A new four-dimensional chaotic system with first Lyapunov exponent of about 22, hyperbolic curve and circular paraboloid types of equilibria and its switching synchronization by an adaptive global integral sliding mode control
1. IntroductionLyapunov exponents play an important role in the characteristic of a dynamical system. Many chaotic or hyperchaotic systems have been reported in various fields.[1–8] The disorderness of a chaotic system is defined by its first Lyapunov exponent.[9,10] Applicability of a chaotic/hyperchaotic system is also governed by the magnitude of the highest Lyapunov exponent.[9,11,12] Thus, developing a chaotic system with the largest first Lyapunov exponent is more interesting in comparison with systems having small Lyapunov exponents. The nature of equilibria depicts the dynamical behavior of a system.[13–17] Recently many chaotic and hyperchaotic systems having different shapes of equilibrium points are reported.[13–17] The investigation of different shaped equilibrium points in chaotic systems is also an interesting research area.
Depending on the types of attractors, available chaotic and hyperchaotic systems are classified as: (i) chaotic systems with self-excited attractors[10,18] and (ii) chaotic systems with hidden attractors.[19–21] A self-excited attractor has a basin of attraction that is associated with an unstable equilibrium point, whereas a hidden attractor has a basin of attraction that does not intersect with small neighborhoods of any equilibrium points.[19–21] The chaotic attractors in the systems with only stable equilibria[15,22] or with no equilibria[23] are called hidden attractors.[21,24–26] Recently, the chaotic systems with an infinite number of equilibria (line/curve/plane shaped equilibria) are also classified into the category of hidden attractors chaotic systems.[26] The study of chaotic systems with hidden attractors is important because they can lead to unexpected and disastrous behaviors with small changes in dynamics as in aeroplane wings, electromechanical systems, bridge,[16,27] etc. The literature was searched to find the availability of 4D chaotic/hyperchaotic systems with many equilibria, and the outcome is listed in Table 1. It is observed from Table 1 that the 4D hyperchaotic/chaotic system with a hyperbolic curve and circular paraboloid equilibria is not available in the literature. Thus, designing of a hyperchaotic/chaotic system with a hyperbolic curve and circular paraboloid equilibria is considered as a worthy motivation of the present work.
Table 1.
Table 1.
 Table 1.
Categorization of the reported 4D chaotic and hyperchaotic systems with an infinite number of equilibrium points.
.
| 4D system |
Shape of equilibria of system |
Reference |
| 4D chaotic system |
plane of equilibria |
[28] |
| 4D hyperchaotic system |
line of equilibria |
[29,30] |
| 4D hyperchaotic system |
curve of equilibria |
[31] |
| 4D memristive hyperchaotic system |
line of equilibria |
[33,37] |
| 4D chaotic/hyperchaotic system |
hyperbolic curve and circular paraboloid equilibria |
this work |
| Table 1.
Categorization of the reported 4D chaotic and hyperchaotic systems with an infinite number of equilibrium points.
. |
Four-dimensional chaotic/hyperchaotic systems have complex dynamical structure and perform better in real life applications compared with the 3D chaotic systems because 4D systems have more disorderness and complexity. In the last decade, various hyperchaotic systems are reported with multiple positive Lyapunov exponents.[34,35] It is also seen in the literature that different methods are reported for developing a hyperchaotic system with multiple positive Lyapunov exponents like (i) by using state feedback control,[36] (ii) the addition of an extra term,[37] (iii) coupling,[38], (iv) parameter perturbation,[39] (v) some more,[34,35] etc. Shen at al.[34,35] reported a systematic methodology for constructing a hyperchaotic system with the desired number of positive LEs. However, developing a four-dimensional chaotic system that has the largest first Lyapunov exponent compared with all the existing 3D and 4D chaotic systems is a worthy motivation for this research work.
In this paper, a new 4D chaotic system which also has hyperchaotic behavior is reported. Following are the contributions and novelty of the paper in comparison with the existing literature on this topic:
(i) A new 4D chaotic system is developed which has the highest first Lyapunov exponent in comparison with many reported 3D and 4D chaotic systems in the literature.
(ii) The new system has a hyperbolic curve and a circular paraboloid type of equilibria. Such a system is not available in the literature.
(iii) The new system has all zero eigenvalues at a particular equilibrium point.
