Bifurcation Point Research Articles

Bifurcation analysis, phase portraits and optical soliton solutions of the perturbed temporal evolution equation in optical fibers

The perturbed nonlinear Schrödinger equation plays a crucial role in various scientific and technological fields. This equation, an extension of the classical nonlinear Schrödinger equation, incorporates perturbative effects that are essential for modeling real-world phenomena more accurately. In this paper, we investigate the traveling wave solutions of the perturbed nonlinear Schrödinger equation using the bifurcation theory of dynamical systems. Graphical presentations of the phase portrait are provided, revealing the traveling wave solutions under various conditions. By employing the auxiliary equation method, we derive a variety of solutions including periodic, dark, singular and bright optical solitons. To provide comprehensive and clearer depiction of the model’s behavior 2D, contour and 3D graphical representations are offered. We also highlight specific constraint conditions that ensure the presence of these obtained solutions. This study expands the scope of known exact solutions and their stability qualities which is offering an extensive analytical technique which enhances previous research. The novelty of our research lies in its examination of bifurcation analysis and the auxiliary equation method within the context of a perturbed nonlinear Schrödinger wave equation for the first time. By integrating these two perspectives, this paper contributes to establishing the complex dynamics and stability characteristics of soliton solutions under perturbations.

Dynamic complexity of a discretized predator-prey system with Allee effect and herd behaviour

This paper investigates the dynamics of a discrete-time predator-prey system in which the prey population is impacted by the Allee effect. Possible fixed points in the system are studied for their existence and topological categorization. Moreover, the presence and direction of period-doubling and Neimark-Sacker bifurcations at the interior fixed point are examined through the application of bifurcation theory and the centre manifold theorem. A hybrid approach is adopted to control chaotic behaviour and prevent bifurcations. Numerical examples are provided to substantiate our theoretical findings. The Allee effect has been shown to affect the dynamics of the system using numerical simulations. The moderate Allee effect stabilizes predator and prey populations, facilitating ecological cohabitation and persistence.

Bifurcation and stability analysis of a Leslie–Gower diffusion predator–prey model with prey refuge and Beddington–DeAngelis functional response

In this paper, we consider the spatiotemporal dynamics behaviors of a Leslie–Gower diffusion predator–prey system with prey refuge and Beddington–DeAnglis (B‐D) functional response. By using the Poincaré inequality and topological degree theory, we first investigate the Turing instability of the reaction–diffusion system and prove the existence of nonconstant positive steady‐state solutions. Then we discuss the steady‐state bifurcation and the direction to Hopf bifurcation of the PDE model by the local bifurcation theorem and center manifold theory. Finally, some numerical simulations are presented to supplement the analytic results in one dimension which indicates that changes in prey refuge and diffusion coefficient can increase the complexity of the system.

Stability and Bifurcations in Time-Delay Systems: A Novel and Efficient Blend of Methods

The goal of this paper is to introduce a hybrid technique, combining two existing methods conveniently, to analyze the stability and bifurcations of equilibrium points and periodic orbits in delay-differential equations of retarded type. One method is called the frequency-domain approach, an analytical tool based on control theory allowing to detect and represent periodic solutions emerging from the Hopf bifurcations, via Fourier series and harmonic balances. The second one, the so-called semi-discretization method, provides a numerical scheme to approximate the monodromy operator of a periodic solution, thus permitting to establish the stability and bifurcations of limit cycles as well as analyzing the equilibrium points. It is shown that a proper combination of these methods provides a straightforward strategy to study both local and global bifurcations in time-delay systems. The usefulness of the proposed approach is shown by three examples, where the results are compared with those given by the software DDE-BIFTOOL.

Transient behavior of pumped storage-wind hybrid power system under grid-connected operation condition

The large-scale grid connection of renewable energy represented by wind power threatens the safe and stable operation of electric system. The vigorous development pumped storage power station (PSPS) is a global consensus to support the grid-connected of renewable energy. This paper investigates the transient behavior of pumped storage-wind (PSW) hybrid power system under grid-connected operation condition (GCOC). Firstly, a model of PSW hybrid power system under GCOC is constructed. The stable domain is determined according to Hopf bifurcation theory. The influence of system parameters on the stability of PSW hybrid power system under GCOC is analyzed. A synergetic control strategy is designed to improve the dynamic performance of the system. Then, the mechanism of multi-time-scale dynamic response is revealed. Finally, the transient behavior of PSPS under different application scenarios and the interaction between PSPS and wind power station (WPS) are studied. The results show that the stability of PSW hybrid power system under GCOC can be judged by stable domain. The designed synergetic control strategy can improve the dynamic performance of the system. The dynamic response is composed of three oscillations with different frequencies, namely, mass oscillation at low frequency, water hammer oscillation at medium frequency, and electrical oscillation at high frequency. The principal oscillation is caused by water hammer. The interaction between PSPS and WPS has a significant effect on the transient behavior. By adjusting the interaction between PSPS and WPS, the system can absorb more wind power while ensuring safe and stable operation, which can increase profits and avoid unnecessary operation oscillations. The analytical methods and results presented in this paper are of great significance for improving the ability of power grid to accept renewable energy.

