2+λ+my2+mz-a+cλ+mcm2≤Rλ,m2,where(4)
R
λ
,
m
2
=
b
2
4
a
b
-
a
a
+
c
λ
+
m
c
2
m
,
a
≥
1
,
b
≥
2
a
,
b
2
4
b
-
1
a
+
c
λ
+
m
c
2
m
,
a
>
1
,
b
≥
2
,
a
+
c
λ
+
m
c
2
,
b
<
2
a
,
b
<
2
.Then, Ωλ,m is the ultimate bound and positively invariant set of system (2).Proof.
Define the Lyapunov-like function(5)Vλ,mX=Vλ,mx,y,z=λx2+λ+my2+mz-a+cλ+mcm2,∀m>0,λ>0.Then, the derivative of Vλ,mX is (6)dVλ,mXdt2=2λxdxdt+2m+λydydt+2mz-a+cλ+mcmdzdt,=2λxay-ax+yz+2m+λycx-y-xz+2mz-a+cλ+mcmxy-bz,=-2aλx2-2m+λy2-2mbz2+2ba+cλ+mcz.Let dVλ,mX/dt=0, and we can get a bounded closed set Γ:(7)Γ=x,y,z∣λx2ba+cλ+mc2/4am+m+λy2ba+cλ+mc2/4m+mz-a+cλ+mc/2m2a+cλ+mc2/4m=1.Since chaotic system (2) is bounded, the continuous function (5) can reach its maximum value on the bounded closed set Γ above.
Hence, solutions of chaos system (2) are contained in the set defined by x,y,z∣Vλ,mX≤max⁡Vλ,mX=Rλ,m2,X∈Γ. We will get the maximum value of function (5) on Γ by dealing with the conditional extremum problem below:(8)maxVλ,mX=maxλx2+λ+my2+mz-a+cλ+mcm2,s.t.λx2ba+cλ+mc2/4am+m+λy2ba+cλ+mc2/4m+mz-a+cλ+mc/2m2a+cλ+mc2/4m=1.Denote(9)λx=x1,λ+my=y1,mz=z1,a+cλ+mc2m=l,ba+cλ+mc24am=n2,ba+cλ+mc24m=k2,and then (8) becomes the following form:(10)maxVm,λX=maxx12+y12+z1-2l2,s.t.x12n2+y12k2+z1-l2l2=1.We can solve problem (10) according to the optimization method and get the expression of Rλ,m2 as follows:(11)
R
λ
,
m
2
=
b
2
4
a
b
-
a
a
+
c
λ
+
m
c
2
m
,
a
≥
1
,
b
≥
2
a
,
b
2
4
b
-
1
a
+
c
λ
+
m
c
2
m
,
a
>
1
,
b
≥
2
,
a
+
c
λ
+
m
c
2
,
b
<
2
a
,
b
<
2
.The proof is thus completed.Remark 2.
(i) Let us takem=1 and λ=1 in Theorem 1; then we have that(12)Ω1,1=x,y,z∣x2+2y2+z-a-2c2≤L2is the ultimate bound and positively invariant set of system (2), where(13)
L
2
=
b
2
a
+
2
c
2
4
a
b
-
a
,
a
≥
1
,
b
≥
2
a
,
b
2
a
+
2
c
2
4
b
-
1
,
a
>
1
,
b
≥
2
,
a
+
2
c
2
,
b
<
2
a
,
b
<
2
.Let us take positive parameters values a=35, b=7, and c=25 in Ω1,1 above; then we can conclude that(14)Ω1,1=x,y,z∣x2+2y2+z-20.252≤121.452is the ultimate bound and positively invariant set for system (2). In Figure 2, we show the localization of chaotic attractor of system (2) in xOyz space defined by Ω1,1.
(ii) System (1) in this general form was studied in 1992 by Leonov and Boichenko [31]. Using Lyapunov’s direct method, they proved its dissipativity in the sense of Levinson, that is, the existence of a global bounded absorbing set containing global attractor, and also construct several positively invariant sets by stating the following results.Figure 2
Localization of chaotic attractor of system (2) in xOyz space defined by Ω1,1.Theorem 3.
For an arbitrary solutionxt,yt,zt of system (1), the following estimate is true:(15)limt→∞supy2t+zt-r2≤l2r2,where(16)l=1,b≤2,b2b-1,b>2.Theorem 4.
Let2σ-b≥0 and ab-2≥0, and then, for an arbitrary solution xt,yt,zt of system (1), the following estimate is true:(17)limt→∞inf2σ-arzt-x2t+ay2t≥0.Theorem 5.
All trajectoriesxt,yt,zt of system (1) enter the following ellipsoid:(18)x2+δy2+a+δz-σ+δra+δ2≤R2,where (19)R=b2ca+δσ+δr,δ>-a,c=minσ,1,b2,and stay in it.
(iii) Our approach to construct the ultimate bounded and globally exponential attractivity sets assumes that the corresponding sets depend on some artificial parameters (λ and m); that is, for the fixed parameters of system (2), we have a series of sets depending on λ and m. Let us take λ=1 and m=δ-1(∀δ>1) in (3); then the set Ωλ,m in (3) coincides with (18).Though Theorem1 gives the ultimate bound set of system (2), the global exponential attractive sets of system (2) are still unknown. The global exponential attractive sets of system (2) are described by the following theorem.Theorem 6.
Suppose thata>0, b>0, and c>0, and let us denote(20)Xt=xt,yt,zt,Vλ,mX=Vλ,mx,y,z=λx2+λ+my2+mz-a+cλ+mcm2,∀λ>0,m>0,Lλ,m2=ba+cλ+cm2θm,θ=mina,b,1>0.Then the estimate(21)Vλ,mXt-Lλ,m2≤Vλ,mXt0-Lλ,m2e-θt-t0holds for system (2).
Hence, by definition,Ψλ,m=X∣Vλ,mX≤Lλ,m2 is the globally exponential attractive set of system (2); that is, lim¯t→+∞Vλ,mXt≤Lλ,m2.Proof.
Define(22)Vλ,mX=Vλ,mx,y,z=λx2+λ+my2+mz-a+cλ+mcm2,∀λ>0,m>0,and then the derivative of Vλ,mX is(23)dVλ,mXdt2=2λxdxdt+2m+λydydt+2mz-a+cλ+mcmdzdt,=2λxay-ax+yz+2m+λycx-y-xz+2mz-a+cλ+mcmxy-bz,=-2aλx2-2m+λy2-2mbz2+2ba+cλ+mcz,≤-aλx2-m+λy2-mbz2+2ba+cλ+mcz,=-aλx2-m+λy2-mbz-a+cλ+mcm2+ba+cλ+mc2m,≤-θVλ,mX+ba+cλ+mc2m,≤-θVλ,mX-Lλ,m2.So, we have(24)Vλ,mXt-Lλ,m2≤Vλ,mXt0-Lλ,m2e-θt-t0.And(25)limt→+∞¯Vλ,mXt≤Lλ,m2.By definition, Ψλ,m=X∣Vλ,mX≤Lλ,m2 is the globally exponential attractive set of system (2).
The proof is thus completed.


## 3. Conclusion

The article is devoted to study the global behavior of the 3D Lorenz-like chaotic system. For the considered system, we obtain the positive invariant set (ultimate bound) and globally exponential attractive set using Lyapunov function theory and optimization method. Numerical simulations are consistent with the results of theoretical analysis. It is expected that the basic ideas presented in this paper can be applied to explore the bounds of similar chaotic systems in other papers.



---

*Source: 7812769-2019-07-09.xml*