Periodic Forcing Term Research Articles

A novel perturbation method to approximate the solution of nonlinear ordinary differential equation after being linearized to the Mathieu equation

To save the manipulation cost for seeking a higher order approximation to enhance the accuracy of analytic solution, the present paper develops a novel perturbation method by linearizing the nonlinear ordinary differential equation (ODE) with respect to a zeroth order solution in advance, where a weight factor splits the nonlinear terms into two sides of the ODE. Consequently, a series of linear ODEs are solved sequentially to obtain higher order approximate analytic solutions, and meanwhile the frequency can be determined explicitly by solving a frequency equation. When the nonlinear problems are linearized to the Mathieu equations endowing with periodic forcing terms, we develop a novel homotopy perturbation method to determine their solutions, and then provide accurate formulas for nonlinear oscillators. For Duffing oscillator as an example, the accuracy of frequency obtained by the linearized homotopy perturbation method can be raised to 10−8, and even for a huge value of nonlinear coefficient, the error is of the order 10−5. A numerical procedure is developed to implement the proposed method, where the computed order of convergence reveals a linear convergence that the accuracy of nth order approximate solution is better than 10−(n+1). The super- and sub-harmonic periodic solutions are exhibited for the forced Duffing equation.

Further properties of Stepanov--Orlicz almost periodic functions

We revisit the concept of Stepanov--Orlicz almost periodic functions introduced by Hillmann in terms of Bochner transform. Some structural properties of these functions are investigated. A particular attention is paid to the Nemytskii operator between spaces of Stepanov--Orlicz almost periodic functions. Finally, we establish an existence and uniqueness result of Bohr almost periodic mild solution to a class of semilinear evolution equations with Stepanov--Orlicz almost periodic forcing term.

Analytic approximate solutions of the cubic–quintic Duffing–van​ der Pol equation with two-external periodic forcing terms: Stability analysis

In the light of the potential applications in engineering, electronics, physics, chemistry, and biology, the current work applies several techniques to achieve analytic approximate and numerical solutions of the cubic–quintic Duffing–van der Pol equation. This equation represents a second-order ordinary differential equation with quintic nonlinearity and includes two external periodic forcing terms. A classical approximate solution involves the secular terms is obtained. Unfortunately, this traditional method does not enable us to ignore these secular terms. Additionally, along with the concept of the expanded frequency, a bounded approximate solution is achieved. The Homotopy perturbation method is utilized to obtain an approximate solution with an artificial frequency of the given system. Near the equilibrium points, in the case of the autonomous system, the linearized stability is accomplished. Furthermore, in the case of the non-autonomous system, by means of the multiple time scales, the stability analysis is effectuated, together with the resonance and the non-resonance cases. Numerical computations are performed to demonstrate, graphically, the perturbed solutions as well as the stability/instability regions. Various numerical solutions to initial–boundary value problems are deduced via a three-step finite difference scheme. These are plotted and discussed to show the chaotic nature of solutions.

Weighted Stepanov-Like Pseudo Almost Periodicity on Time Scales and Applications

By using the measure theory on time scales, we extend the notion of weighted Stepanov-like pseudo almost periodicity to time scales and study some of its basic properties. To illustrate our abstract results, we study the existence and uniqueness of weighted pseudo almost periodic solutions to some classes of nonautonomous dynamic equations involving weighted Stepanov-like pseudo almost periodic forcing terms on time scales.

Crystalline evolutions with rapidly oscillating forcing terms

We consider the evolution by crystalline curvature of a planar set in a stratified medium, modeled by a periodic forcing term. We characterize the limit evolution law as the period of the oscillations tends to zero. Even if the model is very simple, the limit evolution problem is quite rich, and we discuss some properties such as uniqueness, comparison principle and pinning/depinning phenomena.

A new oscillator with mega-stability and its Hamilton energy: Infinite coexisting hidden and self-excited attractors.

In this paper, we introduce an interesting new megastable oscillator with infinite coexisting hidden and self-excited attractors (generated by stable fixed points and unstable ones), which are fixed points and limit cycles stable states. Additionally, by adding a temporally periodic forcing term, we design a new two-dimensional non-autonomous chaotic system with an infinite number of coexisting strange attractors, limit cycles, and torus. The computation of the Hamiltonian energy shows that it depends on all variables of the megastable system and, therefore, enough energy is critical to keep continuous oscillating behaviors. PSpice based simulations are conducted and henceforth validate the mathematical model.

