A NEW HYPERCHAOTIC SYSTEM WITH COEXISTING ATTRACTORS: ITS CONTROL, SYNCHRONIZATION AND SECURE COMMUNICATION

International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.13, No.2/3/4, December 2024 DOI: 10.5121/ijccms.2024.13401 1 A NEW HYPERCHAOTIC SYSTEM WITH COEXISTING ATTRACTORS: ITS CONTROL, SYNCHRONIZATION AND SECURE COMMUNICATION Onma, O. S1,2,*., Adelaja, A. D3., Lasisi, A. M4., Idowu B. A5., Opeifa, S. T1., Okunlola, O. A2,6., Ogabi, C. O5. 1Department of Physics, Federal University of Agriculture Abeokuta, Nigeria. 2Department of Computer Science, Dominion University, Ibadan-Lagos Expressway, Nigeria. 3Department of Physics, Tai Solarin University of Education Ijagun, Ogun State, Nigeria 4Department of Physics, Ajayi Crowther University, Oyo, Oyo State, Nigeria 5Department of Physics, Faculty of Science, Lagos State University, Ojo Lagos, Nigeria 6Department of Computer Science, University of Ibadan, Nigeria ABSTRACT A new hyperchaotic system with coexisting attractors based on Sprott B chaotic system is proposed in this work. A novel feature of this new hyperchaotic system under investigation is that it has two-wing and fourwing coexisting attractors for two sets of different initial conditions. Thus, the new hyperchaotic system has hidden attractors. Interestingly, the proposed designed control function )(tui using adaptive control method was able to control and globally synchronizes two identical new hyperchaotic systems evolving from different initial conditions with uncertain parameters. The adaptive synchronization scheme was applied to secure communication. Finally, the numerical simulation results presented demonstrated the effectiveness of the analytical results of the designed scheme. KEYWORDS hyperchaotic system, coexisting attractors, adaptive control, uncertain parameters, synchronization, secure communication. 1. INTRODUCTION Recently, chaos theory has becomes a focal point of discussion among the expert and researcher due to its potential applications in: physics, chemical and biological sciences [1], finances [2-3], economic [4-6], telecommunication and secure communication [7-11], high performance electric circuit design [12-14]. Historically, the first hyperchaotic system popularly known as four-dimensional hyperchaotic Rossler system was reported in 1979 [15]. Hyperchaotic system is more prominent over the chaotic system, because chaotic system has only one positive Lyapunov exponent while hyperchaotic system has at least two positive Lyapunov exponents. This feature make it more complex and unpredictable than chaotic system, hence give room to wide range of potential applications compared to 3D chaotic system [16-17]. International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.13, No.2/3/4, December 2024 2 Numerous techniques have been developed and reported in the literature to achieve chaos control and synchronization. Some of these methods are: active control [18-20], adaptive control [21-27], backstepping technique [28-30], sliding mode control [31-32]. The main focus in chaotic or hyperchaotic synchronization is to design the effective control feedback function )(tui that will force the state variables of the response (slave) system to track the corresponding trajectories of the state variables of the drive (master) system asymptotically with time. In most practical applications, the unknown parameters in the drive or response state or both states at time usually destroyed the desired synchronization. Therefore, the convectional synchronization techniques are not effective in such situation [33]. Thus, the synchronization technique for chaotic or hyperchaotic systems with uncertain parameter is an interesting challenge that has attracted great attention in a recent time. As the results, the synchronization method for unknown parameter in chaotic systems remains a significant point among the researchers. The synchronization of chaotic system is motivated by its potential applications in secure communication, information security and privacy protection. To improve the security of the aforementioned applications, more complex chaotic dynamical behaviors are used. Consequently, coexisting attractors with more complex dynamical behaviors are more important compared to generated chaotic attractor. To improve the information security and reduce the probability of information being decoded, coexisting attractors are more reliable [34-35]. Coexistence of attractors also known as multistability refer to the systems that neither stable nor totally unstable but alternate between two or more mutually exclusive attractor with time [36]. Coexistence (multistability) is a unique property of a chaotic and hyperchaotic system indicated by the presence of two or more coexisting attractors for the same set of system parameter but different sets of initial conditions [37]. The most important application of chaos synchronization in engineering is in secure communication. The basic idea is to use a chaotic oscillator as a broadband signal generation. The chaotic signal is mask (encrypt) the information signal to produce unpredictable signal which is transmitted from the drive to the response (see refs. [30] and [8]), [38]. At the response, the pseudo-random is generated through the