Approximate Periodic Solutions Research Articles

Sudden Transition from Equilibrium to Hybrid Chaos and Periodic Oscillations of the State-Dependent Round-Trip Delayed Nonsmooth Compound TCP with GRED Congestion Control System via HB-AFT

In this paper, the nonsmooth compound transmission control protocol (TCP) with the gentle random early detection (GRED) system with the state-dependent round-trip delay is investigated in detail. Uniqueness of the positive equilibrium is proved firstly. Then, the closed approximate periodic solutions in this state-dependent delayed nonsmooth compound TCP with the GRED model are obtained by employing the harmonic balance and alternating frequency/time domain (HB-AFT) method. Then, we compare the results generated by numerical simulations with those of the closed approximate expressions obtained by HB-AFT. It indicates that HB-AFT is simple, correct, and powerful for state-dependent delayed nonsmooth dynamical systems. Finally, we find the complicated dynamic: chaos. It is a grazing chaos with a hybrid property, i.e., where usually w oscillates at a very low frequency and q oscillates at a very high frequency. And, the route to chaos is a very rare route, i.e., the instantaneous and local transition of stable equilibrium to chaos. So, to the end of stability and good performance, we should adjust the parameters carefully to avoid the periodic and chaotic oscillations.

Stability Analysis of a Damped Nonlinear Wave Equation

The current manuscript is concerned with extracting an analytical approximate periodic solution of a damped cubic nonlinear Klein-Gordon equation. The Riemann-Liouville fractional calculus is utilized to obtain an analytic approximate solution. The Homotopy technique is absorbed in the multiple time-spatial scales. The approved scheme yields a generalization of the Homotopy equation; whereas, two different small parameters are adapted. The first parameter concerns with the temporal perturbation, simultaneously, the second one is accompanied by the spatial one. Therefore, the analytic approximate solution needs the two perturbation expansions. This approach conducts more advantages in handling the classical multiple scales method. Furthermore, the initial conditions are included throughout the multiple scale method to achieve a special solution of the governing equation of motion. The analysis ends up deriving two first-order equations within the extended variables and their actual solution is achieved. The procedure adopted here is very promising and powerful in managing similar numerous nonlinear problems arising in physics and engineering. Furthermore, the linearized stability of the corresponding ordinary Duffing differential equation is analyzed. Additionally, some phase portraits are shown.

Approximate Analytical Periodic Solutions to the Restricted Three-Body Problem with Perturbation, Oblateness, Radiation and Varying Mass

Against the background of a restricted three-body problem consisting of a supergiant eclipsing binary system, the two primaries are composed of a pair of bright oblate stars whose mass changes with time. The zero-velocity surface and curve of the problem are numerically studied to describe the third body’s motion area, and the corresponding five libration points are obtained. Moreover, the effect of small perturbations, Coriolis and centrifugal forces, radiative pressure, and the oblateness and mass parameters of the two primaries on the third body’s dynamic behavior is discussed through the bifurcation diagram. Furthermore, the second- and third-order approximate analytical periodic solutions around the collinear solution point L3 in two-dimensional plane and three-dimensional spaces are presented by using the Lindstedt-Poincaré perturbation method.

A Periodic Solution of the Newell-Whitehead-Segel (NWS) Wave ‎Equation via Fractional Calculus

The Newell-Whitehead-Segel (NWS) equation is one of the most significant amplitude equations with a wider practical applications in engineering and applied physics. It describes several line patterns; for instance, see lines from seashells and ripples in the sand. In addition, it has several applications in mathematical, chemical, and mechanical physics, as well as bio-engineering and fluid mechanics. Therefore, the current research is concerned with obtaining an approximate periodic solution of a nonlinear dynamical NWS wave model at three different powers. The fractional calculus via the Riemann-Liouville is adopted to calculate an analytical periodic approximate solution. The analysis aims to transform the original partial differential equation into a nonlinear damping Duffing oscillator. Then, the latter equation has been solved by utilizing a modified Homotopy perturbation method (HPM). The obtained results revealed that the present technique is a powerful, promising, and effective one to analyze a class of damping nonlinear equations that appears in physical and engineering situations.