Chaos in a Periodically Perturbed Second-Order Equation with Signum Nonlinearity

We study the periodically perturbed Duffing-type equation [Formula: see text] and its damped counterpart [Formula: see text] The main feature of our model is the presence of a “signum term” in [Formula: see text] We prove the existence of infinitely many subharmonic solutions as well as the presence of chaotic dynamics for some [Formula: see text]-periodic forcing terms.

Emergence of oscillations in a simple epidemic model with demographic data

A simple susceptible–infectious–removed epidemic model for smallpox, with birth and death rates based on historical data, produces oscillatory dynamics with remarkably accurate periodicity. Stochastic population data cause oscillations to be sustained rather than damped, and data analysis regarding the oscillations provides insights into the same set of population data. Notably, oscillations arise naturally from the model, instead of from a periodic forcing term or other exogenous mechanism that guarantees oscillation: the model has no such mechanism. These emergent natural oscillations display appropriate periodicity for smallpox, even when the model is applied to different locations and populations. The model and datasets, in turn, offer new observations about disease dynamics and solution trajectories. These results call for renewed attention to relatively simple models, in combination with datasets from real outbreaks.

On the solvability of the periodically forced relativistic pendulum equation on time scales

Fil: Amster, Pablo Gustavo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Ciudad Universitaria. Instituto de Investigaciones Matematicas Luis A. Santalo. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matematicas Luis A. Santalo; Argentina

Random attractors for stochastic time-dependent damped wave equation with critical exponents

We study the asymptotic behavior of solutions of a stochastic time-dependent damped wave equation. With the critical growth restrictions on the nonlinearity $ f $ and the time-dependent damped term, we prove the global existence of solutions and characterize their long-time behavior. We show the existence of random attractors with finite fractal dimension in $ H^1_0(U)\times L^2(U) $. In particular, the periodicity of random attractors is also obtained with periodic force term and coefficient function. Furthermore, we construct the pullback random exponential attractors.

Chaos in periodically forced reversible vector fields

We discuss the appearance of chaos in time-periodic perturbations of reversible vector fields in the plane. We use the normal forms of codimension~$1$ reversible vector fields and discuss the ways a time-dependent periodic forcing term of pulse form may be added to them to yield topological chaotic behaviour. Chaos here means that the resulting dynamics is semi-conjugate to a shift in a finite alphabet. The results rely on the classification of reversible vector fields and on the theory of topological horseshoes. This work is part of a project of studying periodic forcing of symmetric vector fields.

Switching dynamics analysis of forest-pest model describing effects of external periodic disturbance.

A periodically forced Filippov forest-pest model incorporating threshold policy control and integrated pest management is proposed. It is very natural and reasonable to introduce Filippov non-smooth system into the ecosystem since there were many disadvantageous factors in pest control at fixed time and the threshold control according to state variable showed rewarding characteristics. The main aim of this paper is to quest the association between pests dynamics and system parameters especially the economical threshold ET, the amplitude and frequency of periodic forcing term. From the view of pest control, if the maximum amplitude of the sliding periodic solution does not exceed economic injury level(EIL), the sliding periodic solution is a desired result for pest control. The Filippov forest-pest model exhibits the rich dynamic behaviors including multiple attractors coexistence, period-adding bifurcation, quasi-periodic feature and chaos. At certain frequency of periodic forcing, the varying system initial densities trigger the system state switch between different attractors with diverse amplitudes and periods. Besides, parameters sensitivity analysis shows that the pest could be controlled at a certain level by choosing suitable parameters.