Liver anatomy for totally laparoscopic segment 7 resection from a caudal magnified view-a 3-dimensional computed tomographic analysis.

Laparoscopic resection of hepatic segment 7 is considered particularly difficult. We analyzed anatomic variation of this segment in caudally oriented 3-dimensional (3D) magnified computed tomographic (CT) images obtained prior to liver resection. Analysis included 105 patients with preoperative 3D CT evaluation preceding liver resection for hepatobiliary malignancies between April 2021 and April 2024. Five ramification patterns were evident from a caudal magnified view. Some patients who had multiple segment 7 (S7) portal pedicles and an S7 pedicle branching ventrally posed difficulty in performing segmentectomy for the exact extent of S7. Distance from the point where a perpendicular line from the right rim of the inferior vena cava (IVC) intersected the right posterior portal pedicle to the point of bifurcation of the S6 and 7 pedicles was 34.6mm (range, 3.9-78.8; mean ± standard deviation, 35.2 ± 14.8mm). The median angle between the perpendicular line from the right rim of the IVC and the line from the root of the S7 pedicle to the right rim of the IVC was 77° (10-140); the mean ± standard deviation was 75.3° ± 28.1. Differences among ramification patterns also were evident. The angle between the right posterior portal pedicle and the S7 pedicle was 143° (79-215) or 143.3 ± 26.7, and that between S7 and S6 pedicles was 71°(15-123) or 75.5 ± 21.7°, representing relatively little variation. Understanding these details of caudal-view anatomy may resolve difficulties and clarify access required for exposing S7 portal pedicles.

Anatomical analysis of the abdominal aorta in a South African sample: influence of age and sex.

The anatomy of the abdominal aorta (AA) varies with age and sex; however, limited studies exist from South Africa. Given the increased incidence of endovascular treatment of the AA, reference values are relevant for interventionalists for improving the safety of endovascular procedures. Therefore, the study aimed to determine the lengths, diameters and tortuosity of the AA and their association with age and sex in a South African sample. After ethical approval, 97 computed tomography angiography (CTA) scans from an adult sample (54 male and 43 female), mean age 48.5 ± 17.2 years were analysed. The aortic length was measured from the origin of the coeliac trunk to the bifurcation point of the AA. The lumen diameters of the aorta were measured at three landmarks. Tortuosity of the AA was quantified with the tortuosity index and its prevalence was determined. The AA was longer in males and showed a significant weak positive correlation with age. The mean diameters of the AA were larger in males and had a significant strong positive correlation with age in both sexes (p < .001). There was a strong positive correlation between age and tortuosity in both sexes (p < .001). The prevalence of a tortuous c-shaped-curve phenotype was 8.2%, with a 7:1 male-to-female ratio. The dimensions and tortuosity differed between sexes and varied significantly with age. These findings may contribute towards reference values in the South African setting, inform patient selection and complement decision-making of endovascular treatment strategies.

A Novel Method for Analyzing Global Bifurcation and Global Hysteresis of FGM Truncated Conical Shell Structure

In the dynamic bifurcation and dynamic hysteresis of a system, our first consideration is the tiny changes in time that can lead to variations in the vector field, potentially causing the occurrence of system dynamic bifurcation and dynamic hysteresis. This paper presents a novel method for analyzing the global bifurcation and hysteresis of Functionally Graded Material (FGM) truncated conical shell structure under aerodynamics and in-plane force along meridian near internal resonances. This method involves introducing the ratio of vector fields as a periodic perturbation parameter in the system’s nonlinear second-order ordinary differential governing equations. This allows for the analysis of the nonlinear ordinary differential equations as algebraic equations, yielding algebraic expressions that only involve parameters and do not contain state variables. Subsequently, the global bifurcation set and global hysteresis set of FGM truncated conical shell structure are obtained, providing the parameters interval ranges for the stability and instability of FGM truncated conical shell structure. By utilizing MAPLE software for numerical simulation, the images of the global bifurcation set and global hysteresis set about the amplitude of in-plane load are first simulated numerically. Subsequently, the amplitude graphs of the equilibrium points are numerically simulated for validation. The results demonstrate a perfect alignment between the images of the global bifurcation set and global hysteresis set and the data from the amplitude graphs of the equilibrium point. Due to the periodicity of the periodic perturbation parameter, it shuttles between different persistent regions as other parameters change, leading to the generation of chaotic solutions in the governing equations. The validity of the obtained results is confirmed through comparisons with existing literature.

