External Periodic Perturbation Research Articles

Investigating the invariant solutions of (1+1)-dimensional Sawada–Kotera model using Lie symmetries analysis

Here attention is focused on the (1+1)-dimensional Sawada–Kotera (SK) model that is prominent in mathematical physics and engineering to analyze plasmas and coherent systems for communication. Our techniques offer fresh perspectives on the model’s attributes and structure, deepening our comprehension of the underlying dynamics. First, the SK model is reduced to ordinary differential equations by constructing the Lie symmetries and using the associated transformation. Graphs are used to build and display invariant solutions. This strategy has caused the revelation of novel constant solutions that have not been found in the previous works. We offer new understandings of nature and changes observed during the SK derivation by taking advantage of the Lie symmetries powerful tools. Next, the fluctuating layout of proposed framework is examined from several perspectives such as sensitivity and bifurcation analysis. We examined the bifurcation analysis of planar dynamical system by using bifurcation theory. We also include an external periodic perturbation term that breaks regular patterns in the perturbed dynamical system. Graphical structures are provided to display the invariant solutions. The sensitivity of the SK model is determined to be strong after sensitivity analysis under different initial conditions. These results are fascinating, fresh, and conceptually useful for understanding the suggested framework. In mathematics and the applied sciences, forecasting and learning about new technologies are greatly aided by the dynamic aspect of system processing.

Dynamic effects on traveling wave solutions of the space-fractional long-short-wave interaction system with multiplicative white noise

In this paper, the stochastic space-fractional long-short-wave interaction system (SF-LSWIS) with multiplicative white noise is considered. The stochastic exact solutions, including triangular function solutions, hyperbolic function solutions, and Jacobi elliptic function solutions, are obtained via the complete discriminant system for polynomial method. For this purpose, the traveling wave transformation is applied to carry out the ordinary differential equation of this model, which is also converted to a dynamical system. Thereafter, qualitative behavior and bifurcation of the phase portraits of the dynamical system are investigated using the qualitative theory of dynamical systems. The chaotic behaviors of the system considering external periodic perturbation are studied. Furthermore, the 3D surface, graphs of contour plots, and level curves of some exact solutions are depicted by choosing proper parameter values. The influence and effect of fractional derivatives and noise strength on the solutions are also studied.

Dynamics and optical solitons in polarization-preserving fibers for the cubic–quartic complex Ginzburg–Landau equation with quadratic–cubic law nonlinearity

In this paper, the cubic–quartic complex Ginzburg–Landau (CGL) equation is investigated by using the trial function method. The traveling wave hypothesis is applied to convert the CGL equation to an ordinary differential equation (ODE), which is equivalent to a dynamic system. The qualitative behavior, bifurcation of the phase portraits for this system is studied. Furthermore, the chaotic motions in system involving external periodic perturbation are considered. Some new optical solitons, such as Jacobian elliptic function solutions, for CGL equation with quadratic–cubic law nonlinearity are constructed using the complete discrimination system for polynomial method. Moreover, two-dimensional graphs, three-dimensional, corresponding contour and density plots for some acquired solutions are depicted to carry out the physical properties of optical pulse propagation.

Numerical and dynamical behaviors of nonlinear traveling wave solutions of the Kudryashov–Sinelshchikov equation

In this article, a mathematical model symbolizing numerical solutions of the nonlinear Kudryashov–Sinelshchikov (KS) equation has been investigated. For this aim, we use collocation finite element method with septic B-spline functions and appropriate initial and boundary conditions on uniform mesh points. The proficiency and effectiveness of the presented method are examined in four test examples, consisting; behavior of a single soliton, collision of two solitons, Maxwellian initial condition and growth of a wavy hole. L2 and L∞ error norms are used to validate practicality and reliability of our numerical algorithm. In addition, two invariants are calculated to determine the conservation properties of the presented algorithm and the numerical results have been shown to be unconditionally stable. Obtained numerical results have been illustrated with tables and graphics for easy visualization of properties of the problem modeled. Numerical calculations to solve such problems prove the accuracy and effectiveness of the finite element method. In collocation method, using B-spline functions gives rise to a technique that needs only the unknown parameters at certain node points to create the solution. Collocation method has two excellent advantages: establishing method does not include integrations and the resulting matrix system is banded with small band width. Consequently, B-splines associate with the collocation provide a simple solution procedure of linear and nonlinear partial differential equations (PDEs). After that, we have analyzed error estimation for the Galerkin finite element method. Furthermore, dynamical properties of traveling waves of the KS equation are investigated in absence and presence of external periodic perturbation. In absence of perturbation, periodic traveling wave solution of the KS equation are studied. On the other hand, a collection of various quasiperiodic wave features of the KS equation is presented in presence of periodic perturbation. Therefore, it can be said that the numerical technique is effective and valid in obtaining numerical solutions of such PDEs. In addition, the study on dynamical properties of the KS equation provides qualitative behavior of the corresponding wave features.