A modified harmonic balance method to obtain higher-order approximations to strongly nonlinear oscillators

We propose a new method, namely, the modified harmonic balance method. This paper also analyses and offers the high-order approximate periodic solutions to the strongly nonlinear oscillator with cubic and harmonic restoring force. The existing harmonic balance method cannot be applied directly to such kind of nonlinear oscillators in the presence of forcing term. It is possible if we rewrite the original form of the nonlinear oscillators. If we do so, the results are valid only for small values of amplitude of the oscillation. Moreover, after applying the existing harmonic balance method, a set of complicated higher-order nonlinear algebraic equations are obtained. Analytical investigation of these equations is cumbersome especially when the amplitude of the oscillation is large. These limitations are removed in the proposed method. In addition, a suitable truncation principle has also been used in which the solution achieves better results than existing solutions. The approximate results agree well with numerically obtained exact solutions. Highly accurate results and a simple solution procedure are the advantages of this proposed method, which could be applied to other nonlinear oscillatory problems arising in nonlinear science and engineering.

Response Analysis of Nonlinear Free Vibration of Parallel-Plate MEMS Actuators: An Analytical Approximate Method

In this work, nonlinear free vibration of electrostatically actuated micromechanical system has been investigated. These nonlinear solutions are analytically derived. An undamped spring-mass model and dynamic equation of motion are employed for the parallel-plate microelectrostatic actuator. The first analytical approximate period and periodic solution of the mentioned problems are constructed with the help of Galerkin’s method and coupling of Galerkin’s with Newton’s method. In addition, the second and third analytical approximate solutions are also derived. The “exact” solutions are obtained from direct integration. In this work, the well-known static and dynamic pull-in parameters are discussed. Phase trajectory becomes non-periodic with the dynamic pull-in point. The approximate and “exact” solutions of period, phase trajectory, and time-dependent displacement of microelectrostatic actuators at various physical parameters are obtained and compared. It is found that the analytical approximate solutions are not only an explicit expression but also have excellent precision for a wide range of nonlinear vibration with different amplitudes.

Stability Analysis of a Parametric Duffing Oscillator

The Duffing oscillator represents an important example to describe, mathematically, the nonlinear behavior of several phenomena in physics and engineering. In view of its diverse applications in science and engineering, the current work investigates the stability analysis of the parametric Duffing oscillator. This equation represents a second-order ordinary differential equation with cubic nonlinearity and periodic coefficient. A coupling of the homotopy perturbation method (HPM) and Laplace transform (LT), an approximate solution is derived. The HPM is adapted to find another accurate approximate solution. The latter analysis reveals the exact solution of the cubic Duffing equation. In addition, an expanded frequency parameter is achieved to find another approximate periodic solution. Therefore, this method is exercised to govern the stability criteria of the problem. Finally, the multiple time scales with the HPM is used to judge the stability criteria. The analyses include the resonance as well as nonresonance cases. Numerical estimations are performed to confirm, graphically, the perturbed solutions together with the stability examination. It is shown that the damped parameter and the cubic stiffness parameter have a destabilizing influence. In contrast, the natural and parametric frequencies are of stabilizing influences.

Approximate periodic solution and qualitative analysis of nonnatural oscillators based on the restoring force

Non-natural oscillators are nonlinear conservative systems in which the kinetic energy is not a pure quadratic function of velocity and therefore, they do not possess a constant Hamiltonian function. They can be found in a lot of artificial systems and some predictive models for natural phenomena. The conventional form of their governing equation does not show the restoring force, which is important for analysis of the qualitative behavior of nonlinear oscillators and for obtaining their periodic solution. The purpose of this paper is to derive general solutions for the restoring force of non-natural systems and to demonstrate how the restoring force could be used to conduct qualitative analysis and derive periodic solutions. The general solutions for the restoring force were derived for two cases of non-natural oscillators namely: non-natural oscillators leading to (a) exact differential equation and (b) inexact differential equation. One example of each case of non-natural oscillator was solved and analysed using the general solutions for the restoring force and approximate analytical methods for periodic solution that rely on knowledge of the restoring force. The analysis showed that the general restoring force solutions and the continuous piecewise linearization method can be used to derive very accurate periodic solutions for non-natural oscillators.