(iv) The new system exhibits multistability.
(v) Switching synchronization phenomenon is proposed for the synchronization of the new system.
(vi) An adaptive global integral sliding mode control (SMC) is designed for the switching synchronization of the proposed system.
The rest of this paper is organized as follows. Section 2 describes the dynamics of the new chaotic system. Numerical analyses of the new system are shown in section 3. Circuit implementation of the new system is discussed in section 4. In section 5 we discuss the switching synchronization phenomena for the proposed system. Results and discussions for the switching synchronization are presented in section 6. Finally, the conclusions of this paper are summarized in section 7.
2. A new chaotic system with the highest first Lyapunov exponentThis section describes the generation of the new chaotic system having the largest positive LE. Various analyses using theoretical and numerical tools for the new system are also given in this section.
2.1. Dynamics of the new systemLi et al.[40] reported a 3D chaotic system with a line of equilibria whose dynamics is described in the following equation
The system in Eq. (
1) has chaotic behavior with
a = 1,
b = 4, and
c=0.3. It has two equilibrium points at
and
Thus, system (
1) has a line of equilibria for
. A new hyperchaotic system is generated by using a state feedback control in the third state of system (
1). The dynamics of the new system is given in the following
The system (
2) has the largest positive first Lyapunov exponent with parameters
a = 40,
b = 105,
c=0.152,
d=0.2, and
p=0.1. Lyapunov exponents of the new system for the above sets of parameters are
and the Kaplan–Yorke dimension is
. The positive LE of 22.1301 is the largest among all the reported 3D and 4D chaotic systems in the literature.
The divergence of system (2) is given as follows:
Thus, the hyper-volume of system (
2) contracts exponentially with respect to time at a rate of (1+2
c)
z, for
and converges to zero as time tends to infinity.
System (2) is invariant under the coordinate transformation
. Therefore, system (2) has symmetry about the z axis. Thus, the system may have a symmetric pair of coexisting attractors.
2.2. Equilibrium pointsEquilibrium points of system (2) are obtained by equating the derivative of each state to zero. After some calculations, we obtained the equilibrium point as
. Thus, the equilibrium points are
The shape of equilibria using
and
is shown in Figs.
1(a) and
1(b), respectively. The equilibria
have a shape similar to
(hyperbolic curve). Equilibria
has a shape similar to
(circular paraboloid).
[41]The Jacobian matrix of system (2) is defined as
The characteristics polynomials corresponding to the equilibria
and
using the Jacobian
J are written as
The calculation of the eigenvalues corresponding to the equilibria
of the new system is shown below. From Eq. (6), we can obtain
or it can be written as
where
Now suppose
The calculations of eigenvalues is shown in the following cases.
Case 1
Case 2
Case 3
Case 4
The calculation of eigenvalues for
P2 can be done in a similar manner. It is seen that the nature of the equilibria
depends upon the values of state variables
z and
y, respectively. A special case of equilibria
when
is
, where the eigenvalues of the system are
. Thus, system (
2) has an interesting feature having all zero eigenvalues at a particular case of equilibria
.
3. Numerical analysesDynamical behavior of system (2) with the variation of a parameter and keeping other parameters fixed is evaluated using the Lyapunov spectrum plots. The spectrum is generated by finding the Lyapunov exponents with observation time
fixed initial conditions
and by using Wolf et al. algorithm.[42] The resulted Lyapunov spectrums are shown in Figs. 2–4. The new system has only chaotic behavior with the variation of parameter p. Thus, its effect on variation is not shown in the paper. It is seen from Figs. 2–4 that the system has different dynamical behaviors like chaotic, periodic, quasi-periodic, and hyperchaotic.
Here in Figs. 2–4, it is noticed that the system has the largest first Lyapunov exponent. A comparison of the reported highest first Lyapunov exponent systems along with the proposed system is given in Table 2.
Table 2.
Table 2.
| Table 2.
Comparison of the first positive Lyapunov exponent based on reported works.
. |
It is seen from Table 2 that system (2) has the largest first positive Lyapunov exponent in comparison with the reported three-dimensional and four-dimensional chaotic systems. It needs to be mentioned that only those chaotic systems are considered for comparison which have the first Lyapunov exponent greater than 10. Further, the highest first Lyapunov exponent of a system is invariant for scaling up or scaling down of the system.