ДОСЛІДЖЕННЯ НЕРЕЗОНАНСНИХ КОЛИВАНЬ СИСТЕМИ «ТЕЛЕСКОПІЧНИЙ ГВИНТ – СИПКЕ СЕРЕДОВИЩЕ»

In the article it is substantiated the value of the angular speeds of rotation of the auger screw, which leads to the breakdown of its lateral vibrations. The dependences describing the law of change of amplitude or natural frequency at slowly variable length of the telescopic screw are deduced. Based on the Van der Paul’s method, in the developed system differential equations are obtained that determine the laws of change of amplitude and frequency of the wave process in the system of a telescopic propeller. It is established that for nonresonant oscillations for this system the main parameters of bending oscillations are a continuous flow of bulk medium - the screw does not depend on its small torsional oscillations and external periodic perturbation. The analysis of the given regression equations shows that to reduce the torque of the auger it is necessary to reduce the frequency of its rotation and the angle of the conveyor. The constructive diagram and the results of theoretical calculations for assessing the influence of constructive-kinematic parameters on the torque indicators of the telescopic screw conveyor are presented.

Robust control of a hub-spoke tethered formation system of microsatellites using Hamilton-Jacobi inequality

The problem of controlling a rotating hub-spoke tethered formation system in low Earth orbit is considered, in which microsatellites are located radially around the central spacecraft (hub) and connected to it by tethers (spokes). To analyze the dynamics of the tethered system, a mathematical model is developed in the orbital coordinate system by Lagrange method, in which the central spacecraft is regarded as a rigid body. In the proposed control scheme, the spin motion of the central body is regulated by the control moment, and tether deployment control law is proposed by robust approach, which is carried out by regulating the tether tensions and low thrusts acting on the microsatellites. The robustness and stability of the system are investigated using Lyapunov theory and Hamilton-Jacobi inequality, which is used to determine the robustness index of the control system. The results of numerical calculations are presented, which confirm that the proposed control scheme is effective when taking into account periodic gravitational perturbations, external perturbations and perturbations associated with uncertainty in the initial states of the system and with the rotation of the central body.

Periodic modes in an automatic control system with a three-position hysteresis relay

We consider an automatic control system with a three-position hysteresis relay and a continuous periodic external perturbation. The existence theorem is established that provide domains in the space of system parameters, including the parameters of the relay and perturbation, to which periodic modes having a pass to saturation zones with given periods and a specified configuration of phase trajectories correspond.

Study of Parametric Interactions in the Nuclear Reactor Control with Feedback

This article considers the principal theoretical possibility of regulating a nuclear power reactor under changing operating modes conditions when external periodic disturbances take place in conditions of changing the operating mode. By the external periodic perturbation a downward change in the conditions of the heat sink was meant. The magnitude of the changes was preliminarily calculated in such a way that the operating conditions of the power plant did not exceed the boundaries of the safe operation zone of the reactor. In the case of approaching the operation parameters to the critical ones, the heat sink was increased until the working conditions returned to their previous state. In this work the amplitude frequency response of a non-linearly enhanced system in the nuclear power plant operating conditions when non-linearly reacting to external periodic influences has been studied. The external cyclic disturbances effect produced on the reactor that initially existed under stationary operating conditions has been considered. The research was carried out by numerical simulation of the competition between processes occurring in a nuclear power plant and determined by the system’s reaction time and relaxation time while responding to periodic external influences. Calculations of the relaxation time dependence on the fixed frequency-revealing external influence’s temperature are presented. Also, the relaxation time dependence on the frequency of external influence at a fixed temperature for systems with various relaxation periods was calculated. It is determined that when the dependence between system temperature and the external influence time is calculated, there exists a wide range of possible frequency control. To evaluate the behavior of a nuclear power reactor under conditions of operating modes changes, a fundamental physical mathematical model of the reactor’s state under external harmonic influence is presented. It is based on the nonlinear Riccati equation. The external harmonic effect was simulated by changing the heat supply and heat removal conditions near the critical point.

