Periodic Solutions Research Articles

Dynamic characteristics analysis and vibration control of coupled bending-torsional system for axial flow hydraulic generating set

Aiming at the vibration problems of shaft system for axial flow hydraulic generating sets caused by complex excitations, the coupled bending-torsional dynamic equations of rotor-runner system are established considering multi-effect including hydraulic, mechanical, electrical and other external vibration sources. Whether taking the torsional degree of freedom into account, the stability and disparate destabilization laws of system periodic solutions are analyzed using the shooting method combined with Floquet theory. On this basis, the influence of coupling bending-torsional effect on system dynamic behaviors is discussed through bifurcation diagrams, Poincaré maps, trajectories, time domains and frequency spectrums. Furthermore, in order to alleviate the unsteady vibration of rotor as well as runner, the magnetorheological fluid damper is introduced into the shaft system, and numerical simulations are performed on models of coupled bending-torsional systems with or without the magnetorheological fluid damper. The analyses indicate that the interaction between bending and torsional vibration of the shaft system will expand the chaotic sustained range for rotor. Meanwhile, the amplitude of the rotor-runner system is magnified owing to the torsional vibration, which leads to local rubbing faults of the system under specific operating conditions, while the intervention of the magnetorheological fluid damper can significantly suppress the nonlinear motion of rotor and runner, reduce the amplitude of system vibration, so as to avoid the occurrence of friction defects effectively. The above research results can provide conducive references for fault diagnosis, optimized design, and vibration control of axial flow hydraulic generating sets.

Exact wave solutions of truncated M-fractional Boussinesq-Burgers system via an effective method

Abstract In this paper, we present distinct types of exact wave soliton solutions of an important fluid flow dynamic system called the truncated M-fractional (1+1)-dimensional nonlinear Boussinesq-Burgers system (BBS). This model is used to explain ocean waves, matter-wave pulses, waves in ferromagnetic media, the proliferation of waves in shallow water, etc. We transform the nonlinear fractional system into a nonlinear ordinary differential equation by using a fractional transformation to obtain dark, bright, singular, dark-bright, dark-singular, bright-singular and periodic type solitons solutions by employing the modified extended tanh function method (METhFM). The use of fractional derivatives makes the solutions different from the existing solutions. The obtained results are useful in the optical fibers, fluid dynamics, ocean engineering and other related fields. To visualize the system’s behavior, some of the solutions are represented by two- and three-dimensional graphs which are obtained and verified with the help of Mathematica. The achieved results provide a better understanding of the behavior of the nonlinear fractional partial differential equations and the dynamics of BBS, which are not present in the literature and are helpful in future studies of the concerned system.

Kinetic analysis of an optically pumped rare gas lasing medium with a nanosecond repetitively pulsed discharge in Ar-He mixture

A kinetic model of a lasing medium for an optically pumped rare gas laser (OPRGL) was developed that accounts for the production of excited argon atoms Ar* in a nanosecond repetitively pulsed discharge (NRPD). Analytical formulas were derived for the frequency of Ar* radiation losses, population of the energy levels involved in the laser cycle and specific powers for pump absorption and lasing in the limit of bleaching of pump and lasing transitions. The formulas use the kinetic constants of the model as parameters and Ar* number density as the independent variable. Periodic solutions for number density of plasma components, pump absorption and small signal gain, specific pump absorption and lasing were obtained numerically. Mean over the NRPD period values of Ar* number density, specific pump absorption and lasing were calculated as the functions of the preset peak values of the reduced electric field E/N. Triangular electric field pulses of 80 ns FWHM duration and 200 kHz frequency were considered. Optical pumping increases Ar* loss by more than a factor of 10, due to excess spontaneous emission from Ar* levels that populates the 1s4 state of argon, resulting in a reduced NRPD plasma ionization and Ar* production. As a result, without auxiliary optical pumping from the 1s4 state that compensates this loss, the averaged across the discharge period Ar* number density decreases by a factor of 30. Moreover, the average specific heat release in plasma becomes almost equal to the specific lasing power in the acceptable operating modes, making auxiliary optical pumping mandatory for scaling of an OPRGL. The sensitivity of the results to the values of kinetic constants in the model is analyzed.

Periodic solutions of Lagrangian systems under small perturbations

In this paper, we consider the existence of periodic solutions of Lagrangian systems of the form [Formula: see text], where [Formula: see text] is a [Formula: see text]-function in the sense of Trudinger, [Formula: see text] are [Formula: see text]-smooth, [Formula: see text]-periodic in the time variable [Formula: see text] and [Formula: see text] is a real parameter. We prove the existence of a [Formula: see text]-periodic solution [Formula: see text] for any sufficiently small [Formula: see text], and show that the found solutions converge to a [Formula: see text]-periodic solution of the unperturbed system if [Formula: see text] tends to [Formula: see text]. Let us stress that our theorem only makes a natural assumption on the potential [Formula: see text] and no additional assumption on [Formula: see text] except that it is continuously differentiable. Its proof requires to work in a rather unusual (mixed) Orlicz–Sobolev space setting, which bears several challenges.