inverse operation and the original signal is retrieved. In this paper, a new hyperchaotic system with two-wing and four-wing attrctors that displayed multistability for two different sets of initial conditions is discussed. The next task is to design a control function )(tui to control as well as to synchronize the drive and response systems; design parameter update law to identify the unknown system parameters and to apply the synchronization scheme to secure communication. To the best of our knowledge, adaptive control and synchronization with application to secure communication for Sprott B-based hyperchaotic system is reported here for the first time. 2. NUMERICAL DESCRIPTION OF THE MODELS The mathematical formulation investigated in this paper is the modified Sprott B chaotic system constructed by adding a state-feedback controller on the Sprott B chaotic system and is given in equation (1). International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.13, No.2/3/4, December 2024 3 cwyzw xybz wxzy xyax −= −= += −=    )( (1) In equation (1), x , y , z and w are the state-variables of the system, where a , b and c are the real positive constant system parameters. System (1) displayed hyperchaotic behavior with the real positive constant parameters; 6=a , 11=b and 5=c via numerical simulation. The strange attractors of the Sprott B-based hyperchaotic system (1) are displayed in figure 1. Figure 1 (a, b and c) displayed 2-wing attractor and figure 1(d) shown 4-wing attractor at the same time. Figure 1: The two-wing and four-wing attractors for hyperchaotic system (1) 3. COEXISTENCE OF ATTRACTORS System (1) is invariant under the transformation ),,,(: wzyxS  ),,,( wzyx −−− . Hence, any projection of the attractor has rotational symmetry in the z-axis. Thus, system (1) may likely display coexisting attractors. It is cleared from figure 2 that system (1) exhibited coexisting attractors with respect to two sets of different initial conditions; )8.0,6.0,6.0,5.0(),,,( =wzyx plotted in blue color and )0.9,2.0,0.1,0.9(),,,( −=wzyx plotted in red color through numerical simulation. Thus, system (1) has hidden attractors. International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.13, No.2/3/4, December 2024 4 Figure 2: Two-wing and four-wing coexistence of attractors of the hyperchaotic system (1) with two sets of initial conditions. 4. ADAPTIVE CONTROL FOR THE NEW HYPERCHAOTIC SYSTEM In this section, we applied the adaptive control method to designed the control function )(tui that converge the state variables ( wzyx ,,, ) asymptotically to the origin with at time according to Lyapunov stability theory [39]. 4.1. Design of Adaptive control input )(tui for system (1) The assumption here is that the positive real parameters of the system; a , b and c are uncertain. Therefore, adaptive control technique is used to design the control input )(tui as well as the parameter update law to identify the unknown system parameters. Then, the controlled system is considered as follows: 4 3 2 1 )( ucwyzw uxybz uwxzy uxyax +−= +−= ++= +−=     (2) Where )(tui ( 4,3,2,1=i ) are the control functions to design appropriately. The Lyapunov stability theory (ref. [39]) is used to validate the result of system (2) by selecting a Lyapunov function as: International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.13, No.2/3/4, December 2024 5 ( ) 2222222 ~ ~ ~ 2 1cbawzyxV ++++++= (3) Where aaa −= ~ , bbb −= ~ and ccc −= ~ are the estimated values of the assumed unknown parameters a , b and c respectively. The time derivative of equation (3) above is given in equation (4) below. ccbbaawwzzyyxxV       ~~ ~~ ~~ ++++++= (4) In order to ensure that the control function )(tui in equation (2) converge the state variables of system (1) to the origin asymptotically, the control input )(tui is selected from equation (2) as follows: ( ) wcwyzu zxybu ywxzu xxyau −+−= −+−= −−−= −−−= 4 3 2 1 (5) The Substitution of equation (2) into equation (4) yielded equation (6).         ][ ~ ][ ~ ][ ~ )( 22 4321 wcczbbxyxaa ucwyzwuxybzuwxzyuxyaxV −−++−++−− ++−++−+++++−=     (6) The parameter update laws are estimated from equation (6) and presented in equation (7). 2 2 wc zb xyxa −= = +−=    (7) Substituting equations (5) and (7) respectively into equation (4) give: 0 ~ ~ ~2222222 −−−−−−−= cbawzyxV  (8) Hence, V is a quadratic positive definite Lyapunov function (see equation (3)) and its time derivative ( ) V  is a quadratic negative definite as reflected in equation (8). According to the Lyapunov stability theory, system (2) can converge to the origin asymptotically with the control input )(tui ( 4,3,2,1=i ) as defined in equation (5) and the parameter estimated update laws in equation (7). International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.13, No.2/3/4, December 2024 6 4.2. Numerical Simulation Results To studies the time response of the new Sprott B-based hyperchaotic system with coexisting attractors as described in system (1), classical fourth-order RungeKuta routine with time step 001.0=h is adopted in the numerical simulation. Fixing the parameters value     0.5,0.11,0.6,, =cba in that order and the initial conditions )1.0,6.0,5.0,0.0(),,,( =wzyx , the state variables move hyperchaotically with the control function )(tui deactivated and converges asymptotically to the origin when the control function )(tui is activated at 50=t according to the Lyapunov stability theory. Figure 3 show the results for the time responses of the state variables ),,,( wzyx of the new hyperchaotic system (1). Figure 3: Time responses of the state variables ( wzyx ,,, ) for new hyperchaotic system (1) via adaptive control. 