Positron-acoustic traveling waves solutions and quasi-periodic route to chaos in magnetoplasmas featuring Cairns nonthermal distribution

Dynamics of the positron acoustic waves (PAWs) in magnetoplasmas following Cairns non-thermal distribution is studied on the frameworks of the Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations. The reductive perturbation technique is used to derive the KdV and mKdV equations. Bifurcations of positron acoustic traveling waves of these equations are addressed by employing the bifurcation theory of planar dynamical systems. It is found that the KdV equation supports compressive positron acoustic solitary waves (PASWs), while the mKdV equation supports both compressive and rarefactive PASWs. Using numerical simulations, effect of the nonthermal parameter ( $$\beta $$ ), temperature ratio of hot electron to hot positron ( $$\sigma $$ ), magnetic field ( $$\omega _{c_{1}}$$ ), ratio of hot electron to cold positron concentration ( $$\mu _{e}$$ ), and ratio of hot to cold positron concentration ( $$\mu _{p}$$ ) are discussed on the PASWs solutions of the KdV and mKdV equations. The criterion of chaos for these perturbed equations under the external periodic perturbation are obtained through quasi-periodic route to chaos. It is in fact shown that transition to chaos in our system depends on the frequency $$\omega $$ and the strength of the external periodic perturbation $$f_{0}$$ . These parameters control the dynamic behavior of the PAWs. The relevance of this work may be useful to understand the qualitative changes in the dynamics of perturbed PAWs appearing in auroral acceleration region as well as the astrophysical and laboratory plasma, where static external magnetic field and nonthermal parameter are present.

DIRECTIONS OF IMPROVING THE EFFICIENCY OF VIBRATION TECHNOLOGICAL MACHINES

The article considers the directions of increasing the efficiency of vibrating technological machines. The processing of information on the development of vibration technologies is quite diverse and is represented by numerous scientific and technical developments related to improving the efficiency of vibration machines and ensuring the reliability of their work. The analytical basis for previous research in the creation of new technology are mathematical models that reflect the properties of mechanical oscillatory systems with several degrees of freedom, which perform small oscillations under the action of a system of periodic external perturbations that create vibration fields of one configuration. The dynamic properties of the working bodies of machines in a detailed form are revealed as the distribution of the amplitudes of oscillations of the points of the working bodies. In many cases, this distribution is linear, which is due to the manifestations of the properties of the simplest movements of the working bodies. Approaches are proposed in which the possible structural mathematical modeling is realized on the idea that a linear mechanical oscillatory system with concentrated parameters and several degrees of freedom can be compared with the structural scheme of the automatic control system. Particular attention is paid to the study and evaluation of the possibility of new dynamic effects associated with the simultaneous action of several working bodies of machines, as well as - modes of dynamic damping of oscillations. An important role in ensuring such developments is played by areas of research focused on the development of methods of mathematical modeling. Based on the research it is shown that the vibration field of the vibrating technological machine is formed under the influence of several factors, which are determined by the simultaneous action of several force perturbations, asymmetry of inertial and elastic properties of the mechanical system, the presence of additional connections. The introduction of additional links in the structure of the mechanical oscillating system of the vibrating machine, can significantly affect the structure of the vibration field, providing the choice of conditions for rational organization of the technological process of vibration processing, such as vibration hardening, crushing, transportation, screening.

Superperiodicity, chaos and coexisting orbits of ion-acoustic waves in a four-component nonextensive plasma

Superperiodicity, chaos and coexisting orbits of ion-acoustic waves (IAWs) are studied in a multi-component plasma consisting of fluid ions, q-nonextensive cold and hot electrons and Maxwellian hot positrons. The significant impacts of the system parameters on superperiodic and nonlinear periodic IAWs are presented. Considering an external periodic perturbation various types of quasiperiodic and chaotic features for IAWs are studied in different parametric ranges through time series’ plots, phase spaces and Lyapunov exponents. It has been observed that there exist some coexisting orbits for IAWs. Coexisting orbits for IAWs in a classical electron–positron–ion plasma system are reported.

Rectified phase-matching equation for fiber mode converter gratings using two-mode interference

The resonance wavelength, where a fiber mode converter grating written using periodic external perturbations achieves phase matching, is both a critical design parameter and a device parameter. However, a method to precisely predict the resonance wavelength for any new fiber and grating writing apparatus has been missing so far. The missing link was the lack of direct experimental methods to estimate the modified intermodal phase after writing with external perturbations. The presented method can make this estimation from a single experiment, over a broad wavelength range, based on a novel mathematical connection between two-mode interference and mode conversion. Using the novel methods, experimentally measured resonance wavelengths for different pitch and irradiation conditions have been predicted within relative errors of 4×10-3.