Long-living periodic solutions of complex cubic-quintic Ginzburg-Landau equation in the presence of intrapulse Raman scattering: A bifurcation and numerical study.

We have found long-living periodic solutions of the complex cubic-quintic Ginzburg-Landau equation (CCQGLE) perturbed with intrapulse Raman scattering. To achieve this we have applied a model system of ordinary differential equations (SODE). A set of the fixed points of the system has been described. A complete phase portrait as well as phase portraits near the fixed points have been built for a proper choice of parameters. The behavior of the model system near the fixed points has been determined. We have presented a detailed description of the subcritical Poincaré-Andronov-Hopf bifurcation due to the intrapulse Raman scattering that appears at one of the fixed points. We have established that there appears an unstable limit cycle in the SODE. To check the validity of the obtained results from the model system we have compared them with the results of the numerical solution of the CCQGLE perturbed with intrapulse Raman scattering. There has been found a remarkable correspondence between the obtained numerical results for the amplitude and frequency of the soliton pulses and the results for these parameters of the bifurcation theory. We have observed that the numerical characteristics of the propagating solitonlike pulses-amplitude, frequency, width, and position-periodically change if we change the distance with a period determined by the bifurcation analysis.

Fractional Differential Equations with Impulsive Effects

This paper discusses impulsive effects on fractional differential equations. Two approaches are taken to obtain our results: either with fixed or changing lower limits in Caputo fractional derivatives. First, we derive an existence result for periodic solutions of fractional differential equations with periodically changing lower limits. Then, the impulsive effects are modeled for fractional differential equations regarding the nonlinearities rather than the initial value conditions. The proposed impulsive model differs from common discontinuous and nonsmooth dynamical systems.

Hopf bifurcation analysis of a two‐delayed diffusive predator–prey model with spatial memory of prey

In this paper, we consider a diffusive predator–prey model with spatial memory of prey and gestation delay of predator. For the system without delays, we study the stability of the positive equilibrium in the case of diffusion and no diffusion, respectively. For the delayed model without diffusions, the existence of Hopf bifurcation is discussed. Further, we investigate the stability switches of the model with delays and diffusions when two delays change simultaneously by calculating the stability switching curves and obtain the existence of Hopf bifurcation. We also calculate the normal form of Hopf bifurcation to determine the direction of Hopf bifurcation and the stability of bifurcation periodic solutions. Finally, numerical simulations verify the theoretical results.

On the boundedness of solutions of a forced discontinuous oscillator

We study the global boundedness of the solutions of a non-smooth forced oscillator with a periodic and real analytic forcing. We show that the impact map associated with this discontinuous equation becomes a real analytic and exact symplectic map when written in suitable canonical coordinates. By an accurate study of the behaviour of the map for large amplitudes and by employing a parametrization KAM theorem, we show that the periodic solutions of the unperturbed oscillator persist as two-dimensional tori under conditions that depend on the Diophantine conditions of the frequency, but are independent on both the amplitude of the orbit and of the specific value of the frequency. This allows the construction of a sequence of nested invariant tori of increasing amplitude that confine the solutions within them, ensuring their boundedness. The same construction may be useful to address such persistence problem for a larger class of non-smooth forced oscillators.

BIFURCATION AND EXACT TRAVELING WAVE SOLUTIONS OF FRACTIONAL DATE–JIMBO–KASHIWARA–MIWA EQUATION

Based on the bifurcation theory, the exact solutions of fractional Date–Jimbo–Kashiwara–Miwa equation are constructed. In terms of [Formula: see text]-derivative, a nonlinear integer-order ODE is obtained, the equation is transformed into a plane dynamical system. The phase portraits are obtained under different bifurcation conditions. According to the orbits of the phase portraits, new bright and dark solitary solutions, periodic solutions, periodic-singular solutions, and singular solutions are established, enriching the diversity of solutions. In addition, the dynamical behaviors of various types of solutions are shown by selecting appropriate parameters, which is useful for understanding the fluctuation behavior in physical models. Compared with previous literature, this paper focuses on the combination of qualitative analyses and qualitative calculations, which makes the paper more systematic and comprehensive, and fills the gap of using bifurcation theory to solve the FDJKM equation. The dynamical behavior of new solutions obtained will provide more and more attractive and complex physical phenomena to explore more mysteries.