Complex Dynamics in a Love-Triangle Model with Single-Frequency and Multiple-Frequency External Forcing

In this paper, a new version of the nonlinear model with two women competing vehemently for an attractive man in a competitive love-triangle is proposed, which provides more insight into the dynamical behaviors of the complex love-triangle relationship. A notable and important feature of this model is that the effects of the competition, single-frequency and multiple-frequency external periodic forcing terms are introduced into the model, as environmental fluctuation on every individual in the love-triangle relationship. By analyzing the dynamical behaviors of the competitive love-triangle model through analytical techniques and numerical simulation, we find that under some simple parametric conditions, with the effects of external periodic forcing, the model gives rise to the appearance of various phenomena such as stability, quasi-limit cycles, and resonant responses. The results show that system parameters and external periodic forcing play a key role in the love-triangle relationship: system parameters can be the control parameters and their different appropriate values can produce different phenomena, while external periodic forcing can enhance or weaken the oscillations of the feeling density, and moreover, the interesting phenomenon called resonant response can occur under some suitable conditions. In this way, complex types of love feelings can be described and it is closer to the reality.

Nonlinear Acoustics: Blackstock–Crighton Equations with a Periodic Forcing Term

Blackstock–Crighton equations describe the motion of a viscous, heat-conducting, compressible fluid. They are used as models for acoustic wave propagation in a medium in which both nonlinear and dissipative effects are taken into account. In this article, a mathematical analysis of the Blackstock–Crighton equations with a time-periodic forcing term is carried out. For time-periodic data sufficiently restricted in size it is shown that a time-periodic solution of the same period always exists. This implies that the dissipative effects are sufficient to avoid resonance within the Blackstock–Crighton models. The equations are considered in a bounded domain with both non-homogeneous Dirichlet and Neumann boundary values. Existence of a solution is obtained via a fixed-point argument based on appropriate a priori estimates for the linearized equations.

On almost periodicity of solutions of second-order differential equations involving reflection of the argument

We study almost periodic solutions for a class of nonlinear second-order differential equations involving reflection of the argument. We establish existence results of almost periodic solutions as critical points by a variational approach. We also prove structure results on the set of strong almost periodic solutions, existence results of weak almost periodic solutions, and a density result on the almost periodic forcing term for which the equation possesses usual almost periodic solutions.

Uniform attractor of shallow arch motion under moving points load

We study the long time behavior of arches and membrane-like structures in the strong damping case. Our main goal is to incorporate moving points load, and to establish the existence of a non-empty, compact, uniform attractor for the associated semi-flows. The existence of such an attractor is established for almost periodic forcing terms. The method employs Kuratowski's measure of non-compactness. The presentation is mostly self-contained. Examples simulating a vehicular traffic across a long bridge are considered.

Pseudo almost periodic solutions for some parabolic evolution equations with Stepanov-like pseudo almost periodic forcing terms

The aim of this work is to study the existence and uniqueness of μ-pseudo almost periodic solutions to a class of semilinear parabolic evolution equations with inhomogeneous boundary conditions, under the assumption that the forcing terms of the equation are μ-pseudo almost periodic in the sense of Stepanov. We also provide a new composition result of μ-pseudo almost periodic functions in the sense of Stepanov. An example is given for illustration.

Crystalline evolutions in chessboard-like microstructures

We describe the macroscopic behavior of evolutions by crystalline curvature of planar sets in a chessboard-like medium, modeled by a periodic forcing term. We show that the underlying microstructure may produce both pinning and confinement effects on the geometric motion.

Nonlinear wave equation with damping: Periodic forcing and non‐resonant solutions to the Kuznetsov equation

AbstractExistence of non‐resonant solutions of time‐periodic type is established for the Kuznetsov equation with a periodic forcing term. The equation is considered in a three‐dimensional whole‐space, half‐space and bounded domain, and with both non‐homogeneous Dirichlet and Neumann boundary values. A method based on estimates for the corresponding linearization, namely the wave equation with Kelvin‐Voigt damping, is employed.

Behaviors of the modified nonlinear Schrödinger system in an inhomogeneous alpha helical protein

In this paper, under investigation is the modified nonlinear Schrödinger system, which could be used to describe the behaviors of solitons in an inhomogeneous alpha helical protein. We derive the travelling wave solutions of the system by transforming it to an ordinary differential equation. With those solutions, we analyze the stability of each equilibrium point through the phase-plane analysis. Upon the introduction of a periodic external forcing term, we observe two kinds of the chaotic motions, i.e., weak and developed chaos. We also find that the two chaotic motions can be transformed to each other when we change the strength of the external forcing term. Furthermore, we obtain the periodic motion of the system by balancing the external forcing term and nonlinear term.