Dynamic analysis of a modified Leslie-Gower model with Michaelis-Menten prey harvesting and additive allee effect on predator population

In the current study, the dynamics of a Leslie-Gower model have been studied considering the Allee effect in the growth of the predator and the Michaelis-Menten type harvesting effect on the prey from biological and theoretical perspectives. Key findings highlight how these effects influence the existence of positive equilibria, with local stability properties and potential bifurcation behaviors examined using the center manifold theorem and linearization. Criteria for saddle-node bifurcation are established through Sotomayor’s theorem, while the direction and stability of Hopf bifurcation are explored via the first Lyapunov exponent. An optimal control model is developed to promote sustainable ecosystem development, utilizing the harvesting rate as a control parameter. Further, a few numerical simulations validate our significant findings using phase portrait diagrams.

The effects of toxin and mutual interference among zooplankton on a diffusive plankton–fish model with Crowley–Martin functional response

A diffusive phytoplankton–zooplankton–fish model with toxin and Crowley–Martin functional response is considered. By the global bifurcation theorem and the fixed point index theory, the existence and multiplicity of coexistence states are discussed. Secondly, we investigate the bifurcation from a double eigenvalue by virtue of Lyapunov–Schmidt procedure and implicit function theorem. Then, the effect of large k, which measures the magnitude of interference among zooplankton, is studied by means of the combination of the perturbation theory and topological degree theory. The results indicate that if k is large enough, this system has only a unique asymptotically stable coexistence state provided that toxin is properly small and the maximal growth rates of phytoplankton and fish are suitably large. Furthermore, the extinction and permanence of the time-dependent system are determined by virtue of the comparison principle. Finally, we make some numerical simulations to validate and complement the theoretical analysis and exhibit the critical role of toxin, spatial diffusion and magnitude of interference among zooplankton in the dynamics. The findings suggest that the spatiotemporal dynamics of the systems with toxin and Crowley–Martin functional response are richer and more complex, and toxin and spatial diffusion have significant effects on the coexistence of phytoplankton–zooplankton–fish species.

Optical dromions for M-fractional Kuralay equation via complete discrimination system approach along with sensitivity analysis and quasi-periodic behavior

This paper investigates new analytical wave solutions for the fractional Kuralay-IIB equations (FK-IIBE), including a new definition of the fractional derivative. This model works in disciplines such as ferromagnetic materials, optical fibers, wave mixing, and nonlinear optics. The integrable motion of space curves can be determined by our governing model. This research holds great importance in the area of optical fibers, exact and analytical solitons solution. The polynomial method’s complete discrimination system (CDSPM) is used to identify a variety of exact and analytical soliton solutions such as periodic, hyperbolic, rational, Jacobian elliptic (JE), and trigonometric functions. Additionally, we also convert the JE function into a solitary wave (SW) solution. To provide a comprehensive understanding of the dynamic aspects of the system, we also address bifurcation points, quasi-periodic behavior, sensitivity analysis, Poincarè maps, time series profile, and critical solution conditions. The research offers straightforward, efficient algorithms that can consistently and effectively address FK-IIBE. Machine learning technologies are also utilized to satisfy the criteria defined by dynamic analysis. Python regression learner tools are used for reverse engineering to analyze the mathematical model’s equilibrium based on parameter intervals and thresholds determined by dynamical analysis.

Navier-Stokes: Singularities and Bifurcations

This article presents substantial advances in the analysis of the Navier-Stokes equations for both compressible and incompressible fluids, focusing on the formation of singularities, hypercomplex bifurcations, and regularity in Sobolev and Besov spaces. Through new theorems, we extend the theory of singularities in fluid dynamics and introduce quaternionic bifurcations, representing an innovative extension of classical bifurcation theory. Moreover, we delve into the investigation of the regularity of compressible fluids, exploring the conditions under which solutions remain smooth or develop singularities. These contributions are fundamental to the understanding of global regularity issues, directly linked to the renowned Millennium Prize problem, which seeks definitive answers on the existence and smoothness of solutions to the Navier-Stokes equations. Additionally, we discuss how these theoretical advancements offer new approaches to unresolved problems related to the formation of singularities in turbulent flows and the multiscale behavior of solutions, which are crucial for a comprehensive understanding of fluid dynamics. This work not only broadens the scope of traditional mathematical analysis of the Navier-Stokes equations but also establishes a robust theoretical framework for the investigation of bifurcations and regularity in advanced functional spaces, fostering a deeper understanding of global regularity phenomena and the complex dynamics governing fluid systems.

