Search Results (311)

Macroeconomic mathematical models are practical instruments structured to carry out theoretical analyses of macroeconomic developments. In this manuscript, the Caputo-like fractional operator of variable order is used to introduce and investigate the mechanism of the discrete macroeconomic model. The nature of the dynamics was established, and the emergence of chaos using a distinct variable fractional order, especially the stability of the trivial solution, is examined. The findings reveal that the variable-order discrete macroeconomic model displays chaotic dynamics employing bifurcation, the Largest Lyapunov exponent ($L{E}_{max}$), the phase portraits, and the 0–1 test. Furthermore, a complexity analysis is performed to demonstrate the existence of chaos and quantify its complexity using $C_0$ complexity and spectral entropy ($SE$). These studies show that the suggested variable-order fractional discrete macroeconomic model has more complex features than integer and constant fractional orders. Finally, MATLAB R2024b simulations are run to exemplify the outcomes of this study. Full article

The study of economic maps has consistently attracted researchers due to their rich dynamics and practical relevance. A deeper understanding of these systems enables the development of more effective control strategies. In this work, we examine the influence of the fractional order $\upsilon$ with the Caputo fractional difference on an economic market map. The primary contribution is the comprehensive analysis of how both commensurate and incommensurate fractional orders affect the stability and complexity of the map. Numerical investigations, including phase portraits, largest Lyapunov exponents, and bifurcation analysis, reveal that the system undergoes a cascade of period-doubling bifurcations before transitioning into chaos. To further characterize the dynamics, complexity is evaluated using the 0–1 test and $C_0$ complexity, both confirming chaotic behavior. Furthermore, two-dimensional control schemes are introduced and theoretically validated to both stabilize the chaotic economic market map and achieve synchronization with a combined response map. The theoretical and numerical results are validated through MATLAB 2016 simulations. Full article

This paper introduces an enhanced five-dimensional Chaotic Supply Chain Model (5DCSCM) by incorporating a transport lag variable into a previously established four-dimensional model. The newly added differential equation in the transit dynamics of the supply chain model captures the inherent lag between customer demand and the physical response in transportation, modeled as a first-order transport lag system. Through comprehensive numerical simulations, the influence of various system parameters—including customer demand rate, delivery efficiency, information distortion, contingency reserve, safety stock, and transportation lag—are examined. The study utilizes bifurcation diagrams and a Lyapunov Exponent (LE) to investigate tran-sitions between periodic and chaotic behavior. Additionally, the model is extended with offset boosting control, allowing for controlled amplitude adjustment without altering the underlying chaotic dynamics. Offset boosting control (OBC) is useful in chaotic supply chain systems because it stabilizes inventory and order fluctuations by counter-acting the amplification of small disturbances, reducing the bullwhip effect, and im-proving overall system reliability and responsiveness. As an application, integral sliding mode control (ISMC) technique has been applied to achieve complete synchronization between a pair of the 5DCSCM. Synchronization based on ISMC is useful in chaotic supply chain systems because it ensures robust coordination between different tiers, suppresses chaos-induced fluctuations, and maintains stable inventory and order patterns even under disturbances and uncertainties. Full article

The geometric content of chaos in nonlinear systems with multiple stabilities of high order is a challenge to computation. We introduce a single algorithmic framework to overcome this difficulty in the present study, where a parametrically forced oscillator with cubic–quintic nonlinearities is considered as an example. The framework starts with the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm, which is a self-learned algorithm that extracts an interpretable and correct model by simply analyzing time-series data. The resulting parsimonious model is well-validated, and besides being highly predictive, it also offers a solid base on which one can conduct further investigations. Based on this tested paradigm, we propose a unified diagnostic pathway that includes bifurcation analysis, computation of the Lyapunov exponent, power spectral analysis, and recurrence mapping to formally describe the dynamical features of the system. The main characteristic of the framework is an effective algorithm of computational basin analysis, which is able to display attractor basins and expose the fine scale riddled structures and fractal structures that are the indicators of extreme sensitivity to initial conditions. The primary contribution of this work is a comprehensive dynamical analysis of the DM-CQDO, revealing the intricate structure of its stability landscape and multi-stability. This integrated workflow identifies the period-doubling cascade as the primary route to chaos and quantifies the stabilizing effects of key system parameters. This study demonstrates a systematic methodology for applying a combination of data-driven discovery and classical analysis to investigate the complex dynamics of parametrically forced, high-order nonlinear systems. Full article

