The Caputo fractional-order differential operator is used in epidemiological models, but its accuracy benefits are typically ignored. We validated the suggested fractional epidemiological seasonal influenza model of the SVEIHR type to demonstrate the Caputo operator’s relevance. We analysed the model using fractional calculus, revealing its basic properties and enhancing our understanding of disease progression. Furthermore, the positivity, bounds, and symmetry of the numerical scheme were examined. Adjusting the Caputo fractional-order parameter $\mathsf{\alpha }$ = 0.99 provided the best fit for epidemiological data on infection rates. We compared the suggested model with the Caputo fractional-order system and the integer-order equivalent model. The fractional-order model had lower absolute mean errors, suggesting that it could better represent sickness transmission and development. The results underline the relevance of using the Caputo fractional-order operator to improve epidemiological models’ precision and forecasting. Integrating fractional calculus within the framework of symmetry helps us build more reliable models that improve public health interventions and policies.

This article investigates the chaotic features of a novel variable-order fractional Rössler system built with Liouville–Caputo derivatives of variable order. Variable-order fractional (VOF) operators incorporated in the system render its dynamics more flexible and richer with memory and hereditary effects. We run numerical simulations to see how different fractional-order functions alter the qualitative behavior of the system. We demonstrate this via phase portraits and time-series responses. The research analyzes bifurcation development, chaotic oscillations, and stability transition and demonstrates dynamic patterns impossible to describe with integer-order models. Lyapunov exponent analysis also demonstrates system sensitivity to initial conditions and small disturbances. The outcomes confirm that the variable-order procedure provides a faithful representation of nonlinear and intricate processes of engineering and physical sciences, pointing out the dominant role of memory effects on the transitions among periodic, quasi-periodic, and chaotic regimes.

In this study, we analyze the solution characteristics and dynamics of a variable-order fractional (V-OF) Arneodo system using the Liouville–Caputo fractional operator with variable order. The V-OF operator is used to describe the time-dependent memory effect in the system, which leads to more complex and diverse dynamics compared to integer-order systems. In this work, numerical simulations are performed to observe the effect of the order functions on the dynamic behaviors of the system. In addition, the phase portraits, time series graphs, and three-dimensional diagrams are used to analyze the dynamic behaviors and different types of oscillations present in the system. Furthermore, the bifurcations, chaotic behaviors, and stability of the system with variable orders are studied, and it is found that the system has more complex dynamics compared to the integer-order case. In this case, the Lyapunov exponents indicate that the system under investigation is sensitive to the initial conditions, and the memory effect can control the chaotic oscillation depending on the order of the functions.

In this study, an analysis of fractional-order Lü systems is performed through a framework approach consisting of analytical solution strategies in combination with numerical methods. On the analytical methodology front, the recently developed form of the new generalized differential transform method (NGDTM) is adopted for its efficiency in providing an approximate solution with high capability in tracking the behavior of these systems. On the other hand, the Grünwald–Letnikov via Riemann–Liouville scheme (GLNS) is adopted within this study as one of its tools in confirming whether chaos exists within these systems. The performance and accuracy of the proposed method are also rigorously tested, and comparisons are made numerically with the Adams–Bashforth–Moulton method, which is used here as a standard method for validation purposes. It is clear from the results that the combination of analytical and numerical methods can greatly enhance both the speed of computation and the accuracy of results. Additionally, the proposed method or approach is found to be quite robust and accurate and can thus be employed for analyzing various fractional dynamical systems that display chaotic attractors. The proposed method can also be expanded upon in the future for analyzing complex models in science and engineering.