Electron-exchange potential correction on dynamics of multidimensional ion acoustic waves in quantum plasmas

We describe the propagation of arbitrary amplitude ion acoustic waves with electron exchange-correlation effects for two-dimensional quantum plasmas by using the quantum hydrodynamic model. The evolution of nonlinear waves in such plasmas is described by deriving a pseudoenergy-balance like equation, involving a Sagdeev-type pseudopotential. The effects of the key plasma configuration parameters, viz., quantum diffraction, electron exchange-correlation, and the angle of propagation of the wave, on the periodic and solitonic characteristics are studied in detail by employing the concept of dynamical systems. Also, we extend our investigation by considering an external periodic perturbation in a modified pseudoforce model. It is found that the dynamics of nonlinear ion acoustic oscillations in quantum plasma support periodic and quasiperiodic behavior depending on the external pseudofrequency. The implications of our results may have relevance in various dense astrophysical environments as well as in laboratory plasmas.

Multistability and dynamical properties of ion-acoustic wave for the nonlinear Schrödinger equation in an electron–ion quantum plasma

Multistability and dynamical properties of quantum ion-acoustic (QIA) waves are examined in a quantum plasma consisting of warm ions and inertialess electrons. Using phase plane analysis, QIA periodic, a pair of solitonic and superperiodic wave solutions are reported. The physical parameters viz., ion–electron Fermi temperature ratio (ρ), quantum parameter (H), travelling wave velocity (V) and β affect significantly on these wave solutions. Considering an external periodic perturbation various types of QIA coexisting hidden attractors (periodic, quasiperiodic and chaotic attractors) are observed in parametric space through phase plots, time series plots and Lyapunov exponents. Coexisting QIA hidden attractors are reported for the first time in electron–ion quantum plasmas.

Bursting oscillation analysis and synergetic control of permanent magnet synchronous motor

The main purpose of this paper is to reveal the evolution mechanism of the bursting oscillation and suppress the bursting oscillation. The permanent magnet synchronous motor (PMSM) system is taken as a research object, and the case of the PMSM with periodic external load perturbation is considered. The first part in this paper is for the analysis of bursting oscillation. First, a mathematical model of the non-autonomous PMSM system with external load perturbation is established, and the frequency of the external load perturbation is set to be far less than the natural frequency of the PMSM system, so that the PMSM system has a fast-slow coupling effect. Then, the non-autonomous PMSM system with external load perturbation is transformed into a generalized autonomous PMSM system by taking the external load perturbation as a slow-varying parameter of the PMSM system. In order to obtain the bifurcation behaviors and different equilibrium types of the PMSM system, the time series diagram, the equilibrium point distribution curve that changes with slow-varying parameter, and the transformed phase portrait are analyzed. Finally, the evolution mechanism of bursting oscillation is revealed by analyzing the overlay of the equilibrium point distribution curve and the transformed phase portrait, and it is found that the change of the equilibrium type and the corresponding bifurcation behavior will cause the PMSM system to exhibit “periodic symmetrical subcritical Hopf bursting oscillation”. The second part focuses on the control of the bursting oscillation. First, a macro-variable is defined by using the synergetic control strategy, which is a linear combination of all state variables of the PMSM system. Then, the synergetic controller is designed based on the constraint that the macro-variable converges to the invariant manifold. When the macro-variable converges to the invariant manifold, the PMSM system is also stabilized to the equilibrium. In addition, in order to explore the influence of controller parameters, a large number of simulation experiments are carried out, and the relationship between the control parameters with the response speed of the PMSM system is obtained. Finally, the effectiveness of the synergetic control strategy is verified by changing the amplitude of the external load perturbation. The simulation results show that the synergetic control strategy has a continuous control law when the system has external load perturbations, and can effectively suppress the bursting oscillation phenomenon of the PMSM system, so that the PMSM system runs stably.

Bifurcation analysis of ion-acoustic waves in an adiabatic trapped electron and warm ion plasma

Bifurcation analysis of ion-acoustic waves in complex plasmas in the presence of adiabatic trapped electrons and warm ions is studied. Using bifurcation theory of dynamical structure, the Hamiltonian system inculpated electrostatic potential is derived. Effects of physical parameters, such as T and , are shown on the analytical solitary wave solution. The numerical results show that parameters T and affect significantly on nonlinear electrostatic solitary waves. By adding an external periodic perturbation to unperturbed Hamiltonian system, we investigate quasiperiodic structure of the perturbed Hamiltonian system.