Computer viruses remain a persistent technological challenge in information security. They require mathematical frameworks that realistically capture their propagation in digital networks. Classical continuous-time, integer-order models often overlook two key aspects of cyber environments: their inherently discrete nature and the memory-dependent effects of networked interactions. In this work, we introduce a fractional-order discrete computer virus (FDCV) model, derived from a three-dimensional continuous integer-order formulation and reformulated into a two-dimensional fractional discrete framework. We analyze its rich dynamical behaviors under both commensurate and incommensurate fractional orders. Leveraging a comprehensive toolbox including bifurcation diagrams, Lyapunov spectra, phase portraits, the 0–1 test for chaos, spectral entropy, and $C_0$ complexity measures, we demonstrate that the FDCV system exhibits persistent chaos and high dynamical complexity across broad parameter regimes. Our findings reveal that fractional-order discrete models not only enhance the dynamical richness compared to integer-order counterparts but also provide a more realistic representation of malware propagation. These insights advance the theoretical study of fractional discrete systems, supporting the development of potential technologies for cybersecurity modeling, detection, and prevention strategies. Full article

Computer viruses continue to threaten the security of digital networks, and their complex propagation dynamics require refined modelling tools. Most existing models rely on integer-order dynamics or assume uniform memory effects, which limit their ability to capture heterogeneous behaviours observed in practice. To address this gap, we propose a discrete incommensurate fractional-order virus model based on Caputo-like delta differences, where each compartment is assigned a distinct fractional order to represent mismatched time scales. The model’s dynamics are analysed in terms of stability, bifurcation, and chaos. Numerical results reveal the emergence of rich chaotic attractors, emphasizing the impact of fractional memory on system evolution. To quantify complexity, we employ Approximate Entropy and Spectral Entropy and relate these indicators to the maximum Lyapunov exponent, confirming the system’s sensitivity and unpredictability. All numerical simulations and visualizations were performed using MATLAB (R2015a). The findings highlight the importance of heterogeneous memory in computer-virus modeling and offer new insights for developing theoretical foundations of robust cybersecurity strategies. Full article

Chaos-based wireless communication systems can enhance the physical-layer security of IoT devices, but their reliability depends on stable chaotic behavior under real conditions. We investigate a modified Colpitts oscillator with a tunable base bias voltage, introduced as an independent control parameter to flexibly adjust nonlinear regimes. Using numerical studies, SPICE simulations, and hardware experiments, we show that simplified numerical models predict only a DC offset shift, whereas realistic implementations reveal qualitative changes in the dynamics, highlighting the need for experimental validation. We further demonstrate hybrid synchronization between the analog oscillator and an FPGA-based digital model. Despite model simplifications and non-idealities, synchronization is successfully achieved using the Pecora–Carroll method, showing that preserving the core dynamic structure is more critical than exact waveform replication. These results clarify the constraints of idealized models for predicting dynamical patterns while confirming the robustness of hybrid synchronization for secure, resource-constrained communication systems. Full article

This study presents a qualitative analysis of the fractional coupled nonlinear Schrödinger equation (FCNSE) to obtain its complete set of solutions. An appropriate wave transformation is applied to reduce the FCNSE to a fourth-order dynamical system. Due to its non-Hamiltonian nature, this system poses significant analytical challenges. To overcome this complexity, the dynamical behavior is examined within a specific phase–space subspace, where the system simplifies to a two-dimensional, single-degree-of-freedom Hamiltonian system. The qualitative theory of planar dynamical systems is then employed to characterize the corresponding phase portraits. Bifurcation analysis identifies the physical parameter conditions that give rise to super-periodic, periodic, and solitary wave solutions. These solutions are derived analytically and illustrated graphically to highlight the influence of the fractional derivative order on their spatial and temporal evolution. Furthermore, when an external generalized periodic force is introduced, the model exhibits quasi-periodic behavior followed by chaotic dynamics. Both configurations are depicted through 3D and 2D phase portraits in addition to the time-series graphs. The presence of chaos is quantitatively verified by calculating the Lyapunov exponents. Numerical simulations demonstrate that the system’s behavior is highly sensitive to variations in the frequency and amplitude of the external force. Full article

This paper investigates the nonlinear dynamics behavior and practical realization of a V/Heart-shape chaotic system. Nonlinear analysis contemporary tools, including bifurcation diagram, Lyapunov exponents, phase portraits, power spectral density (PSD) bicoherence, and spectral entropy (SE), are employed to investigate the system’s complex dynamical behaviors. To discover the system’s versatility, two case studies are presented by varying key system parameters, revealing various strange attractors. The system is modeled and implemented using an industrial-grade programmable logic controller (PLC) with structured text (ST) language, enabling robust hardware execution. The dynamics of the chaotic system are simulated, and the results are rigorously compared with experimental data from laboratory hardware implementations, demonstrating excellent agreement. The results indicate the potential usage of the proposed chaotic system for advanced industrial applications, secure communication, and dynamic system analysis. The findings confirm the successful realization of the V-shape and Heart-shape Chaotic Systems on PLC hardware, demonstrating consistent chaotic behavior across varying parameters. This practical implementation bridges the gap between theoretical chaos research and real-world industrial applications. Full article

