In this paper, a novel synchronization scheme is investigated for a class of chaotic systems. Themultiswitching synchronization scheme is extended to the combination–combination synchronization scheme such that the combination of state variables of two drive systems synchronize with different combination of state variables of two response systems, simultaneously. The new scheme, multiswitching combination–combination synchronization (MSCCS), is a notable extension of the earlier multiswitching schemes concerning only the single drive–response system model. Various multiswitching modified projective synchronization schemes are obtained as special cases of MSCCS, for a suitable choice of scaling factors. Suitable controllers have been designed and using Lyapunov stability theory sufficient condition is obtained to achieve MSCCS between four hyperchaotic systems and the corresponding theoretical proof is given. Numerical simulations are performed to validate the theoretical results.

In this paper, the methodology to achieve combination synchronization of time-delay chaotic system via robust adaptive sliding mode control is introduced. The methodology is implemented by taking identical time-delayLorenz chaotic system. The selection of switching surface and the design of control law is also discussed, which is an important issue. By utilizing rigorous mathematical theory, sufficient condition is drawn for the stability of error dynamics based on Lyapunov stability theory. Theoretical results are supported with the numerical simulations. The complexity of this methodology is useful to strengthen the security of communication. The hidden message can be partitioned into several parts loaded in two master systems to improve the accuracy of communication.

Based on three drive–one response system, in this article, the authors investigate a novel synchronization scheme for a class of chaotic systems. The new scheme, multiswitching compound antisynchronization (MSCoAS), is a notable extension of the earlier multiswitching schemes concerning only one drive–one response system model. The concept of multiswitching synchronization is extended to compound synchronization scheme such that the statevariables of three drive systems antisynchronize with different state variables of the response system, simultaneously. The study involving multiswitching of three drive systems and one response system is first of its kind. Various switched modified function projective antisynchronization schemes are obtained as special cases of MSCoAS, for a suitable choice of scaling factors. Using suitable controllers and Lyapunov stability theory, sufficient condition is obtained to achieve MSCoAS between four chaotic systems and the corresponding theoretical proof is given.Numerical simulations are performed using Lorenz system in MATLAB to demonstrate the validity of the presented method.

In this paper,we study the qualitative behaviour of satellite systems using bifurcation diagrams, Poincaré section, Lyapunov exponents, dissipation, equilibrium points, Kaplan–Yorke dimension etc. Bifurcation diagrams with respect to the known parameters of satellite systems are analysed. Poincaré sections with different sowing axes of the satellite are drawn. Eigenvalues of Jacobian matrices for the satellite system at different equilibrium points are calculated to justify the unstable regions. Lyapunov exponents are estimated. From these studies, chaosin satellite system has been established. Solution of equations of motion of the satellite system are drawn in the form of three-dimensional, two-dimensional and time series phase portraits. Phase portraits and time series display the chaotic nature of the considered system.

In this paper, we have studied the hybrid projective synchronisation for incommensurate, integer and commensurate fractional-order financial systems with unknown disturbance. To tackle the problem of unknown bounded disturbance, fractional-order disturbance observer is designed to approximate the unknown disturbance. Further, we have introduced simple sliding mode surface and designed adaptive sliding mode controllers incorporating with the designed fractional-order disturbance observer to achieve a bounded hybrid projective synchronisation between two identical fractional-order financial model with different initial conditions. It is shown that the slave system with disturbance can be synchronised with the projection of the master system generated through state transformation. Simulation results are presented to ensure the validity and effectiveness of the proposed sliding mode control scheme in the presence of external bounded unknown disturbance. Also, synchronisation error for commensurate, integer and incommensurate fractional-order financial systems is studied in numerical simulation.

In this paper, multiswitching combination synchronisation (MSCS) scheme has been investigated in a class of three non-identical fractional-order chaotic systems. The fractional-order Lorenz and Chen systems are taken as the drive systems. The combination of multidrive systems is then synchronised with the fractional-orderLü chaotic system. In MSCS, the state variables of the two drive systems synchronise with different state variables of the response system, simultaneously. Based on the stability of fractional-order chaotic systems, the MSCS of three fractional-order non-identical systems has been investigated. For the synchronisation of three non-identical fractional-order chaotic systems, suitable controllers have been designed. Theoretical analysis and numerical results are presented to demonstrate the validity and feasibility of the applied method.

In this article, an adaptive control method is proposed to study the combination–combination synchronisation phenomenon of four non-identical time-delayed chaotic systems for fully unknown parameters with parametric uncertainties and external disturbances. Based on the Lyapunov–Krasovskii functional theory, an appropriate adaptive controller is constructed so that a globally and asymptotically stable synchronisation state can be established between the master and the slave systems. Unknown parameters are identified by designing suitable parameter update laws. To elaborate the presented scheme, double-delay Rossler and time-delay Chen systems are considered as the master systems and time-delay Shimizu–Morioka and time-delay modified Lorenz systems are considered as the slave systems. Numerical simulations are presented to justify the theoretical analysis.

In this paper, we analyse the chaotic satellite system using dissipativity, equilibrium points, bifurcation diagrams, Poincare section maps, Lyapunov exponents and Kaplan–Yorke dimension. We obtain the equilibrium points of chaotic satellite system and at each equilibrium point, we obtain the eigenvalue of Jacobian matrix of the satellite system to verify the unstable region.We calculate the Kaplan–Yorke dimension, which ensures the strange behaviour of the system. We observe closely the three-dimensional (3D) phase portraits of the satellite system at different parameter values. We plot the Lyapunov exponent graphs corresponding to every 3D phase portrait of satellite systems, to verify the chaoticity of satellite systems. We establish time-delay synchronisation for twoidentical satellite systems. The simulated and qualitative results are in an excellent agreement.

In this paper, a novel synchronisation scheme involving six chaotic systems is proposed. The proposed scheme is named as ‘compound–compound synchronisation’. Instead of using a scaling system in compound synchronisation, a compound scaling signal is applied to the proposed scheme. The phenomenon of multiswitching synchronisation and the proposed scheme are combined together.Appropriate controllers are designed by employing nonlinear control method and Lyapunov stability theory to achieve asymptotically stable synchronisation states. An example of identical Chen systems is presented to demonstrate the proposed methodology. The proposed scheme is very different and complex in comparison with the previous schemes, as this scheme is first of its kind having five drive systems and one response system. Computational results are presented to justify the theoretical analysis. Numerical results and theoretical studies converge to the same conclusions.