A Memristor-Based Complex Lorenz System and Its Modified Projective Synchronization

The aim of this paper is to introduce and investigate a novel complex Lorenz system with a flux-controlled memristor, and to realize its synchronization. The system has an infinite number of stable and unstable equilibrium points, and can generate abundant dynamical behaviors with different parameters and initial conditions, such as limit cycle, torus, chaos, transient phenomena, etc., which are explored by means of time-domain waveforms, phase portraits, bifurcation diagrams, and Lyapunov exponents. Furthermore, an active controller is designed to achieve modified projective synchronization (MPS) of this system based on Lyapunov stability theory. The corresponding numerical simulations agree well with the theoretical analysis, and demonstrate that the response system is asymptotically synchronized with the drive system within a short time.

Motivated by the above discussions, this paper aims to construct a novel complex Lorenz system with a flux-controlled memristor, and to realize modified projective synchronization (MPS) of two identical memristor-based complex Lorenz systems.The system is generated from a 4D (four-dimensional) memristor-based real Lorenz system by replacing two real variables to complex variables, which can be expressed as a 6D (six-dimensional) real ordinary differential equations (ODEs).Symmetry and invariance, dissipation, equilibria and stability of the system are analyzed theoretically.Dynamical behaviors are numerically investigated by means of bifurcation diagram, Lyapunov exponent, time-domain waveform, and phase portrait.Fixed points, limit cycles, torus, chaos, and transient behaviors are revealed in the system with different parameters and initial conditions.Furthermore, MPS is designed to synchronize the memristor-based Lorenz systems quickly and efficiently, which is verified by numerical simulations.
The rest of this paper is organized as follows.In Section 2, a novel memristor-based complex Lorenz system is proposed, and its properties are analyzed theoretically.In Section 3, dynamical behaviors of the system are explored numerically.In Section 4, an active controller is designed to realize MPS for identical memristor-based complex Lorenz systems, and numerical simulations are presented to illustrate the effectiveness of the proposed scheme.Finally, some conclusions are drawn in Section 5.

A Novel Memristor-Based Complex Lorenz System and Its Properties
In this section, we construct a novel memristor-based complex Lorenz system and analyze its properties including symmetry, dissipation, equilibria and stability.

Memristor Model
Memristor, a nonlinear resistor with memory effect, is defined by ( , ) 0 f q   , which includes flux-controlled memristor and charge-controlled memristor.Equations ( 1) and ( 2) describe flux-controlled memristor and charge-controlled memristor respectively [42]: where  and q denote magnetic flux and charge; i and v denote the device terminal current and voltage; ( ) W  and ( ) M q denote memductance and memristance which are characterized by a piece-wise linear function [1], smooth continuous cubic nonlinear function [13], Chebyshev polynomials [19], etc.
In this paper, a flux-controlled memristor with cubic nonlinear characteristics is considered to be: where  and  are positive parameters.

Memristor-Based Complex Lorenz System
In [8], a modified Lorenz system with a flux-controlled memristor is described by: where 1 2 3 4 , , , a a a a are positive parameters, ( ) 3 in this paper.By replacing real variables 1 2 , x x with complex variables, we construct a memristor-based complex Lorenz system of the form: (1/ 2)( ) where 1

Properties of the
is also a solution of the system.

Dissipation
The divergence of system ( 6) is: where 1 4 , , , a a   are positive parameters as mentioned, which can guarantee 0   V , so the system ( 6) is dissipative.

Equilibria and Stability
By solving 1 2 , we can obtain the equilibria of system (6): {( , , , , , where , , , R m n p c  , and The system has infinite equilibrium points, in which, 01 E represents the equilibrium points on 6 u axes, obviously.For convenient description, we study the stability of 01 E , and the Jacobian matrix of system (6) at 01 E is calculated as: The characteristic polynomial of Equation ( 9) is: The eigenvalues of Equation ( 10) are: Since 1 2 3 4 , , , , , a a a a   are positive, the system of ( 6) has a zero eigenvalue ( 1  ) and three negative eigenvalues ( 2 3 4 , ,    ).According to the Routh-Hurwitz theorem, the necessary and sufficient condition for the other roots to have negative real parts is if and only if if the inequality constraint is satisfied and the equilibrium points are stable, and vice versa.Therefore, the system has infinite stable and unstable equilibrium points with different parameters.