Symmetry Breaking and Modal Localization in a System of Parametrically Excited Microbeam Resonators

In this work, we study the nonlinear dynamics of parametrically excited bending vibrations of two weakly coupled beam microresonators under electrothermal excitation. A steady-state harmonic temperature distribution in the volume of the resonators in the frequency domain was obtained. A system of equations for mechanically coupled beam resonators is derived, considering the deposited particle on one of them. Using asymptotic methods of nonlinear dynamics, equations in slow variables were obtained, which were studied by methods of the theory of bifurcations. It is shown that in a perfectly symmetrical system in a certain frequency range, the effect of symmetry breaking is observed – the emergence of a mode with different amplitudes of oscillations of two beam resonators, which can be the basis for a new principle of high-precision measurements of weak disturbances of various physical natures, in particular – measurements of ultra-low masses of deposited particles.

Stability and Bifurcation Analysis of a Symmetric Fractional-Order Epidemic Mathematical Model with Time Delay and Non-Monotonic Incidence Rates for Two Viral Strains

This study investigates a symmetric fractional-order epidemic model with time delays and non-monotonic incidence rates, considering two viral strains. By confirming the existence, uniqueness, and boundedness of the system’s solutions, the research ensures the model’s well-posedness, guaranteeing its mathematical soundness and practical relevance. The study calculates and evaluates the equilibrium points and the basic reproduction numbers R01 and R02 to understand the dynamic behavior of the model under different parameter settings. Through the application of the Lyapunov method, the research examines the asymptotic global stability of the system, determining whether it will converge to a particular equilibrium state over time. Furthermore, Hopf bifurcation theory is employed to investigate potential periodic solutions and bifurcation scenarios, highlighting how the system might shift from stability to periodic oscillations under certain conditions. By utilizing the Adams-Bashforth-Moulton numerical simulation method, the theoretical results are validated, reinforcing the conclusions and demonstrating the model’s applicability in real-world contexts. It emphasizes the importance of fractional-order models in addressing epidemiological issues related to time delays (τ), individual heterogeneity (m, k), and memory effects (θ), offering greater accuracy compared with traditional integer-order models. In summary, this research provides a theoretical foundation and practical insights, enhancing the understanding and management of epidemic dynamics through fractional-order epidemic models.

Scalarization of Taub-NUT black holes in extended scalar-tensor-Gauss-Bonnet theory

Recently, the scalarization of the Schwarzschild black hole has been extensively studied. In this work, we explore the scalarization of the Taub-NUT black hole within the context of the extended scalar-tensor-Gauss-Bonnet theory, which admits a Ricci-flat Taub-NUT black hole as a solution. We carried out an analysis of the probe scalar field to identify the mass parameter and NUT parameter (m, n) where hairy black holes begin to emerge. Subsequently, we used the shooting method to construct the scalarized Taub-NUT black hole numerically. Unlike the Schwarzschild case, there are two branches of new hairy black holes that are smoothly connected. We calculated the entropy of the scalarized black holes and compared these entropies with those of scalar-free Taub-NUT black holes, finding that the entropies of the new hairy black holes are larger. A novel phenomenon emerges in this system: the entropy of the black holes at the bifurcation point is constant for a positive mass parameter. We then conjecture a maximal entropy bound for all scalarized black holes whose mass parameter at the bifurcation point is greater than zero.

Existence, stability and the number of two-dimensional invariant manifolds for the convective Cahn–Hilliard equation

We study the well-known generalised version of the nonlinear Cahn–Hilliard evolution equation, supplemented with periodic boundary conditions. We study local bifurcations in the vicinity of spatially homogeneous equilibrium states. We show the possibility of the existence of a finite or countable set of equilibrium states of the boundary value problem under study, in the vicinity of which, if appropriate conditions are met, there exist two-dimensional invariant manifolds filled with solutions that are periodic in the evolutionary variable. Moreover, we derive asymptotic formulas for these periodic solutions. Finally, we study the stability of invariant manifolds and the solutions belonging to them.In order to analyse the bifurcation problem, we used methods from the theory of dynamical systems with infinite-dimensional phase, namely the method of invariant manifolds and the method of normal forms.