Similarity transformations and exact solutions of the (3+1)-dimensional nonlinear Schrödinger equation with spatiotemporally varying coefficients

In this paper, an extended (3+1)-dimensional nonlinear Schrödinger equation is studied. By using similarity transformation, some exact solutions of this equation are obtained, which include soliton solutions and periodic function solutions, its nonlinear spatial modulation and external potential are affected by time and space. Based on the solution obtained previously, some figures are displayed.

Exploring Solitons Solutions of a (3+1)-Dimensional Fractional mKdV-ZK Equation

This study presents the application of the ϕ6 model expansion technique to find exact solutions for the (3+1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation under Jumarie’s modified Riemann–Liouville derivative (JMRLD). The suggested method captures dark, periodic, traveling, and singular soliton solutions, providing deep insights into wave behavior. Clear graphics demonstrate that the solutions are greatly affected by changes in the fractional order, deepening our understanding and revealing the hidden dynamics of wave propagation. The considered equation has several applications in fluid dynamics, plasma physics, and nonlinear optics.

Onset of Spontaneous Filling and Voiding Cycles in the Lower Urinary Tract: A Modeling Study.

Spontaneous filling and voiding cycles represent a key dynamical feature of the healthy lower urinary tract. Some urinary tract dysfunctions, such as over-flow incontinence, may alter the natural occurrence of these cycles. As the function of the lower urinary tract arises from the interplay of a multitude of factors, it is difficult to determine which of them can be modulated to regain spontaneous cycles. In this study, we develop a mathematical model of the lower urinary tract that can capture filling and voiding cycles in the form of periodic solutions of a system of ordinary differential equations. After experimental validation, we utilize this model to study the effect that several physiological quantities have on the onset of cycles. We find that some parameters have an associated numerical threshold that determines whether the system exhibits healthy cycles or settles in a state of constant overflow.

A non-autonomous fractional granular model: Multi-shock, Breather, Periodic, Hybrid solutions and Soliton interactions

This paper explores a novel generalized one-dimensional fractional order Granular equation with the effect of periodic forced term. This type of equation arises in different area of mathematical physics with several rough materials in engineering applications. By taking into account of external forces in combination with Hertz constant law and the long wave approximation principle, we construct the fractional order one-dimensional crystalline chain of elastic spheres. A suitable transformation is implemented to convert the fractional order equation to a regular equation. The Hirota’s bilinear approach is used to secure solutions for kink and anti-kink types shock solutions. In order to find periodic and solitary wave solutions, the Granular model is converted to approximate KdV model. The newly developed solutions exhibit a range of interesting dynamics due to the existence of an external force and the roughness effect. Many hybrid solutions are created by taking a long wave limit of a fraction of the exponential and trigonometrical functions in the bilinear form of the granular model. The hybrid solutions show different superposed wave shapes with lumps, kinks and breathers. The dynamical interaction of these solutions are further illustrated graphically. Furthermore, the system’s stability is assessed using perturbation techniques in order to comprehend its resilience and possible uses in granular particles in real-world situations, guaranteeing their dependability in a range of circumstances.

Existence of uncountably many periodic solutions for second-order superlinear difference equations with continuous time

Existence of uncountably many periodic solutions for second-order superlinear difference equations with continuous time

A neural network model for goat gait.

In this paper, our main objective was to investigate the central pattern generator (CPG) neural network model for quadruped gait with time delay. First, we computed the normal form of the model on the center manifold, the bifurcation direction, and stability conditions of the bifurcating periodic solutions. Second, we applied the CPG model for quadruped gait to obtain reference models for goat's diagonal trotting gait on the flat ground and walking gait on the 18 degree slope through the trust region inversion algorithm. Finally, we performed numerical simulations to support theoretical analysis.

Dynamical investigation of the perturbed Chen–Lee–Liu model with conformable fractional derivative

Abstract This study focuses on the investigation of the perturbed Chen–Lee–Liu model with conformable fractional derivative by the implementation of the generalized projective Riccati equations technique. The proposed method uses symbolic computations to provide a dynamic and powerful mathematical tool for addressing the governing model and yielding significant results. Numerous analytical solutions of the governing model, including bell-shaped soliton solutions, anti-kink soliton solutions, periodic solitary wave solutions and other solutions, have been constructed effectively utilizing this effective technique. The findings acquired from the governing model utilizing the suggested technique demonstrate that all results are novel and presented for the first time in this study. Solitons are of immense significance in the domain of nonlinear optics due to their inherent ability to preserve their shape and velocity during propagation. The study of the propagation and the dynamical behaviour of the derived results have been explored by representing them graphically through 3D, density, and contour plots with different selections of arbitrary parameter values. The solitons acquired from the proposed model can provide significant advantages in the field of fiber-optic transmission technology. The obtained results demonstrate that the suggested approach is extremely promising, straightforward, and efficient. Furthermore, this approach may be effectively used in numerous emerging nonlinear models found in the fields of applied sciences and engineering.

