This paper proposes a numerical technique to study dynamical systems and uncover new behaviors in chaotic fractional-order models, a field that continues to attract significant research interest due to its broad applicability and the ongoing development of innovative methods. Through various types of simulations, this approach is able to uncover novel dynamic behaviors that were previously undiscovered. The results guarantee that initial conditions and fractional-order derivatives have a significant contribution to system dynamics, thus distinguishing fractional systems from traditional integer-order models. The approach demonstrated has excellent consistency with traditional approaches for integer-order systems while offering higher accuracy for fractional orders. Consequently, this approach serves as a powerful and efficient tool for studying complex chaotic models. Fractional-order dynamical systems (FDSs) are particularly noteworthy for their ability to model memory and hereditary characteristics. The method identifies new complex phenomena, including new chaos, unusual attractors, and complex time-series patterns, not documented in the existing literature. We use Lyapunov exponents, bifurcation analysis, and Poincaré sections to thoroughly investigate the system dynamics, with particular emphasis on the effect of fractional-order and initial conditions. Compared to traditional integer-order approaches, our approach is more accurate and gives a more efficient device for facilitating research on fractional-order chaos.

This study investigates the complex dynamics of a predator–prey system governed by the classical Lotka–Volterra model incorporating a Holling‐type III. To capture environmental variability, the prey’s carrying capacity is modeled as a periodic function, introducing a time‐dependent forcing into the system. The resulting nonautonomous system is transformed into an equivalent autonomous system by coupling it with an auxiliary oscillator. Through numerical continuation techniques and bifurcation analysis using MATCONT, we explore the emergence of rich dynamical behaviors, including Hopf and limit point cycle bifurcations, Neimark–Sacker and period‐doubling bifurcations, and two different routes to chaos. The results reveal that fluctuations in the carrying capacity significantly influence the system’s long‐term behavior, leading to differently‐periodic and quasiperiodic solutions, bistability, and chaos via two routes: torus destruction and cascade of period‐doublings. These findings provide new insights into the ecological implications of environmental forcing in predator–prey interactions.

This paper presents a new fractional‐order model that captures the propagation dynamics of computer viruses within digital networks by incorporating memory effects and nonlocal interactions via the Caputo derivative. The network is partitioned into four functional compartments: susceptible, latently infected, breaking‐out (actively infectious), and recovered computers. The analytical framework establishes the fundamental mathematical properties of the model—existence, uniqueness, nonnegativity, and boundedness of solutions—ensuring its well‐posedness. The stability of the infection‐free and endemic equilibria is investigated using Matignon’s theorem for fractional‐order systems, Lyapunov functionals, and the fractional LaSalle invariance principle. The basic reproduction number is derived via the next‐generation matrix approach to determine threshold conditions governing the transition between virus eradication and persistence. To approximate the fractional dynamics, two robust numerical algorithms are implemented and compared: the multistep generalized differential transform method (MSGDTM) and the Jacobi spectral collocation scheme (JSCS). The model is extended to a stochastic framework to account for random fluctuations inherent in real‐world network environments, with both deterministic and stochastic solvers evaluated through Monte Carlo simulations. The stochastic analysis reveals key phenomena including noise‐induced transitions, enhanced extinction probability, and memory–noise interplay that are not captured by the deterministic model. Numerical experiments reveal that the fractional order strongly influences infection intensity and system memory, with smaller orders yielding delayed peaks and prolonged transients. While MSGDTM provides computational simplicity and flexibility for nonlinear problems, JSCS attains superior spectral accuracy and convergence efficiency. The comparative results highlight the role of fractional modeling in representing long‐memory effects in cyber‐epidemics and offer quantitative guidance for designing antivirus strategies in memory‐dependent network environments.

This paper develops a fractional-order mathematical framework to investigate the co-infection dynamics of Pneumonia and Typhoid fever using the Caputo–Fabrizio (CF) fractional derivative. The CF operator, characterized by a non-singular exponential kernel, captures the fading memory and hereditary effects inherent in biological processes, providing a more realistic representation of disease transmission. The total human population is divided into eight interacting compartments—susceptible, singly infected (with either Pneumonia or Typhoid), co-infected, recovered groups, and the environmental bacterial concentration. A system of CF-based fractional differential equations is formulated to describe the temporal evolution of each compartment. Theoretical analysis establishes the positivity, boundedness, and Hyers–Ulam stability of the system. Existence and uniqueness of solutions are rigorously proved using Lipschitz continuity and fixed-point theorems in Banach spaces. A modified fractional Euler method tailored for the CF operator is employed to obtain numerical solutions and assess convergence behavior. Numerical simulations under various fractional orders demonstrate that decreasing the fractional order enhances memory effects, leading to slower epidemic progression, attenuated infection peaks, and delayed recovery. The results confirm that memory-driven fractional dynamics significantly influence co-infection behavior and bacterial persistence. The proposed CF–Euler framework improves the biological realism of epidemic modeling and provides a foundation for developing memory-based control strategies in managing co-infectious diseases.

The regulation of blood glucose and insulin in the human body is a highly nonlinear physiological process influenced by hereditary memory, feedback mechanisms, and intrinsic delays. Classical integer-order models often fail to capture these memory-dependent and nonlocal effects. In this study, we introduce a novel fractional-order glucose–insulin regulatory model formulated with the Caputo–Fabrizio (CF) derivative, which employs a nonsingular exponential kernel to describe fading memory effects more realistically. The model accounts for essential physiological parameters and incorporates the interplay between blood glucose concentration, insulin secretion, and an auxiliary control variable representing insulin action or resistance. We establish the well-posedness of the proposed CF-based system by proving existence and uniqueness of solutions via fixed-point theory under Lipschitz continuity and boundedness conditions on the nonlinear operators. To approximate the solution, we design a two-step Adams–Bashforth scheme tailored to the CF operator and provide a rigorous analysis of its convergence and stability. Numerical simulations confirm the accuracy and robustness of the method, demonstrating its ability to reproduce physiologically consistent glucose–insulin trajectories that outperform classical integer-order models. The results highlight the effectiveness of Caputo–Fabrizio fractional calculus in capturing hereditary and fading-memory effects in glucose–insulin dynamics, offering a robust mathematical framework for parameter estimation, optimal control, and personalized strategies in diabetes management.