Local Dynamics Analysis of a Four-Dimensional Hyperchaotic Lorenz System ()
1. Introduction
In 1963, Lorenz, an American meteorologist, discovered the first chaotic attractor while studying the atmospheric motion, which started the study of chaos. The Lorenz model also has become a classical model in the field of chaos.
Based on Lorenz system, many scholars proposed Lorenz-like systems, which are widely used in image encryption, secure communication and many other fields. Perevoznikov et al. [1] considered two methods of stability analysis of systems described by dynamical equations through Lorenz and Rossler Model Problems. Liu et al. [2] gave new insights into a new Lorenz-like chaotic system from the literature. Wu et al. [3] investigated the stability of equilibrium points and Hopf bifurcation of a Lorenz-like system with the normal form theory. The conditions guaranteeing the Hopf bifurcation were derived. Hu et al. [4] studied stability and Hopf bifurcation of a new four-dimensional quadratic autonomous system both analytically and numerically. Aziz et al. [5] designed a new electronic circuit as an engineering application on a four-dimensional chaotic system. Fu et al. [6] combined a three-dimensional autonomous system with a complex network, and used the chaotic synchronization of this system to apply it to confidential communication. Wang et al. [7] suggested a brand-new Julia-fractal system in three-dimensional to increase the randomness of chaotic sequences and widen the key space and then a fresh quantum circuit for Fibonacci scrambling was created.
The bifurcation problem also has a profound application background. Zhang et al. [8] investigated the bifurcation characteristics of the active magnetic bearing-rotor system subjected to the external excitation analytically. Wang et al. [9] presented a two-dimensional simplified Hodgkin-Huxley model under exposure to electric fields, and the Hopf bifurcation of this model was analyzed both analytically and numerically. Wen et al. [10] obtained the Hopf bifurcation associated with positive non-homogeneous steady states of a reaction-diffusion-advection logistic model, with two non-local delayed density-dependent terms and zero-Dirichlet boundary conditions. Li et al. [11] studied a four-dimensional inertial two-nervous system with delay and obtained the critical value of zero-Hopf bifurcation by analyzing the distribution of eigenvalues.
The following manuscript is organized as follows: a four-dimensional hyperchaotic Lorenz system is presented in the next section. In Section 3, the equilibrium points and their stability are studied. The pitchfork bifurcation and Hopf bifurcation are analyzed in Section 4. In the last section, numerical simulations are presented to verify the theoretical results.
2. Dynamic Modeling
Wang et al. [12] proposed a four-dimensional hyperchaotic system by adding the nonlinear controller to Lorenz system, and analyzed its basic dynamic characteristics. On the basis of these works, Xie et al. [13] proposed a plaintext related color image encryption algorithm based on multiple chaotic maps to solve the problem of color image encryption and the local dynamics analysis of this system is carried out. The specific model is shown in the following:
(1)
Where
are the state variables,
are controllable parameters. When
,
,
,
, the system is in a hyperchaotic state [13] . Its hyperchaotic state can be observed by Figure 1.
Figure 1. The projections of system (1) when
,
,
,
. (a) The three-dimensional projection of x-z-w plane; (b) The two-dimensional projection of x-y plane; (c) The two-dimensional projection of y-z plane; (d) The two-dimensional projection of x-z plane.
3. Equilibrium Analysis
In this section, the stability of the equilibrium points of this system will be analyzed.
Theorem 1. Let
,
,
,
,
, the system has only one equilibrium point
as long as it meets any of the following conditions: 1)
i.e.
; 2)
and also
; 3)
and also
. And the system has three equilibrium points as long as it meets any of the following conditions: 1)
; 2)
and also
,
; 3)
and also
,
. The system has five equilibrium points as long as it meets the condition: both
and
,
.
Proof 1. Solving the equations to find the equilibrium points of system (1):
(2)
when
, it’s easy to obtain the equilibrium point
; and when
,we have
(3)
let
,
, where
.
When
the system has only one equilibrium point S0.
When
and also
, the system has equilibrium point S0. And when
and the
, the system has three equilibrium points as following: S0;
;
. Where
,
.
When
or
, the system has three equilibrium points. The equilibrium points in the former case are
;
,
;
,
respectively. And the equilibrium points in the other case are S0;
,
;
,
respectively.
When
, the system has five equilibrium points. The equilibrium points are S0, M1, M2, N1, N2. Therefore, Theorem 1 is proved.
Then the stability will be discussed. Choosing case
as an example and the analysis process for other situations are similar. Based on Theorem 1, the system has equilibrium points S0, M1, M2, N1, N2 when
. Since the system has symmetry, the stability of M2, N2 are same as the M1, N1. The Routh-Hurwitz criterion is applied to analyze stability of S0, M1, N1, and the following theorems are obtained.
Theorem 2. When
,
,
,
; or
and
,
,
the equilibrium point
is asymptotic stable.
Proof 2. The Jacobian matrix of system (1) at S0 is as following:
It’s characteristic polynomial is
. And characteristic roots are
,
,
.
All characteristic roots have negative real parts if these parameters satisfy
,
,
,
. Applying the Routh-Hurwitz criterion [14] we can know S0 is asymptotically stable. Similarly, when the parameters simultaneously satisfy
and
,
,
, the equilibrium point is also asymptotically stable.
Theorem 3. When
,
,
,
, the equilibrium point
is asymptotically stable.
Proof 3. The Jacobian matrix of system (1) at M1 is as following:
The characteristic polynomial of this matrix is
where
Calculating the following determinants:
,
,
,
. According to Routh-Hurwitz critetion, the equilibrium point M1 is asymptotically stable.
The analysis process of equilibrium N1 is the same as that of M1. Simply replace the value of t1 with t2 in the coordinate of equation M1, then the stability of point N1 can be obtained.
