Search Results (276)

This work explores the qualitative dynamics of the modified unstable time-fractional nonlinear Schrödinger equation (mUNLSE), a model applicable to nonlinear wave propagation in plasma and optical fiber media. By transforming the governing equation into a planar conservative Hamiltonian system, a detailed bifurcation study is carried out, and the associated equilibrium points are classified using Lagrange’s theorem and phase-plane analysis. A family of exact wave solutions is then constructed in terms of both trigonometric and Jacobi elliptic functions, with solitary, kink/anti-kink, periodic, and super-periodic profiles emerging under suitable parameter regimes and linked directly to the type of the phase plane orbits. The validity of the solutions is discussed through the degeneracy property which is equivalent to the transmission between the phase orbits. The influence of the fractional derivative order on amplitude, localization, and dispersion is illustrated through graphical simulations, exploring the memory impacts in the wave evolution. In addition, an externally periodic force is allowed to act on the mUNLSE model, which is reduced to a perturbed non-autonomous dynamical system. The response to periodic driving is examined, showing transitions from periodic motion to quasi-periodic and chaotic regimes, which are further confirmed by Lyapunov exponent calculations. These findings deepen the theoretical understanding of fractional Schrödinger-type models and offer new insight into complex nonlinear wave phenomena in plasma physics and optical fiber systems. Full article

This paper considers the solution behavior and dynamical properties of the variable-order fractional Newton–Leipnik system defined via Liouville–Caputo derivatives of variable order. In contrast to integer-order models, the presence of variable-order fractional operators in the Newton–Leipnik structure enriches the model by providing memory-dependent effects that vary with time; hence, it is capable of a broader and more flexible range of nonlinear responses. Numerical simulations have been conducted to study how different order functions influence the trajectory and qualitative dynamics: clear transitions in oscillatory patterns have been identified by phase portraits, time-series profiles, and three-dimensional state evolution. The work goes further by considering the development of bifurcations and chaotic regimes and stability shifts and confirms the occurrence of several phenomena unattainable in fixed-order and/or integer-order formulations. Analysis of Lyapunov exponents confirms strong sensitivity to the initial conditions and further details how the memory effects either reinforce or prevent chaotic oscillations according to the type of order function. The results, in fact, show that the variable-order fractional Newton–Leipnik framework allows for more expressive and realistic modeling of complex nonlinear phenomena and points out the crucial role played by evolving memory in controlling how the system moves between periodic, quasi-periodic, and chaotic states. Full article

The global phase portrait (GPP) classification of polynomial planar Hamiltonian systems with finitely many isotropic points is a challenging problem. Only homogeneous Hamiltonian systems of degrees up to five have been dealt with in existing literature. In this paper, through a polar coordinate compactification, we prove that the GPP of a homogeneous planar Hamiltonian system is uniquely determined by the phase portrait around its isotropic point, referred to as the local phase portrait (LPP). Thus, the global classification can be reduced to the local classification. Secondly, two distinct approaches, topological index analysis and algebraic factorization, are proposed to establish both the local classification and the global one. And finally, the corresponding physical flows are discussed, and the consistency of results from the two approaches is validated through four examples. Full article

In this paper, we study the time-space truncated M-fractional model of shallow water waves in a weakly nonlinear dispersive media that characterizes the nano-solitons of ionic wave propagation along microtubules in living cells. A fractional wave transformation is applied, reducing the model to a third-order differential equation formulated as a conservative Hamiltonian system. The stability of the equilibrium points is analyzed, and the corresponding phase portraits are constructed, providing valuable insights into the expected types of solutions. Utilizing the dynamical systems approach, a variety of predicted exact fractional solutions are successfully derived, including solitary, periodic and unbounded singular solutions. One of the most notable features of this approach is its ability to identify the real propagation regions of the desired waves from both physical and mathematical perspectives. The impacts of the fractional order and gravitational force variations on the solution profiles are systematically analyzed and graphically illustrated. Moreover, the quasi-periodic dynamics and chaotic behavior of the model are explored. Full article