Dynamics, stability and bifurcations of a planar three-link swimmer with passive visco-elastic joint using “ideal fluid” model

Articulated swimming robots have a promising potential for various marine applications. A common theoretical model assumes ideal fluid, where the viscosity is negligible and the swimmer–fluid interaction is induced by reactive forces originating from added mass effect. Some previous works used this model to study planar multi-link swimmers under kinematic input prescribing all joint angles. Inspired by biological swimmers in nature that utilize body flexibility, in this work we consider an underactuated three-link swimmer where one joint is periodically actuated while the other joint is passive and viscoelastic. Analysis of the swimmer’s nonlinear dynamics reveals that its motion depends significantly on the amplitude and frequency of the actuated joint angle. Optimal frequency is found where the swimmer’s net displacement per cycle is maximized, under symmetric periodic oscillations of the passive joint. In addition, upon crossing critical values of amplitude or frequency, the system undergoes a bifurcation where the symmetric periodic solution loses stability and asymmetric solutions evolve, for which the swimmer moves along an arc. We analyze these phenomena using numerical simulations and analytical methods of perturbation expansion, harmonic balance, Floquet theory, and Hill’s determinant. The results demonstrate the important role of parametric excitation in stability and bifurcations of motion for flexible underactuated locomotion.

Two point boundary value problems for ordinary differential systems with generalized variable exponents operators

In recent years an increasing interest in more general operators generated by Musielak–Orlicz functions is under development since they provided, in principle, a unified treatment to deal with ordinary and partial differential equations with operators containing the p-Laplace operator, the ϕ-Laplace operator, operators with variable exponents and the double phase operators. These kind of consideration lead us in García-Huidobro et al. (2024), to consider problems containing the operator (S(t,u′))′, where ′=ddt and look for period solutions of systems of nonlinear systems of differential equations. In this paper we extend our approach to deal with systems of differential equations containing the operator (S(t,u′))′ this time under Dirichlet, mixed and Neumann boundary conditions. As in García-Huidobro et al. (2024) our approach is to work in C1 spaces to obtain suitable abstract fixed points theorems from which several applications are obtained, including problems of Liénard and Hartman type.

Discrete mechanical model for nonlinear dynamical analysis of planar guyed towers considering the unilateral contact of cables

Towers are widely used for power line transmission, wind power plants, TV and radio broadcasting, and telecommunications. To enhance their stability, cables are often employed to anchor these towers to the ground. In this study, we investigate the nonlinear static and dynamic responses of a planar guyed tower in which the unilateral constraints on the cables are considered. A representative discrete mechanical model with two degrees of freedom is developed to simulate the central mast of the tower, and the cables are modeled as unilateral springs with linear stiffness. The nonlinear equilibrium equations are derived using an energy approach that incorporates the dissipative forces, total potential, and kinetic energies into the Euler-Lagrange equations. Unilateral cable contact is directly included in the nonlinear equilibrium equation for the guyed tower, allowing for numerical analysis without the need to evaluate the contact point at each time or load step. Several numerical strategies are employed to obtain nonlinear static equilibrium paths, bifurcation diagrams, phase portraits, and Poincaré sections. Our analyses provide novel results for the influence of unilateral cable contact in nonlinear static and dynamic analysis, evaluating the effects of unilateral contact and prestressing on the results. A parametric analysis reveals that cable contact affects nonlinear oscillations, bifurcation, and stability. Our numerical results indicate that unilateral cable contact introduces less structural stiffness compared to bilateral contact, thereby significantly affecting the static and dynamic stability of a planar guyed tower. This is evidenced by a decrease in the static limit load and alterations in the bifurcation diagrams, where unilateral contact destroys the trivial solutions, leading to periodic and quasi-periodic solutions at low levels of vertical load.

Higher-order breather and interaction solutions to the [formula omitted]-dimensional Mel’nikov equation

In this paper, we construct the high-order breather and interaction solutions of the (3+1)-dimensional Mel’nikov equation using the KP hierarchy reduction approach and express them in a concise determinant form. Our solutions show that the two breathers, two periodic waves, and the hybrid mode of the breather and periodic wave are all mutually parallel. Furthermore, by examining the long wave limit of the periodic wave solutions, a variety of rational solutions (lumps) and mixed solutions are obtained. Notably, the interaction between the lump and breather is found to be elastic. These novel results provide deeper insights into the interactions among different solution types.