The hyperchaotic attractors of the system (2) with parameters a = 40, b = 105, c=0.152, and d=0.2 are shown in the following Fig. 5.
The new system shows coexistence of chaotic attractors with the change of initial conditions. The coexistence of chaotic attractors of system (2) are shown in Fig. 6.
Figure 6 is simulated with the following initial conditions
The Poincaré maps of the system (
2) across different sections of planes are shown in Fig.
7. It is evidently seen that the results shown in Figs.
5–
7 validate the chaotic behavior of the new system.
Basins of attraction of the new system across x–y and z–w planes are shown in Fig. 8.
4. Circuit implementationThis section describes the design and construction of an electronic circuit realization of system (2). To validate the mathematical expression of a chaotic system and its applicability in real life, hardware implementation is important and desirable.[26]
Here, the circuit is designed and implemented by using NI multisim software. Four integration lines are considered corresponding to four states of the system. Circuit implementation of system (2) is shown in Fig. 9. The circuit is designed with parameters a = 5, b = 40, c=0.2, d=0.2, and p=0.1. The circuit equation similar to system (2) can be written as the following form
where
x,
y,
z, and
w are the outcomes of the integrators U1A, U7A, U5A, U3A, respectively. The circuit equation (
15) is similar to system (
2) with
,
R = 100 kΩ,
,
,
,
,
,
,
,
,
,
,
a = 5,
b = 40,
c=0.2,
d=0.2, and
p=0.1.
Circuit implementation results are shown in Fig. 10. It is observed from this figure that the attractors plot has chaotic behavior and validates with the MATLAB simulation results.
5. Switching synchronization of the new systemSynchronization of a hyperchaotic/chaotic system is one of the most focused applications of hyperchaotic/chaotic systems and studied by various researchers as seen in Refs. [44] and [45]. Various types of synchronization techniques like complete,[46] anti,[46] hybrid,[47] phase,[48] lag,[49] projective synchronization,[50] etc., are reported in the literature. Most of the papers deal with one of the aforesaid synchronization techniques. However, the concept of switching synchronization may be considered as an interesting research problem. Thus, designing of switching synchronization, which makes the system behavior switch among chaotic states, stable states, periodic states, synchronization and anti-synchronization with a master system, is the motivation of the present research work.
Different control techniques are proposed for the synchronization of a hyperchaotic/chaotic system like active control,[46] sliding mode control (SMC),[51] observer-based control,[52] and hybrid control techniques,[53] etc. Recently, some papers reported with linear and composite control of chaotic systems.[54,55] Among these control techniques, an SMC is considered as a better and effective controller because of its many inherent advantages.[56] Recently, more attention has been paid for overcoming the inherent disadvantages of an SMC like high gain and the chattering problem. An integral SMC is used to solve the high gain problem[56] and reaching time of the sliding surface is reduced by a global SMC.[45] Motivated with the above-stated literature, the aim of this section is to design an adaptive global integral SMC for the multiple switching synchronization of the new system.
The proposed system in Eq. (2) is considered as both master and slave systems. Two control inputs (
) are added in the first and third states of the slave system. The master and slave systems are described in Eqs. (16) and (17), respectively.
where
d1,
d2,
d3, and
d4 are the bounded matched disturbances. The synchronization error dynamics between the master and slave systems (
16) and (
17) is defined as:
Asymptotic stability of error dynamics in Eq. (
18) is ensured by considering the following integral sliding surfaces as
where
k1 and
k2 are positive constants to be decided by the designer of the controller. When the system operates in the sliding surface, it satisfies
. Therefore, using this condition along with Eqs. (
18) and (
19), the equivalent sliding mode dynamics is written as
[56]The stability of Eq. (
20) is shown by considering a Lyapunov function candidate as
The time derivative of
using Eq. (
20) is written as
It is apparent that equation (
21) is a negative-definite function. Thus, according to the Lyapunov stability theory, the sliding motion on the sliding surface is asymptotically stable and ensures the convergence of error dynamics to the origin.
In order to achieve the smoother, and chattering free control input and sliding surface, global sliding surfaces are designed as[45]
where
G1,
, and
a1,
are the gain constants.
and
are the initial states of the sliding surfaces
s1 and
s2, respectively.
Now, in order to bring the system trajectory onto the sliding surfaces
, sliding mode control laws are proposed as:
where
M1,
M2,
,
,
, and
are positive gains constants.