Modeling and analysis of nonlinear spacecraft relative motion via harmonic balance and Lyapunov function

The harmonic balance method, an approximate method for the study of nonlinear oscillating systems described by ordinary nonlinear differential equations, is applied to the cubic polynomial approximation of the relative motion to develop approximate periodic solutions of the motion. Firstly, the cubic nonlinear terms neglected in the formulation of linear time invariant Clohessy-Wilshire (CW) equations are obtained from the original nonlinear equation of motion. Secondly, using the CW solutions as the generating solutions, harmonic balance technique is applied to the cubic nonlinear terms and correction terms are obtained. The new and improved harmonically linearized model contains CW equations and the correction terms. Using total energy as a Lyapunov function, the stability of the new model is determined. The additional terms in the new harmonically linearized spacecraft relative motion enabled us to obtain excellent approximate periodic solutions of the nonlinear equations as evidenced from the numerical simulation results.

A Generalization of Lindstedt–Poincaré Perturbation Method to Strongly Mixed-Parity Nonlinear Oscillators

In this paper, generalization for the Lindstedt–Poincare perturbation method for nonlinear oscillators to a class of strongly mixed-parity oscillating system is established. In this extended and enhanced approach, two new odd nonlinear oscillators are introduced in terms of the mixed-parity oscillator. By combining the analytical approximate solutions corresponding to the two new systems, the accurate approximate solutions of the original mixed-parity oscillator are obtained. Comparing with the existing methods such as the perturbation method, the new solution methodology for the singular nonlinear system introduced is not only simple, but the combinatory solution is straight forward and it yields very accurate and physically insightful solutions. Using two typical examples, we demonstrated that this new proposed approach is capable of establishing highly accurate approximate analytical frequency and periodic solutions for small as well as large amplitude of oscillation. The new analytical methodology established will potentially shed new insights to the physical interpretations of strongly nonlinear oscillators including optoelectronic oscillators, pendulums and spring-masses.

Stability and Approximate Analytical Periodic Solution of a Structurally Orthotropic Stringer Shell

Combined with real engineering, we mainly study the stability of the equilibrium state at the coordinate center of an orthotropic stringer shell system, and the approximate analytic periodic motion is presented around the studied equilibrium state. In addition the rationality of the obtained results is verified by numerical simulation. Furthermore we found that there are actually only four kinds of phase portrait of this type of shell.

A novelty to the nonlinear rotating Rayleigh–Taylor instability

This paper presents a novel approach for studying the nonlinear Rayleigh–Taylor instability (RTI). The system deals with two rotating superposed infinite hydromagnetic Darcian flows through porous media under the influence of a uniform tangential magnetic field. The field allows the presence of currents on the surface of separation. The appropriate linear governing equations are solved and confirmed with the corresponding nonlinear boundary conditions. A nonlinear characteristic of the surface deflection is deducted. Away from the traditional techniques of the stability analysis, the work introduces a new one. The analysis depends mainly on the homotopy perturbation method (HPM). To achieve an analytical approximate periodic solution of the surface deflection, the secular terms are removed. This cancellation resulted in well-known amplitude equations. These equations are utilised to achieve stability criteria of the system. Therefore, the stability configuration is exercised in linear as well as nonlinear approaches. The mathematical procedure adopted here is simple, promising and powerful. The method may be used to treat more complicated nonlinear differential equations that arise in science, physics and engineering applications. A numerical calculation is performed to graph the implication of various parameters on the stability picture. In addition, for more convenience, the surface deflection is depicted.