Periodic Solutions of a System of Differential Equations with Hysteresis Nonlinearity in the Presence of Eigenvalue Zero

We study a system of ordinary differential equations of order n containing a nonlinearity of imperfectrelay type with hysteresis and external periodic perturbation. We consider the problem of existence of solutions with periods equal to (or multiple of) the period of external perturbation with two points of switching within the period. The problem is solved in the case where the collection of simple real eigenvalues of the matrix of the system contains an eigenvalue equal to zero. By a nonsingular transformation, the system is reduced to a canonical system of special form, which enables us to perform its analysis by the analytic methods. We propose an approach to finding the points of switching for the representative point of periodic solution and to the choice of the parameters of nonlinearity and the feedback vector. A theorem on necessary conditions for the existence of periodic solutions of the system is proved. Sufficient conditions for the existence of the required solutions are established. We also perform the analysis of stability of solutions by using the point mapping and the fixed-point method. We present an example that confirms the established results.

Nonplanar dust-acoustic waves and chaotic motions in Thomas Fermi dusty plasmas

The properties of linear and nonlinear nonplanar dust acoustic (DA) solitary waves and chaotic behavior are investigated in an unmagnetized Thomas Fermi dusty plasma, whose components are degenerate electrons, ions, and negatively charged inertial cold dust grains. A linear dispersion relation is obtained and solved numerically. It has been observed that linear excitation characteristics are influenced by radial distance r, geometric term ν, and ion-to-electron Fermi temperature ratio σi. We have also noted that the addition of a geometrical term in dispersion relation gives damping along the radial axis. A modified Korteweg-de Vries (KdV) equation is derived by employing the reductive perturbation technique, and its numerical solutions are obtained. The modified KdV equation is discussed for cold dust grains in planar and nonplanar frameworks. Upon the introduction of external periodic perturbation, the perturbed modified KdV equation is studied in planar geometry via some qualitative and quantitative approaches. The perturbed KdV equation can give rise to the periodic, quasiperiodic, and chaotic motions for DA waves. The strength of the external perturbation and dust concentration h play the major role of the switching parameter in the transition of dynamic motion. The developed chaos can be weakened with the variation of dust concentration h. It has been observed that the dust concentration affects the dynamics of DA waves in planar geometry which is an important observation in this study.

Resonant phenomena of elastic bodies that perform bending and torsion vibrations

The method of study of the influence of torsional oscillations of one-dimensional models of nonlinear elastic bodies, along which moves with a constant velocity continuous flow of inelastic homogeneous medium, into bending, is developed. It is believed that information on torsional oscillations is known from empirical studies. Based on the latter, the refined model of the dynamics of the process of the investigated object is constructed. The latter is a boundary value problem for nonlinear nonautonomous differential equations with partial derivatives. The imposed restrictions on power factors and the main parameters of torsional oscillations allow for the analytical study of the dynamics of the process to use the basic ideas of the asymptotic integration of equations with partial derivatives. With their help, we obtain a two-parameter set of solutions that describe the determinant parameters of bending vibrations of an elastic body. It is established that for the considered elastic body there can be resonance oscillations, which are caused not only by external factors, but also by internal – torsional oscillations. Regarding the law of the change in the basic parameters of the dynamics of the bending motion of an elastic body, its rotation around the vertical axis reduces the frequency of its own flexural oscillations of the body, and even small torsional oscillations cause an additional periodic action on the transverse. In connection with the above bending vibrations of the elastic body, which performs complex oscillations (torsion and bending), resonances are possible both at the frequency of the external periodic perturbation and at the frequencies of the torsional oscillations (internal resonances). The amplitude of the transition through the resonance: a) at the basic frequency of external perturbation takes less value for elastic bodies of greater flexural rigidity and for higher values of the relative motion of the medium; b) at the frequency of torsional oscillations for larger values of the angular velocity takes more importance; c) with “fast” transition through resonance at the frequency of external or internal perturbation is less than with “slow”. The obtained results can serve as the basis for the choice of operating parameters of elastic elements of machines that carry out complex oscillations.

Dynamical survey of the dual power Zakharov–Kuznetsov–Burgers equation with external periodic perturbation

We investigate the dynamical behavior of the newly introduced dual power Zakharov–Kuznetsov–Burgers equation. Using bifurcation theory of planar dynamical systems, we study bifurcations of traveling wave solutions of the dual power Zakharov–Kuznetsov–Burgers equation in presence and absence of viscosity (μ) effect. In presence of an external periodic perturbation, we discuss the periodic and chaotic motions of the perturbed dual power Zakharov–Kuznetsov–Burgers equation by analyzing phase projection analysis, time series analysis, Poincaré section and sensitivity analysis. The effect of viscosity (μ) plays an important role in the transition from chaotic motion to periodic motion of the perturbed dual power Zakharov–Kuznetsov–Burgers equation.