5. ADAPTIVE SYNCHRONIZATION FOR THE NEW HYPERCHAOTIC SYSTEM Here, we employed adaptive control techniques base on Lyapunov stability theory (ref. [39]) to achieved complete synchronization of two identical hyperchaotic systems. International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.13, No.2/3/4, December 2024 7 5.1. Design of Adaptive control input )(tui for system (1) In this section, the adaptive control method is used to synchronize two identical hyperchaotic systems emanating from different initial condition. From equation (1), let; 1 xx = , 2 xy = , 3 xz = and 4 xw = . Then, 4324 213 4312 121 )( cxxxx xxbx xxxx xxax −= −= += −=     (9) The equation (9) above is called the master or drive system while equation (10) below is designated as slave or response system. 44324 3213 24312 1121 )( ucyyyy uyyby uyyyy uyyay +−= +−= ++= +−=     (10) Where )(tui ( 4,3,2,1=i ) are the control input to determines. The synchronization error vector between the master (9) and the slave (10) is defined by: 444 333 222 111 xye xye xye xye −= −= −= −= (11) Hence, using the definition of the error vector in equation (11), the error dynamic is calculated as follows: 443223324 3212213 243113312 1121 )( )( uceeeexexe ueeexexe ueeeexexe ueeae +−++= +++−= ++++= +−=     (12) Choosing the Lyapunov function; ( ) 2222 4 2 3 2 2 2 1~ ~ ~ 2 1cbaeeeeV ++++++= and differentiating it with respect to time result in equation (13). International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.13, No.2/3/4, December 2024 8 ccbbaaeeeeeeeeV     ~ ~ ~ 44332211 ++++++= (13) Where aaa −= ~ , bbb −= ~ , and ccc −= ~ are the estimated values of the unknown parameter a , b and c respectively. Then, equation (14) is obtained by substituted equation (12) into equation (13).             )( ~~ ~~ )( ~~ )( )( 2 4121 44322332432112213 2431133121121 eccbbeeeaa uceeeexexeueeexexe ueeeexexeueeaeV −−+       +−− ++−++++++− +++++++−=     (14) From equation (12), the control input )(tui is chosen as: 432233244 32112213 243113312 1121 )( )( )( eeeexexceu eeeexexu eeeeexexu eeeau −++−= −++= −+++−= −−−= (15) And the estimated parameter update law is chosen from equation (14) as follows: cec bb aeeea −−= −= −−= 2 4 121 )(   (16) Substituting equations (15) and (16) respectively into equation (14) gives equation (17). 0 2222 4 2 3 2 2 2 1−−−−−−−= cbaeeeeV  (17) The Lyapunov function V is positively definite with it derivative V  is negatively definite as confirmed by equation (17) above. Hence, the error dynamic variable in equation (12) can converge to the origin asymptotically in line with the Lyapunov stability theory (ref. [39]) and one can conclude that system (12) is globally and exponentially stable. Also the master (drive) and the slave (response) systems (equations (9) and (10)) are globally and exponentially synchronized for all the initial conditions )0( i x and )0( i y , and the estimated update law (16). 5.2. Numerical Simulation Results The main objective of adaptive synchronization is to design an approximate control function )(tui to force the state variables of the response (slave) system to track the trajectories of the drive (master) state variables such that both systems will remain in step throughout the transmission of signal with the parameter update law as well as to stabilize the error function )(tei between the drive (master) and the response (master) systems at the origin )0,0,0,0( at any chosen time. International Journal of Chaos, Control, Modelling and Simulation (IJCCMS) Vol.13, No.2/3/4, December 2024 9 To achieve the stated objective, fourth-order RungeKuta algorithm is used to solve the control law (15) and the estimated parameter update law (16) by fixing the parameter values of the system (1) ]0.5,0.11,0.6[],,[ =cba with the time step 001.0=h . The initial conditions for the drive (master) )( i x and the response (slave) )( i y systems are respectively choosing as )8.0,6.0,6.0,5.0(),,,( 4321 =xxxx and )0.9,2.0,0.1,0.9(),,,( 4321 −=yyyy . The reports of the numerical simulation are: the state variables of the response (slave) system track the dynamics of the drive (master) system when the control function )(tui is activated at 50=t as shown in figure 4; the error function )(tei converges asymptotically to the origin in line with the Lyapunov stability theory, when the controllers are switched on at 50=t as depicted in figure 5 and the synchronization norm 2 4 2 3 2 2 2 1eeeee +++= is displayed in figure 6. For parameter updating, the initial values of the parameter update law (16) are selected as 0.6)0( 1=a , 0.11)0( 1=b and 0.5)0( 1=c . The parameters estimated value a , b and c updated to 0.10=a , 0.13=b and 0.8=c respectively as shown in figure 7 as →t . Figure 4: Time responses for the state variables; drive (master) ),,,( 4321 xxxx and response (slave) ),,,( 432 yyyyi systems for new hyperchaotic system (1) with the controller activated.