This paper presents a nonlinear vertical dynamic model of an electromagnetic suspension (EMS) system in maglev trains regulated by a dual nonlinear saturation controller (DNSC) under simultaneous resonance ($\Omega \cong {\omega }_s, {\omega }_s\cong 2{\omega }_c$). The governing nonlinear differential equations of the system are addressed analytically utilizing the multiple time-scale technique (MTST), concentrating on resonance situations obtained from first-order approximations. The suggested controller incorporates two nonlinear saturation functions in the feedback and feedforward paths to improve system stability, decrease vibration levels, and enhance passenger comfort amidst external disturbances and parameter changes. The dynamic bifurcations caused by DNSC parameters are examined through phase portraits and time history diagrams. The goal of control is to minimize vibration amplitude through the implementation of a dual nonlinear saturation control law based on displacement and velocity feedback signals. A comparative analysis is performed on different controllers such as integral resonance control (IRC), positive position feedback (PPF), nonlinear integrated PPF (NIPPF), proportional integral derivative (PID), and DNSC to determine the best approach for vibration reduction in maglev trains. DNSC serves as an effective control approach designed to minimize vibrations and enhance the stability of suspension systems in maglev trains. Stability evaluation under concurrent resonance is conducted utilizing the Routh–Hurwitz criterion. MATLAB 18.2 numerical simulations (fourth-order Runge–Kutta) are employed to analyze time-history responses, the effects of system parameters, and the performance of controllers. The evaluation of all the derived solutions was conducted to verify the findings. Additionally, quadratic velocity feedback leads to intricate bifurcation dynamics. In the time domain, higher displacement and quadratic velocity feedback may destabilize the system, leading to shifts between periodic and chaotic movements. These results emphasize the substantial impact of DNSC on the dynamic performance of electromagnetic suspension systems. Frequency response, bifurcation, and time-domain evaluations demonstrate that the DNSC successfully reduces nonlinear oscillations and chaotic dynamics in the EMS system while attaining enhanced transient performance and resilience. Full article

Fractional calculus extends conventional differentiation and integration to non-integer orders, facilitating a more suitable modeling framework for complicated dynamical processes characterized by memory and long-range dependence. The fractional and variable-order fractional Genesio-Tesi systems have recently attracted significant interest owing to their rich nonlinear dynamics and the added flexibility introduced by variable fractional orders. This work comparatively studies the stability and chaotic behavior of constant-order versus variable-order formulations of the fractional Genesio–Tesi system. The study of the system dynamics is carried out by numerical simulations, including time series, bifurcation diagrams, Lyapunov exponents, and phase portraits. We identify further stability boundaries and chaotic regimes through analytical investigations based on the Jacobian eigenvalue spectrum. It is found that variable-order derivatives intensify sensitivity and transient responses, disclosing chaotic patterns that contribute to a more profound understanding of fractional nonlinear dynamics. Full article

Over the past thirty years, the focus in singular optics has been on structured beams carrying orbital angular momentum (OAM) for diverse applications in science and technology. However, as practice has shown, the OAM-free structured Gaussian beams with several degrees of freedom are no worse than the OAM beams, especially when propagating through turbulent flows. In this paper, we partially fillthis gap by theoretically and experimentally mapping cyclic changes in vortex-free states (including OAM) as a phase portrait of the beam evolution in an astigmatic optical system. We show that those cyclic variations in the beam parameters are accompanied by the accumulation of the geometric Berry phase, which is an additional degree of freedom. We find also that the geometric phase of cyclic changes in the intensity ellipse shape does not depend on the radial numbers of the Laguerre–Gaussian mode with zero topological charge and is always set by changing the shape of the Gaussian beam. The Stokes parameter formalism was developed to map the beam states’ evolution onto a Poincaré sphere based on physically measurable second-order intensity moments. Theory and experiment are found to be in a good enough agreement. Full article

This paper investigates the dynamical properties of the fractional-order Arneodo system using a Grünwald–Letnikov-based numerical discretization. Fractional-order operators introduce memory and hereditary effects, enabling a more realistic description than classical integer-order models. The local stability of equilibrium points is examined through eigenvalue analysis of the Jacobian matrix, along with dissipativity conditions and the emergence of complex attractors. A comprehensive dynamical investigation is presented through phase portraits, time series, Lyapunov exponents, and bifurcation diagrams for varying fractional orders. Numerical findings demonstrate the emergence of new chaotic and hyperchaotic attractors. The results confirm that the fractional order strongly influences the system’s stability, sensitivity, and complexity. Our results confirm the relevance of fractional-order modeling in applications, such as secure communication, random number generation, and complex system analysis. Full article

