Analysis of a discrete time fractional-order Vallis system

Vallis system is a model describing nonlinear interactions of the atmosphere and temperature fluctuations with a strong influence in the equatorial part of the Pacific Ocean. As the model approaches the fractional order from the integer order, numerical simulations for different situations arise. To see the behavior of the simulations, several cases involving integer analysis with different non-integer values of the Vallis systems were applied. In this work, a fractional mathematical model is constructed using the Caputo derivative. The local asymptotic stability of the equilibrium points of the fractional-order model is obtained from the fundamental production number. The chaotic behavior of this system is studied using the Caputo derivative and Lyapunov stability theory. Hopf bifurcation is used to vary the oscillation of the system in steady and unsteady states. In order to perform these numerical simulations, we apply Grünwald–Letnikov tactics with Binomial coefficients to obtain the effects on the non-integer fractional degree and discrete time vallis system and plot the phase diagrams and phase portraits with the help of MATLAB and MAPLE packages.