Analysis of harmonic vibration synchronization for a nonlinear vibrating system with hysteresis force

Harmonic vibration synchronization of the two excited motors is an important factor affecting the performance of the nonlinear vibration system driven by the two excited motors. From the point of view of the hysteresis force, the nonlinear dynamic models of the nonlinear vibration system driven by the two excited motors are presented for the analysis of the hysteresis force with the asymmetry. An approximate periodic solution for the nonlinear vibration system with the hysteresis force is investigated using the nonlinear models. The condition of harmonic vibration synchronization is theoretically analyzed using the rotor–rotation equations of the two excited motors in the nonlinear dynamic models and the stability condition of harmonic vibration synchronization also is theoretically analyzed using Jacobi matrix of the phase difference equation of two excited motors. Using Matlab/Simlink, harmonic vibration synchronization of the two excited motors and the stability of harmonic vibration synchronization for the nonlinear vibration system with the hysteresis force are analyzed through the selected parameters. Various synchronous processes of the nonlinear vibration system with the hysteresis force are obtained through the difference rates of the two excited motors (including the initial phase difference, the initial rotational speed difference, the difference of the motors parameters). It has been shown that the research results can provide theoretical basis for the design and research of the vibration system driven by the two-excited motors.

Higher-order approximate periodic solution for the oscillator with strong nonlinearity of polynomial type

In this paper the harmonic balance method (HBM) is adopted for solving a special group of oscillators with strong nonlinear damping and elastic forces. The nonlinearity is of polynomial type. The motion is described with a strong nonlinear differential equation, whose approximate solution is assumed as a suitable sum of trigonometric functions. To find the most convenient combination of trigonometric functions as the probe function is the most important part of this investigation. Introducing the procedure of equating the terms with the same order of the trigonometric functions to zero, the problem is transformed into solving a system of nonlinear algebraic equations. Solving these equations the parameters of the solution up to high-order approximation are obtained. In the paper, the suggested solving procedure is applied for two nonlinear oscillator problems: free vibrations of a restrained uniform beam carrying an intermediate lumped mass and of a particle on a rotating parabola. The obtained approximate analytic solutions are compared with the already published results and with the numerically obtained solution. The solution up to third-order approximation is calculated. It is proved that the HBM with the suggested function gives more accurate results than the previous applied ones. Besides, the difference between the third-order approximate analytical solution and the numerical one is negligible. The method works well for different, even high, values of initial amplitudes.

Approximate periodic solution for the large-amplitude oscillations of a simple pendulum

This paper presents approximate periodic solutions to the anharmonic (i.e. not harmonic or non-sinusoidal) response of a simple pendulum undergoing moderate- to large-amplitude oscillations. The approximate solutions were derived by using a modified continuous piecewise linearization method that enabled very accurate solutions to the pendulum oscillations for the entire range of possible amplitudes i.e. [Formula: see text]. The present solution method is very simple and can be used to obtain amplitude-frequency solutions as well as the displacement and velocity histories of the simple pendulum without the need for a complementary method. The purpose of this paper is to present simple and accurate approximate analytical solutions to the large-amplitude oscillations of the simple pendulum that can be applied by undergraduates.

Modified harmonic balance method for solving strongly nonlinear oscillators

In this paper, a modified harmonic balance method is applied to obtain higher-order approximate periodic solutions of strongly nonlinear oscillatory systems having a rational force. Approximate frequency and periodic solution are obtained. Frequency and solutions are analytically analyzed. Obtained results are compared with several authors and corresponding numerical results. Error analysis is carried out and found satisfactory results for the proposed method. The new technique also avoids the necessity of solving sets of equations with very complex nonlinearities numerically as in the classical harmonic balance method. Effectiveness of the method found from comparison with other articles. The method is illustrated by examples.

