In this paper, we have carried out Projective Synchronization(PS) using Backstepping Design and Function Projective Synchronization(FPS) by Adaptive Control Scheme between two identical chaotic systems evolving from different initial conditions. The first method is applied when the two systems are certain in nature whereas when the systems contain uncertain time-varying parameters, second method is found to be successful. Further, comparing the two synchronization phenomena, FPS is a more general form of PS. These methods are applied to a model of chaotic dumbell satellite which is under periodic external disturbances and numerical simulations are carried out which confirm our analytical findings.
Fractional order tumor-immune interaction models are being frequently used for understanding the complex behaviour of immune system and tumor growth. In this paper, a generalized fractional order tumor-immune interaction model has been developed by introducing immunotherapy (IL2) as third variable in the model. The study of generalized model is done by using conformable fractional order derivative. The stability analysis is done for both fractional order tumor model and its conformable fractional order version. By considering some biological fixed points for both versions of the model, the stability analysis around these fixed points shows that both the systems are stable at some fixed point under some stability conditions, which are defined in the model analysis. The numerical and graphical analysis is also done for both the systems by varying two parameters and keeping other parameters fixed for better understanding the dynamics of proposed model.
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