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. Full article

This study conducts an in-depth analysis of the dynamical characteristics and solitary wave solutions of the integrable Zhanbota-IIA equation through the lens of planar dynamic system theory. This research applies Lie symmetry to convert nonlinear partial differential equations into ordinary differential equations, enabling the investigation of bifurcation, phase portraits, and dynamic behaviors within the framework of chaos theory. A variety of analytical instruments, such as chaotic attractors, return maps, recurrence plots, Lyapunov exponents, Poincaré maps, three-dimensional phase portraits, time analysis, and two-dimensional phase portraits, are utilized to scrutinize both perturbed and unperturbed systems. Furthermore, the study examines the power frequency response and the system’s sensitivity to temporal delays. A novel classification framework, predicated on Lyapunov exponents, systematically categorizes the system’s behavior across a spectrum of parameters and initial conditions, thereby elucidating aspects of multistability and sensitivity. The perturbed system exhibits chaotic and quasi-periodic dynamics. The research employs the maximum Lyapunov exponent portrait as a tool for assessing system stability and derives solitary wave solutions accompanied by illustrative visualization diagrams. The methodology presented herein possesses significant implications for applications in optical fibers and various other engineering disciplines. Full article

This paper investigates the nonlinear dynamics of the wheel-side planetary reducer, considering the tooth wear effect. The tooth wear model based on the Archard adhesion wear theory is established, and the impact of tooth wear on meshing stiffness and piecewise-linear backlash of the planetary gear system is discussed. Then, the torsional vibration model and dimensionless differential equations considering tooth wear for the wheel-side planetary reducer are established, in which meshing excitations include time-varying mesh stiffness (TVMS), piecewise-linear backlash, and transmission error. The dynamic responses are numerically solved using the fourth-order Runge–Kutta method. On this basis, the nonlinear dynamics, such as the bifurcation and chaos properties of the wheel-side planetary reducer with tooth wear, are analyzed. Simulation results demonstrate that the existence of tooth wear reduces meshing stiffness and increases backlash. The reduction in the meshing stiffness changes the bifurcation path and chaotic amplitude of the system, inducing chaotic phenomena more easily. The increase in the gear backlash causes a higher amplitude of the relative displacement and more severe vibration. Full article

This study investigates the dynamic behavior of a discrete epidemic model as affected by media coverage through integrated analytical and numerical methods. The main objective is to quantitatively assess the impact of media coverage on disease outbreak models through mathematical modeling. We use the central manifold theorem and bifurcation theory to perform a rigorous analysis of the periodic solutions, focusing on the coefficients and conditions governing the flip bifurcation. On this basis, state feedback and hybrid control are utilized to control the system chaotically. Under certain conditions, the chaos and bifurcation of the system can be stabilized by the control strategy. Numerical simulations further reveal the bifurcation dynamics, chaotic behavior, and control techniques. Our results show that media coverage is a key factor in regulating the intensity and chaos of disease transmission. Control techniques can effectively prevent large-scale outbreaks of epidemics. Notably, enhanced media coverage can effectively increase public awareness and defensive behaviors, thus contributing to mitigating disease spread. Full article

This study investigates the dynamic behaviors of the discrete epidemic model influenced by media coverage through integrated analytical and numerical approaches. The primary objective is to quantitatively assess the impact of media coverage on disease outbreak patterns using mathematical modeling. Firstly, the Euler method is used to discretize the model (2), and the periodic solution is strictly analyzed. Secondly, the coefficients and conditions of restricted flip and Neimark–Sacker bifurcation are studied by using the center manifold theorem and bifurcation theory. By calculating the largest Lyapunov exponent near the critical bifurcation point, the occurrence of chaos and limit cycles is proved. On this basis, the chaotic control of the system is carried out by using state feedback and hybrid control. Under certain conditions, the chaos and bifurcation of the system can be stabilized by control strategies. Numerical simulations further reveal bifurcation dynamics, chaotic behaviors, and control technologies. Our results show that media coverage is a key factor in regulating the intensity of disease transmission and chaos. The control technology can effectively prevent the large-scale outbreak of epidemic diseases. Importantly, enhanced media coverage can effectively promote public awareness and defensive behaviors, thereby contributing to the mitigation of disease transmission. Full article

Differential equations have demonstrated significant practical effectiveness across diverse fields, including physics, chemistry, biological engineering, computer science, electrical power systems, and security cryptography. This study investigates the dynamics of a Caputo fractional discrete-time modified Brusselator model. Conditions for the existence and local stability of the fixed point $FP$ are established. Using bifurcation theory, the occurrence of both period-doubling and Neimark–-Sacker bifurcations is analyzed, revealing that these bifurcations occur at specific values of the bifurcation parameter. Maximum Lyapunov characteristic exponents are computed to assess system sensitivity. Two-dimensional diagrams are presented to illustrate phase portraits, local stability regions, closed invariant curves, bifurcation types, and Lyapunov exponents, while the 0-1 test confirms the presence of chaos in the model. Finally, MATLAB simulations validate the theoretical results. Full article