Analytical and dynamical analysis of nonlinear Riemann wave equation in plasma systems

The Riemann wave equation presents appealing nonlinear equations applicable in sea-water and tsunami wave propagation, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, and homogeneous stationary media. This study focuses on deriving soliton solutions in optics and exploring their physical properties. A wave transformation is used to convert a partial differential equation into an ordinary differential equation, from which soliton solutions are obtained using the generalized Riccati equation mapping approach. The solutions encompass various types of solitons, including bright, dark, periodic, and kink solitons. A comparison of solutions from this analytical method enhances the understanding of the nonlinear model’s behavior, with implications in plasma physics, fluid dynamics, optics, and communication technology. Additionally, 2D and 3D graphs illustrate the physical phenomena of the solutions using appropriate constant parameters. The qualitative analysis of the undisturbed planar system involves examining phase portraits in bifurcation theory, followed by introducing an outward force to induce disruption and reveal chaotic phenomena. Chaotic trajectories in the perturbed system are detected through various plots, including 3D, 2D, power spectrum, and chaotic attractor, alongside Lyapunov exponents. Stability analysis under different initial conditions is conducted, and sensitivity assessments are performed using the Runge–Kutta method. The findings are innovative and have not been previously explored for this system, highlighting the reliability, simplicity, and effectiveness of these techniques in analyzing nonlinear models in mathematical physics and engineering.

Synchronization of Chaotic Satellite Systems with Fractional Derivatives Analysis Using Feedback Active Control Techniques

This research article introduces a novel chaotic satellite system based on fractional derivatives. The study explores the characteristics of various fractional derivative satellite systems through detailed phase portrait analysis and computational simulations, employing fractional calculus. We provide illustrations and tabulate the phase portraits of these satellite systems, highlighting the influence of different fractional derivative orders and parameter values. Notably, our findings reveal that chaos can occur even in systems with fewer than three dimensions. To validate our results, we utilize a range of analytical tools, including equilibrium point analysis, dissipative measures, Lyapunov exponents, and bifurcation diagrams. These methods confirm the presence of chaos and offer insights into the system’s dynamic behavior. Additionally, we demonstrate effective control of chaotic dynamics using feedback active control techniques, providing practical solutions for managing chaos in satellite systems.

Bifurcation enhances temporal information encoding in the olfactory periphery.

Living systems continually respond to signals from the surrounding environment. Survival requires that their responses adapt quickly and robustly to the changes in the environment. One particularly challenging example is olfactory navigation in turbulent plumes, where animals experience highly intermittent odor signals while odor concentration varies over many length- and timescales. Here, we show theoretically that Drosophila olfactory receptor neurons (ORNs) can exploit proximity to a bifurcation point of their firing dynamics to reliably extract information about the timing and intensity of fluctuations in the odor signal, which have been shown to be critical for odor-guided navigation. Close to the bifurcation, the system is intrinsically invariant to signal variance, and information about the timing, duration, and intensity of odor fluctuations is transferred efficiently. Importantly, we find that proximity to the bifurcation is maintained by mean adaptation alone and therefore does not require any additional feedback mechanism or fine-tuning. Using a biophysical model with calcium-based feedback, we demonstrate that this mechanism can explain the measured adaptation characteristics of Drosophila ORNs.

Buckling disappearance via merging/divergence in a nonlinear three-d.o.f. system with linear constitutive law

The phenomenon of buckling disappearance, occurring in a parameter-dependent family of systems admitting a nontrivial fundamental path, is studied. Two different forms of disappearance are detected, namely: (i) the divergence, in which the critical load continuously tends to infinity, and (ii) the merging, in which two critical loads approach each other, coalesce, and then disappear at a finite value of the critical load. It is shown that the two phenomena can be exhibited by the same mechanical system, when a suitable elasto-geometric parameter is varied. More importantly, it is proved that merging continuously changes into divergence when a second parameter is changed. A paradigmatic system is chosen to illustrate the two forms of buckling, i.e., a three degree-of-freedom spherical pendulum, elastically constrained at the ground, loaded by a transverse force and/or a conservative couple, made of two longitudinal potential forces. The springs are taken elastically linear, to stress the fact that divergence not necessarily calls for introducing a nonlinear constitutive law, as also mentioned in literature. Only a linear bifurcation analysis is carried out here, aimed to find the bifurcation points along the nonlinear fundamental path. However, due to the presence of non-negligible prestrains, such a bifurcation problem is governed by nonlinear algebraic equations, whose number of roots cannot be predicted in advance.