Dynamical Behaviors of the Memristor-Based Complex Lorenz System
In this section, we explore dynamical behaviors of the memristor-based complex Lorenz system with different parameters and initial conditions based on conventional numerical methods.

Dynamical Behaviors with Different Parameters
In order to reveal dynamical behaviors of the proposed memristor-based complex Lorenz system, we fix and vary 3 a in range of [0,200] .Based on the Wolf algorithm and local maximum method, Lyapunov exponents and a bifurcation diagram are calculated and plotted with different computational time intervals 0-3000 s and 0-30000 s as shown in Figures 1 and 2, respectively, which can provide an overall perspective view of the system.However, the two figures are not consistent to each other, apparently, and Figure 2  indicates that transient dynamical behaviors exist in such intervals due to a boundary crisis [16].For the sake of exploring the dynamical behaviors in detail, time-domain waveforms, phase portraits, and Lyapunov exponents against time are presented under some fixed parameters as shown in Figures 3-8.(1) Fixed points exist for 3 [0, 18.2] a  .When 3 5 a  , the system converges to a fixed point (0,0,0,0,0, 891.2) E  , and the six Lyapunov exponents are non-positive as shown in Figure 3. (2) Transient chaos to fixed points exist for 3 (18.2, 45.3] a  .When 3 25 a  , the system goes through transient chaos and converges to fixed point (0,0,0,0,0, 2120) E  as shown in Figure 4.
One Lyapunov exponent (i.e., L1) is positive incipiently, and then tends to zero asymptotically.(3) A chaotic zone covers most region of 3 (45.3,161.4) a  .When 3 50 a  , the system operates chaotically with a positive Lyapunov exponent as shown in Figure 5.     Remark 1. Due to transient phenomena, the dynamical behaviors of nonlinear dynamical systems become more complicated and difficult to depict.The time-domain waveform, phase portrait, bifurcation diagram, and Lyapunov exponent should be considered conjunctively, and the computational time interval should be large enough to describe the system behaviors completely and accurately.

MPS of the Memristor-Based Complex Lorenz System
In this section, we study the MPS of two identical memristor-based complex Lorenz systems, and provide numerical simulation results to verify the feasibility and effectiveness of the proposed schemes.

MPS Design
System (5) is our synchronization object, and the drive system and response system can be described as: (1 / 2)( ) where 1 , , , , L v  are control functions, and then the complex systems ( 12) and ( 13) can be described by real Equation ( 14) and Equation ( 15) respectively: Errors between the drive system (12) and the response system (13) are defined as:  , which are errors between systems ( 14) and ( 15), ( 1 6) j k j   denotes projective factor.By substituting Equations ( 14) and ( 15), we can obtain the derivative of ( 1 6) ui e i   as: Theorem 1.The MPS between the drive system (12) and the response system (13) Proof.Choose the Lyapunov function as: Solve the derivative of ( ) V t along the trajectory of error system (16): ( ) [ Substitute Equation (17) into Equation ( 20), then According to the Lyapunov stability theory, since ( ) V t is a positive function and its derivative ( ) ) ) j e j   tend to zero as t   .Therefore, the response system ( 13) is synchronized with the drive system (12) asymptotically.□

Conclusions
In this paper, a novel complex Lorenz system with a flux-controlled memristor has been constructed and investigated.We found that the proposed system has infinite stable and unstable equilibrium points and exhibits abundant dynamical behaviors including fixed points, limit cycles, tori, chaos, and complex transient behaviors.Furthermore, MPS controller has been designed to synchronize the memristor-based Lorenz systems up to different scaling factors quickly and efficiently, which has been demonstrated by numerical simulations.However, the relevant research has just begun, and the memristor-based complex systems and their synchronization will have to be investigated from the viewpoint of applications in the future.

4 )
Transient chaos to Period-5 orbits exist in a narrow interval 3 (52.7479,52.7482) goes through transient chaos and Period-5 orbit intermittently, and enters into the steady state of Period-5 eventually, as shown in Figure6.(5)Transient Period-3 to tours exist for 3 [87.6,92.2] a  .When 3 90 a  , the system operates in Period-3 orbit at first, and then enters into the state of tours as shown in Figure7.(6) Transient Period-1 to chaos exist for 3[161.4,200]   a  .When 3 165 a  , as shown in Figure8, in the beginning the system operates in Period-1 orbit, then enters into the state of tours, and slides into a chaotic state in the end.The Lyapunov exponent L1 changes from zero to a positive number.