Theorem 1 If the slave system in Eq. (17) is controlled by the sliding mode control given in Eq. (23) then its trajectories follow the master system (16) and ensure the asymptotic convergence of the global sliding surfaces
and
to origin
,
.
Proof We now consider a Lyapunov function candidate as follows:
The time derivative of
is written by using Eqs. (
19) and (
23) as the following forms
Using control input (
23), equation (
25 can be written as
Let the disturbances be bounded as
,
,
,
, and suppose
,
, where
Then, using Eq. (
27), equation (
26) can be written as
where
,
,
,
,
,
.
It is seen that the derivative of Lyapunov function V2 is asymptotically stable. Thus, global sliding surfaces (22) are asymptotically stable and ensure that the global sliding surfaces
and
converge to
and
, respectively. Hence trajectories of the slave system (17) follow the master system (16).
Usually it is assumed that the upper bound of disturbances is unknown and hence finding the actual values of the parameters
and
is difficult. Therefore, the adaptive tuning method is designed to determine the upper bound of the disturbances. Hence the control inputs in Eq. (23) are modified as:
where
and
are the estimated values of
and
, respectively. The estimates
and
are done using the following adaptation rule:
where
is a positive constant.
Now, the stability of the sliding surfaces and their convergence to origin is ensured with the help of the following theorem.
Theorem 2 Suppose that the slave system in Eq. (17) is controlled by Eq. (30) together with the global sliding surfaces (22). Then the trajectories of the slave system (17) follow master system (16) and ensure the asymptotic convergence of the global sliding surfaces
and
to origin
,
.
Proof Let a Lyapunov candidate as
, where
and
, with
. The time derivative of
using Eqs. (22) and (31) can be written as
Using the derivative of integral sliding surfaces (
19), equation (
32) can be written as
Now, using error dynamics (
18) and control input (
30), equation (
33) can be written as
Using the boundness condition defined in Eq. (
27), equation (
34) can be written as
Arranging the terms, we can write as
Or equation (
36) can be written as
Using
, equation (
37) can be written as
Since,
,
, and
,
, we can say that according to Lyapunov stability theory, Lyapunov function V3 is asymptotically stable and ensure that the global sliding surfaces
and
converge to
and
, respectively, in the presence of unknown bounded disturbances. The stability analysis of the Lyapunov function V3 presented in the paper is done with a similar stability analysis to the ones available in Refs. [57] and [58].
Switching synchronization technique is achieved by using the following stages:
Stage 1 The chaotic behavior is achieved by the synchronization between the master and slave systems using the technique presented above.
Stage 2 The stable behavior is achieved by the synchronization but considering those parameters of the master system for which it behaves in a stable manner. One such set of parameters is a = 5, b = 40, c=0.001, d=0.2, and p=0.1.
Stage 3 The periodic behavior is achieved by the synchronization but, the value of parameter
is considered for the master system and keeping other parameters the same as before.
Stage 4 The chaotic behavior is used for the hybrid synchronization, i.e., the compete synchronization of the first and fourth states, and anti-synchronization of the second and third states. The error dynamics and control inputs for hybrid synchronization are defined in Eqs. (39) and (40), respectively. It may be noted here that the Stages 1–4 need not be in sequential order, rather it can be in any arbitrary order.
where the structure of the sliding surfaces is similar to Eqs. (
19), (
22) and the stability of the sliding surfaces can be ensured by similar analyses as shown in Eqs. (
31)–(
37).
6. Results and discussion for the switching synchronization between the master system in Eq. (16) and the slave system in Eq. (17)The initial conditions considered here for the master, slave systems, and adaptation laws are
,
, and
, respectively.
The external bounded disturbances are considered as
,
,
,
.
The parameter considered for adaptive global integral SMC for the synchronization are given here as
,
,
,
,
,
,
,
. These parameters are chosen to ensure the faster synchronization but with lesser and smoother control inputs.
The switching synchronization states of the master and slave systems are shown in Figs. 11 and 12, respectively. The switching synchronization errors are given in Figs. 13 and 14, respectively.
The behaviors of the sliding surfaces si and
are shown in Figs. 15 and 16, respectively. The estimation of the gains of the switching control laws for the boundness of the disturbances is shown in Fig. 17.