In this article, we investigate the dynamical analysis and soliton solutions of the microtubule equation. First, the Lie symmetry method is applied to the considered model to reduce the governing partial differential equation into an ordinary differential equation. Next, the multivariate generalized exponential rational integral function method is employed to derive exact soliton solutions. Finally, the bifurcation analysis of the corresponding dynamical system is discussed to explore the qualitative behavior of the obtained solutions. When an external force influences the system, its behavior exhibits chaotic and quasi-periodic phenomena, which are detected using chaos detection tools. We detect the chaotic and quasi-periodic phenomena using 2D phase portrait, time analysis, fractal dimension, return map, chaotic attractor, power spectrum, and multistability. Phase portraits illustrating bifurcation and chaotic patterns are generated using the RK4 algorithm in Matlab version 24.2. These results offer a powerful mathematical framework for addressing various nonlinear wave phenomena. Finally, conservation laws are explored. Full article

The article presents a numerical analysis of a nonlinear seven-degree-of-freedom mechanical system composed of stick–slip-driven masses and magnetically coupled pendulums, emphasizing the influence of friction and magnetic coupling on the system’s dynamics. The objective is to develop a dynamic model, analyze bifurcation structures and synchronization, and examine multistability and sensitivity to initial conditions. The equations of motion are derived using the Lagrangian formalism and expressed in a dimensionless form. Bifurcation diagrams, phase portraits, spectral diagrams, and attraction basins are used to explore system behavior across parameter ranges. Saddle-node, Neimark–Sacker, and period-doubling bifurcations are observed, along with multiple coexisting attractors—periodic, quasiperiodic, and chaotic—indicating pronounced multistability. Small variations in initial conditions or system parameters lead to abrupt transitions between attractors. It has been shown that the mass of the pendulum strongly affects the system’s synchronization capability. Full article

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. Full article

This paper investigates a discrete-time compartmental model for computer virus propagation. The model classifies computers into susceptible, latent, and breaking-out states, with nonlinear dynamics driven by infection, recovery, and breakout processes. Stability is analyzed using the basic reproduction number $R_0$, and chaotic behavior is demonstrated through phase portraits, bifurcation diagrams, and maximum Lyapunov exponents. To further characterize complexity, the $C_0$ complexity measure is computed, confirming the richness of the chaotic regime. In addition, control strategies are designed to stabilize the dynamics, and a master–slave synchronization scheme is proposed and validated. Numerical simulations highlight both the complexity and controllability of the system, underscoring its relevance for understanding and mitigating the propagation of computer viruses. Full article

The (3 + 1)-dimensional KdV–Calogero–Bogoyavlenskii–Schiff equation, a model that describes long-wave interactions and has numerous applications in mathematics, engineering, and physics, is examined in this work. First, a wave transformation is used to reduce the equation to lower dimensions. The modified Khater method is then used to derive different types of solitary wave solutions, such as chirped, kink, periodic, and kink-bright types. By allocating suitable constant parameters, 3D, 2D, and contour plots are created to demonstrate the physical behavior of these solutions. Phase portraits are used to qualitatively analyze the undisturbed planar system using bifurcation theory. The system is then perturbed by an external force, resulting in chaotic dynamics. Chaos in the system is confirmed using multiple diagnostic tools, including time series plots, Poincaré sections, chaotic attractors, return maps, bifurcation diagrams, power spectra, and Lyapunov exponents. The stability of the model is further investigated with varying initial conditions. A bidirectional scatter plot technique, which efficiently reveals overlapping regions using data point distributions, is presented for comparing solution behaviors. Overall, this work offers useful tools for advancing applied mathematics research as well as a deeper understanding of nonlinear wave dynamics. Full article

This work investigates the dynamical properties of the Kolmogorov–Petrovskii–Piskunov (KPP) equation. We begin by establishing its non-integrability through the Painlevé test. Using a traveling wave transformation, we reduce the equation to a planar dynamical system, which we identify as non-conservative. A subsequent bifurcation analysis, supported by Bendixson’s criterion, rules out the existence of periodic orbits and, thus, periodic solutions—a finding further validated by phase portraits. Furthermore, we classify the types and co-dimensions of the bifurcations present in the system. We demonstrate that under certain conditions, the system can exhibit saddle-node, transcritical, and pitchfork bifurcations, while Hopf and Bogdanov–Takens bifurcations cannot occur. This study concludes by systematically deriving a power series solution for the reduced equation. Full article

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