Метод гармонического баланса для отыскания приближённых периодических решений системы Лоренца

We consider the harmonic balance method for finding approximate periodic solutions of the Lorenz system. When developing software that implements the described method, the math package Maxima was chosen. The drawbacks of symbolic calculations for obtaining a system of nonlinear algebraic equations with respect to the cyclic frequency, free terms and amplitudes of the harmonics, that make up the desired solution, are shown. To speed up the calculations, this system was obtained in a general form for the first time. The results of the computational experiment are given: the coefficients of trigonometric polynomials approximating the found periodic solution, the initial condition, and the cycle period. The results obtained were verified using a high-precision method of numerical integration based on the power series method and described earlier in the articles of the authors.

Stability approach for periodic delay Mathieu equation by the He- multiple-scales method

In the present work, the version of homotopy perturbation included time-scales is applied to the governing equation of time-periodic delay Mathieu equation. Periodical structure for the amplitude of the zero-order perturbation is constructed. The stability analysis is accompanied by considering three-time-scales. Approximate periodic solutions are derived to the second accuracy of perturbations at the harmonic resonance case as well as at the non-harmonic resonance case. Stability conditions are derived in both cases. Numerical calculations have been done to illustrate the stability behavior at both resonance and non-resonance case. It is shown that the time-delay has a destabilizing influence. We note that the delayed of the parametric excitation has a great interested and application to the design of nuclear accelerators.

Continuous piecewise linearization method for approximate periodic solution of the relativistic oscillator

This paper presents an approximate periodic solution to the vibration of the relativistic oscillator using a novel analytical method called continuous piecewise linearization method. First, an equivalent conservative equation for the vibration of the relativistic oscillator was derived in a simple straightforward manner that elucidates the physical meaning of the conservative equation. The continuous piecewise linearization method was then applied to derive periodic solutions for the displacement and velocity of the relativistic oscillator based on the conservative equation. The results of the present method were compared with results of published methods and exact numerical solution and the maximum error of the present method was less than 0.002%. The model derivations and the solutions presented in this paper are considerably simple and very accurate and can be used to introduce the relativistic oscillator in relevant undergraduate courses on dynamics. Essentially, knowledge of freshman calculus is sufficient to comprehend and implement the continuous piecewise linearization method for the relativistic oscillator.

Bifurcation and stability analysis of a nonlinear rotor system subjected to constant excitation and rub-impact

Abstract This paper focuses on the mechanism of a complex bifurcation behavior caused by flight maneuvers in an aircraft rub-impact rotor system with Duffing type nonlinearity. The maneuver load during flight maneuvers may induce a rub-impact phenomenon, accompanied by complex nonlinear behaviors such as periodic, sub-harmonic and quasi-periodic motions, but the bifurcation mechanism is not so clear. In this study, the harmonic balance method combined with an alternating frequency/time domain procedure (HB-AFT method) is formulated and used to derive the approximate periodic solutions of the system. In conjunction with the arc-length continuation, the solution branches for both periodic 1-T motion (synchronous oscillation) and periodic 2-T motion (sub-harmonic oscillation) are traced. Then with the aid of the Floquet theory, the stabilities of the obtained periodic solutions are examined. The changes in the number or the stability of the solutions lead to the identification of the bifurcation points, which can be classified qualitatively into three types, i.e. Neimark-Sacker bifurcation point (NSBP), quasi-periodic Hopf bifurcation point (QPHBP) and saddle-node bifurcation point (SNBP). In addition, the constant excitation significantly affects the instability as well as the bifurcation of the rotor system. In the case of a smaller constant excitation, the rotation region with respect to the quasi-periodic motions may disappear, accompanied with a PBP (pitchfork bifurcation point) instead of the two NSBPs and one QPHBP. The bifurcation analysis in this paper provides deep insight into the mechanism of the complex nonlinear phenomenon induced by the constant excitation. The results obtained will also contribute to a better understanding of the nonlinear dynamic behaviors of aircraft rotor systems during flight maneuvers.