It is observed from Fig. 11 and Fig. 12 that during the time unit
, the synchronization is achieved for the chaotic behavior of the master system; during the time unit
, the synchronization is achieved for the stable behavior; during the time unit
, the synchronization is achieved for the periodic behavior; and during the time unit
, the hybrid synchronization is achieved. During hybrid synchronization the first and fourth states are synchronized, and the second and third states are anti-synchronized. It is noted from Fig. 13 and Fig. 14 that the synchronization errors converge to zero in a very short time interval. It is visual from Fig. 15 and Fig. 16 that the reaching time to the sliding surfaces is less and also chattering is negligible.
7. Comparison of the performances of the controller in Ref. [54] and the controller of the present workThis section presents a comparison of the performances of the controller of the present work and the controller reported in Ref. [54].
7.1. Linear synchronization between the master and slave systems as given in Ref. [54]The structure of master system, slave system, errors, and controller designed in Ref. [54] are rewritten respectively in the following equations.
where the initial conditions, and parameters of systems and controllers are considered to be the same as those in Ref. [
54] and are given as
,
,
,
,
,
,
a = 1,
m = 1,
c = 1.
7.2. Synchronization between chaotic systems of Ref. [54] by using the adaptive global integral SMCThe structure of the master system, slave system, errors, sliding surfaces, and controller designed using the adaptive global integral SMC, i.e., the controller designed in the present work, are shown below.
The initial conditions considered for simulation of the synchronization between the master system (
45) and slave system (
46) by using the adaptive global integral SMC (
47) are the same as those in Ref. [
54], i.e.,
,
,
a = 1,
m = 1,
c = 1, and the parameters of the controllers are considered as
,
,
,
, and
.
The performances of these two controllers are compared in the absence and presence of the matched disturbances and are shown below. The disturbances d1, d2, d3 and d2, d3 added in the slaves systems (42) and (46), respectively, are given in the following equation
7.3. Comparison of the synchronization results when
in slave systems (42) and (46)Here, the comparison between the performances of the controller in Ref. [54] and the controller designed in the present work is shown in the absence of disturbances. The synchronization of the states of masters (41), (45), and slaves (42), (46) is given in Fig. 18. The synchronization errors in the absence of disturbances between the states of master systems (41), (45) and slave systems (42), (46) are shown in Fig. 19. It is seen from Figs. 18(a)–18(c) that states of master system (41) and slave system (42) are not synchronized properly by using the controller[54] given in Eq. (44). But, using the controller of the present work, the states of master system (45) and slave system (46) are synchronized properly as shown in Figs. 18(d)–18(f). It is seen from Fig. 19 that the synchronization error is the same by using both the controllers.
7.4. Comparison of the synchronization results in the presence of disturbances for both the controllersThe robustness of synchronization in the presence of disturbances between the controller designed in Ref. [54] and the controller of the present work is compared. Synchronization of the states of master (41), (45, and slave (42), (46) systems is shown in Fig. 20. It is noted from Figs. 20(a)–20(c) that states of master (41) and slave (42) systems are not properly synchronized by using the controller[54] given in Eq. (44). But, using the controller of the present work, the states of master (45) and slave (46) systems, as shown in Figs. 20(d)–20(f), are synchronized properly. It is apparent from Fig. 21 that the steady state synchronization error is very high in the results obtained by using the controller of Ref. [54], whereas the same is absent using the controller designed in the present work. Thus, the controller designed in the present paper performs better than that designed in Ref. [54].
8. ConclusionsIn this paper, a new 4D chaotic system having the largest first positive Lyapunov exponent is reported. The largest first positive Lyapunov exponent of the new system is compared with many reported systems (3D and 4D chaotic systems) in the literature. The proposed system has a hyperbolic curve and circular paraboloid types of equilibria along with coexistence of chaotic attractors and hyperchaotic behavior. The simulation results confirm the claims about the system. Further, the circuit implementation results validate the chaotic nature of the system. The proposed system may be more suitable for secure communication since it has the largest first Lyapunov exponent. Switching synchronization phenomenon is proposed for the new chaotic system. An adaptive global integral sliding mode control is designed for the switching synchronization. The switching synchronization is achieved in four stages. These are (i) chaotic behavior, (ii) stable behavior, (iii), periodic behavior, and (iv) hybrid synchronization.