4. Bifurcation Analysis
4.1. Pitchfork Bifurcation
In this section, the pitchfork bifurcation of equilibrium point S0 is investigated. b is set to the bifurcation parameter, which leads to the following theorem.
Theorem 4. Assuming the parameters satisfy conditions that
,
,
, the pitchfork bifurcation occurs when the parameter b crosses the critical value
.
Proof 4. The Jacobian matrix of S0 is
It’s characteristic polynomial is
and the characteristic roots are
,
,
,
, respectively. In other words, the system has one zero eigenvalue and the others with non-zero real parts. The corresponding eigenvectors of these four eigenvalues are calculated as follows:
,
,
,
Let
Using the following transformation
Then system (1) becomes
(4)
where
Next, the central manifold theorem [15] [16] is used to analyze the stability of point S0 when
. Constructing the following local central manifold represented by the two-parameter family x1 and b.
and
are sufficiently small.
Suppose
have expressions of the following forms:
(5)
(6)
(7)
According to the chain rule for derivatives, we have
(8)
where
Substituting Equations (4)-(6) to Equation (7), and comparing the coefficients of the corresponding terms with equal exponents, we can calculate the expressions of
as follows:
(9)
(10)
(11)
Finally, substituting expressions (8)-(10) to Equation (3), we can obtain the vector field of the system on the central manifold.
(12)
The equilibrium point of this system in this expression is
. To satisfy the condition that pitchfork bifurcation occurs, letting
, we calculate the value at
of the following equations:
According to the pitchfork bifurcation theory [17] , when parameter b crosses the critical value
, the pitchfork bifurcation occurs at the equilibrium point Sp.
4.2. Hopf Bifurcation
This section focuses on Hopf bifurcation with a as the bifurcation parameter.
Theorem 5. Assume
, when the parameter a crosses the critical value −1, the Hopf bifurcation will occur at the point S0.
Proof 5. The characteristic polynomial of Jacobian matrix of the system at S0 is
when
, the eigenvalues are
,
,
. From calculating, we obtain
, which satisfy the two conditions for generating Hopf bifurcation. According to Hopf bifurcation theory the system generates Hopf bifurcation at S0.
Normal form theory [15] [16] [18] and central manifold theorem are used for further analysis. According to the conclusion of Theorem 5, the specific values of the relevant parameters are set to
to satisfy the conditions for the Hopf bifurcation to occur. The corresponding eigenvalues of these parameters are
respectively. Converting the system (1) to the following normal form:
where
The
has Taylor expansion near the S0:
where
Letting
is the complex eigenvector which is corresponding to the eigenvalue
,
is adjoint eigenvector which meets
and the standardized condition
. From calculation we have the value of q and p:
(13)
The critical real eigenspace
corresponds to
and
corresponds to
are both two-dimensional and is spanned by
. For any
,
, where
,
,
. The complex variable z is the coordinate on
, having
(14)
Therefore, system (33) has the following form in the coordinate system (36):
(15)
where
The central manifold
has an expression:
We also have
, by combining the quadratic forms with respect to z and
in the invariance condition in
, we can obtain
,
satisfy the following system of linear equations
(16)
According to all the above formulas, the following results can be obtained:
Then the constraint equation of the original system on
is
Calculating the first Lyapunov coefficient
Therefore, the Hopf bifurcation is subcritical.
5. Numerical Simulations
In this section we present some numerical simulations of system (1) to verify the above theoretical analysis. For showing the stability of the equilibrium point S0 of system (1) under the parameter conditions satisfying Theorem 1, we give the time histories of the solutions by Fourth-order Runge-Kutta methods. The parameters are chosen to be
and
, satisfying the first and second cases of Theorem 1, respectively. According to Figure 2, it can be seen that the equilibrium solution is asymptotically stable when the conditions of Theorem 1 are satisfied.
The bifurcation of the system is analyzed in Section 4. According to the results of the theoretical analysis, choosing
,
,
,
, at this point, the parameter conditions for the system to generate Hopf bifurcation are satisfied. We give the time histories, Lyapunov exponential spectrum, projections of four-dimensional phase portrait diagram of the model at this time.
Figure 3(a) shows that in this case the system is in periodic motion and also in a steady state. From the Lyapunov exponents spectrum in Figure 3(b), we can see the Lyapunov characteristic index satisfy
which means stable closed orbit will occur. Figure 4 and Figure 5 show the two-dimensional and three-dimensional projections of the system when the Hopf bifurcation condition is satisfied, respectively.
Figure 2. Time histories of system (1) for different parameters. (a)
; (b)
.
Figure 3. Bifurcation numerical simulations. (a)
; (b)
.
Figure 4. The two-dimensional projection of system (1). (a) The two-dimensional projection of x-y plane; (b) The two-dimensional projection of z-w plane.
Figure 5. The three-dimensional projection of system (1). (a) The three-dimensional projection of x-y-z space; (b) The three-dimensional projection of x-y-w space.
6. Conclusion
The local dynamics of a new four-dimensional Lorenz-like system is investigated with both analytical and numerical methods in this paper. Starting with a discussion of the number of equilibrium points, the stability is analyzed under different parameters by applying the Routh-Hurwitz criterion. The parameter conditions corresponding to the occurrence of pitchfork bifurcation and Hopf bifurcation, respectively, are given, and the related properties of the bifurcations are analyzed. Finally, numerical simulations verify the correctness of the theoretical parts and show the change in the number of equilibrium points as the pitchfork bifurcation parameter crosses the critical value.
Acknowledgements
The project was supported by National Natural Science Foundation of China (11772148, 12172166) and China Postdoctoral Science Foundation (2013T60531).