A new 4-D hyperchaotic hidden attractor system: Its dynamics, coexisting attractors, synchronization and microcontroller implementation

ABSTRACT


THE NEW SYSTEM
The following system of equations which have two quadratic nonlinearity terms represents the new system: In this system, a and b are two positive real parameters. The system is really simple where just two nonlinearities are presented which is the minimum number of nonlinearities required for chaotic behavior, to our knowledge. To obtain the equilibrium points of a system, the left side of the system equations should be equated to zero: Then, solving for the states of the system (the X={ x1,x2,x3,x4} vector). It is easily noted that our system has no any solution because from 2.1 and 2.4 the states x3 and x4 equal to zero and this result cannot be consistent with 2.2 for non-zero values of b. Then, there are no solution, and hence there are no equilibrium points for this system and any strange attractor generated by the system is a hidden attractor. System 1 shows chaotic behaviors for different values of parameters (a and b). As instant for a=3.5 and b=0.05, the phase portraits clearly show the chaotic nature of the system. Figures 1 and 2, show the 2 dimensional hyperchaotic attractors of the system for x1,x2 and x2,x3 with {0.1,0.1,0.1,0.1} as an initial conditions and 500 sec. simulation time.

Symmetry
It can easily be noted that the system is invariant under the transformation: which means that if 1 , 3 , 1 , 3 are a solution for the system equations, then − 1 , 2 , − 3 , − 4 is also a solution for it.

Dissipation
To investigate whether the system is dissipative or not, the divergence of the system should be calculated. To this end, the system is rewritten as a vector notation: Then, the divergence of the system can be obtained by the following: The negative sign of the divergence value indicates that the system is dissipative. System dissipation means that the system will settle to a set of zero measurement for any asymptotic motion. Which leads to chaotic behavior.

SYSTEM ANALYSIS 3.1. System analysis using Lyapunov exponents
System in (1) shows chaotic behavior for certain sets of values of a and b. To investigate these values and determining system behavior over the full range of a and b, and also to determine whether the system can exhibit hyperchaotic attractor, we used first and second Lyapunov exponents. We selected a=0.02 as a rough value from its chaotic range, then the Lyapunov exponents for b= [1 10] with a step equal to 0.04 has been calculated using {0.1,0.1,0.1,0.1} as an initial condition. Lyapunov exponents have been found by using Alen wolf algorithm [28]. The system has been simulated using Runge-Kuta method for each value of b and for sufficient time (1000 sec.) to reach to exponents settling values. MATLAB 2018a has been used in this analysis. Figure 3 shows the four Lyapunov exponents with b changing. Figure 4 and Figure 5 shows the maximum and the second Lyapunov exponents with b changing. These figures show that the system is chaotic for b=[2. 7 10], and within this range of b the system is hyperchaotic for some subranges. Also, the same process has been repeated for a where b has been fixed to 3.5 and a has been changed from 0 to 0.2 with 0.001 step size using the same IC. Figure 6 shows the resulting plot, and from this figure it is noted that the system is hyperchaotic from 0 to 0.085. As a specific case, we put a=3.5 and b=0.05 and finding Lyapunov exponents for 2000 secs. simulation time, the settling values for Lyapunov exponents in this case are: and since, the first and second exponents are positive, the system is hyperchaotic. The Lyapunov dimension is a measure of system chaotic behavior degree, it can be defined according to Kaplan-York [29] conjecture and as follows: where J should be selected such that ∑ =1 > 0 and ∑ +1 =1 < 0. Then, using the Lyapunov exponents found for a=0.05 and b=3.5, the Lyapunov dimension is:

System analysis using bifurcation diagram
Bifurcation diagrams is a great tool for investigating system behaviours with the change of one or more than one of system parameters. It can be used also to determine some system features like the rout to chaotic and the existing of coexisting attractors which is an interesting feature of the systems. Firstly, the bifurcation diagram for b parameter has been found by fixing a at 0.02 and changing b from 0 to 14 with 0.02 as a step size. At each step, the system is simulated for 500 sec. with IC={0.1,0.1,0.1,0.1} and the local maxima of x2 have been found. The first 150 secs have been ignored to ensure reaching steady state behaviour and cancelling transient effects. Figure 7 shows the resultant plot which clearly shows the chaotic nature of the system for the same range of b obtained previously by using Lyapunov exponents. Also, bifurcation diagram for parameter a change from 0 to 0.2 has been found with 0.0001 step size and b fixed at 3.5 which are the same conditions used in Lyapunov exponents section. Figure 8 shows the resultant plot which clearly shows consistency between the results obtained in this plot and the results obtained from the plot obtained for Lyapunov exponents where the same ranges of chaotic and regular behaviour of system can be noted in the two plots.
It is known from literature that there are three main routes to chaos. The first one is period doubling in which the period of periodic motion of the system is doubled at a specific point of bifurcation parameter and then and at a specific value of bifurcation parameter the motion becomes chaotic. The second route is the intermittency chaos in which periodic motion is replaced by chaotic at a certain value of bifurcation parameter. The last route is quasi periodic to chaos in which the system motion changes from fixed to periodic and quasi periodic and lastly chaotic motion appear, this rout is rear in systems. From Figures 3 and 7, it can be concluded that our system shows unusual rout to chaos where reverse quasi periodic rout is the rout of our system to chaos. The system is chaotic for b<8, then quasi periodic motion appears and for b>13 the system exhibits periodic motion.
To show the capability of the proposed system to exhibit coexisting attractors, we generate bifurcation diagram with two different ICs in each step. The first IC was [0.1,0.10.1,0.1] and it is fixed in each step and the other IC is selected from ten randomly generated ICs, where for each IC the maxima of x2 is determined and compared with those obtained for fixed IC condition to find the one which has more deviation. For the two IC, the maxima of x2 have been found and plotted against the bifurcation parameter which was b. the bifurcation parameter has been changed in the range [0. 5 14] with 0.02 as a step size and a has been fixed to 0.02. The maxima of x2 for each IC have been plotted two different colours: red and green. The resultant plot is shown by Figure 9. From this figure it is clear that the system exhibits coexisting attractors for several ranges of the bifurcation parameter. As example for b=2.2 and a=0.02 and for the ICs I1={0,-3,1,-1.5}, I2={-0.2,-2,1,-1} the system exhibits two chaotic coexisting attractors, but for I3={0,0,1,1} the system reach to stable spiral response after 400 s transient chaos. Table 1 lists some examples of these coexisting attractors and Figures 10-15 shows the phase portraits plots of the attractors.

SYNCHRONIZATION CONTROLLER DESIGN
In this section, a controller to synchronize two identical systems of the proposed system is designed. The system (1) is uncontrolled system or the master and its states output should be tracked or synchronized by the second system which is often called the slave. The slave system is the same as system (1) added to it the control signals = { 1 , 2 , 3 , 4 }. Master system is represented by system 1 with a and b assumed to be unknown, while the slave system is represented by the following system (4): The synchronization error is defined as (5): Using this relation with (1), (4) and ̇=̇−̇ , = 1,2,3,4, the following system of equations is obtained: Our target in the design procedure is to satisfy the following stable error dynamics: Combining (6) and (5) and solving for U, we obtain: where the error between the real parameter values of a and the estimate values an. The parameter estimation will be designed by Lyapunov theory. The following function is selected as a Lyapunov function candidate: Then, and, If we put: Then, Which is a negative semidefinite and the prove of system stability is completed.

SYSTEM IMPLEMENTATION
The proposed system has been implemented using microcontroller platform. We propose implementing of the system using the so-called Arduino MCU board, where Arduino uno board has been used to implement the system. Arduino boards are easy to use MCUs development boards. The used board is the Arduino uno which has 32 kB flash memory for storing the program and 2 kB of sram memory for storing program data and 1 kB EEPOM which can be used for storing static data of the program. These specifications can cover all the requirements of implementing our system. The implementation program uses fourth order Runge-Kuta method to solve the ODE system of equations of our system with step size equal to 0.01 sec. For the purpose of testing the system, we have used Matlab to plot the phase portraits of the system states plane where the MCU board and laptop have been connected via serial communication. Figure 16 and Figure 17 shows the phase portrait of x1, x2 and x2, x3 planes received from MCU board for 100 secs. Run time with a=0.1 and b=4. Figure 16. x1, x2 phase portrait of MCU system Figure 17. x2, x3 phase portrait of MCU system

SIMULATION AND EXPERIMENTAL STUDY
In this section, the theoretical results of synchronization of the proposed hyperchaotic system are investigated by simulation and experimentally. Also, a secure communication using our synchronization system is introduced and tested. As mentioned earlier, MATLAB 2018a script is used to write the simulation programs. The experimental study relayed on the system implemented on the MCU Arduino board and as described in the previous section.

= + 1
and at the receiver: It is clear that if full synchronization is achieved, signal is completely retrieved, i.e. rs=sig. In our test, we used the implemented system on MCU as a transmitter side and the laptop as a receiver side. A sinusoidal signal (sig=2sin(20t)) is selected as a signal to be transmuted. Figure 20 shows the transmitted (sig) signal and the received (rs) signal. From this figure, it can be noted that the transmitted signal is tracked at the receiver with no error and in very short time.

CONCLUSION
In this paper, a new 4-dimensional hidden attractor hyperchaotic system has been proposed. Dynamical properties of the system have been investigated which includes coexisting feature of the system. After that, a synchronization system for the introduced system has been designed assuming unknown system parameters. The design procedure depended on simple controller with Lyapunov theory for the purpose of deriving update lows for the unknown parameters. An efficient and cost-effective implementation basing on Arduino boards shows very efficient implementation tool for chaos implementation and application where it has been used to build secure communication system. Simulation and experimental study have showed the effectiveness of the designed synchronization system and also the effectiveness of the secure communication system which can be designed basing on the designed synchronization system.