Existence Of Oscillatory Solutions Research Articles

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English

Stability and Oscillatory Behavior of the Solutions on a Class of Coupled Van der Pol-Duffing Equations with Delays

In the present paper, a class of coupled van der Pol-Duffing oscillators with a nonlinear friction of higher polynomial order model which involves time delays is investigated. The coefficients of the highest order of the polynomial determine the boundedness of the solutions. With special attention to the boundedness of the solutions and the instability of the unique equilibrium point of linearized system, some sufficient conditions to guarantee the existence of oscillatory solutions for the model are obtained based on the generalized Chafee's criterion. Convergence of the trivial solution is determined by the negative real part of eigenvalues of the linearized system. Examples are provided to demonstrate the reduced conservativeness for the parameters of the proposed results. The results obtained shown that the passive decay rate in the model affects the oscillatory frequency and amplitude. When a permanent oscillation occurred, time delays affect mainly oscillatory frequency and amplitude slightly.

ON OSCILLATION OF SOLUTIONS TO QUASI-LINEAR EMDEN – FOWLER TYPE HIGHER-ORDER DIFFERENTIAL EQUATIONS

Existence and behavior of oscillatory solutions to nonlinear equations with regular and singular power nonlinearity are investigated. In particular, the existence of oscillatory solutions is proved for the equation y(n) + P(x; y; y ; : : : ; y(n1))|y|k sign y = 0; n 2; k R; k 1; P ̸= 0; P C(Rn+1): A criterion is formulated for oscillation of all solutions to the quasilinear even-order differential equation y(n) + n1 i=0 aj(x) y(i) + p(x) |y|ksigny = 0; p C(R); aj C(R); j = 0; : : : ; n 1; k 1; n = 2m; m N; which generalizes the well-known Atkinsons and Kiguradzes criteria. The existence of quasi-periodic solutions is proved both for regular (k 1) and singular (0 k 1) nonlinear equations y(n) + p0 |y|ksigny = 0; n 2; k R; k 0; k ̸= 1; p0 R; with (1)np0 0: A result on the existence of periodic oscillatory solutions is formulated for this equation with n = 4; k 0; k ̸= 1; p0 0:

Oscillation of third order differential equation with damping term

We study asymptotic and oscillatory properties of solutions to the third order differential equation with a damping term $$x'''(t) + q(t)x'(t) + r(t)\left| x \right|^\lambda (t)\operatorname{sgn} x(t) = 0,{\text{ }}t \geqslant 0.$$ We give conditions under which every solution of the equation above is either oscillatory or tends to zero. In case λ ⩽ 1 and if the corresponding second order differential equation h″ + q(t)h = 0 is oscillatory, we also study Kneser solutions vanishing at infinity and the existence of oscillatory solutions.

Sturm theorems for second order linear nonhomogenous differential equations and localization of zeros of the solution

It is shown that Sturm theorems, formulated in the 1830’s ([1], [2], [3] and [4]) and valid for second order linear homogeneous differential equation L(y)≡y″+a(x)y′+b(x)y=0, could as well be formulated for the class of nonhomogeneous linear differential equations L(y)=f(x). Criteria for the existence of oscillatory solutions of nonhomogeneous equations, as well as more exact locations of the zeros are given.

Existence of oscillatory solutions of forced second order delay differential equations

In this paper, under weaker hypothesis, the global existence of oscillatory solutions is established for a forced second order nonlinear delay differential equation. The results of this paper generalize and improve some known results.

Existence of Oscillatory Solutions of Singular Nonlinear Differential Equations

Asymptotic properties of solutions of the singular differential equation(p(t)u′(t))′=p(t)f(u(t))are described. Here,fis Lipschitz continuous on ℝ and has at least two zeros 0 andL>0. The functionpis continuous on [0,∞) and has a positive continuous derivative on (0,∞) andp(0)=0. Further conditions forfandpunder which the equation has oscillatory solutions converging to 0 are given.

Dynamics for Nonlinear Difference Equation xn+1=(αxn−k)/(β+γxn−lp)

We mainly study the global behavior of the nonlinear difference equation in the title, that is, the global asymptotical stability of zero equilibrium, the existence of unbounded solutions, the existence of period two solutions, the existence of oscillatory solutions, the existence, and asymptotic behavior of non-oscillatory solutions of the equation. Our results extend and generalize the known ones.

Brane cosmology as a dynamical system

We investigate the qualitative dynamical properties of a brane model for the flat isotropic universe with a single matter component represented by a scalar field. We study the flat and quadratic potential. Three classes of behaviors of the scale factor are determined. In particular, in the case of the brane with dark negative radiation, via a fine tuning, the existence of oscillatory solutions is shown, which is not possible in the traditional flat FRW model.

Convergence of lindstedt series for the non linear wave equation

We prove the existence of oscillatory solutions of the nonlinear wave equation, under irrationality conditions stronger than the usual Diophantine one, by perturbative techniques inspired by the Lindstedt series method originally introduced in classical mechanics to study the existence of invariant tori in quasi-integrable Hamiltonian systems.

Oscillation and Global Attractivity in Stage-Structured Population Models

AbstractStage-structured models of population growth have been considered in the constant delay and state-dependent delay cases, when modeled by retarded functional differential equations. In the first case we settle a conjecture posed by Aiello and Freedman [1] by showing the existence of oscillatory solutions. In the second case, we show that under suitable criteria, all positive solutions tend to a global attractor.

Chaos and instability in a power system: Subharmonic-resonant case

The response of a single-machine quasi-infinite busbar system to the simultaneous occurrence of principal parametric resonance and subharmonic resonance of order one-half is investigated. By numerical simulations we show the existence of oscillatory solutions (limit cycles), period-doubling bifurcations, chaos, and unbounded motions (loss of synchronism). The method of multiple scales is used to derive a second-order analytical solution that predicts (a) the onset of period-doubling bifurcations, which is a precursor to chaos and unbounded motions (loss of synchronism), and (b) saddle-node bifurcations, which may be precursors to loss of synchronism.

Internal waves in a rotating stratified spherical shell: asymptotic solutions

Summary. Small amplitude oscillations of a rotating, density-stratified fluid bounded by a spherical shell are examined. No restrictions are placed on the thickness of the shell. The internal mode spectrum is examined in the complete rotation-stratification parameter range including the regime that is appropriate for a plausible stratification distribution in the Earth's fluid core. A mathematical model is derived in terms of an eigenvalue PDE of mixed type. The existence of oscillatory solutions is exhibited in the limits of no rotation and no stratification. The frequency spectrum is extended asymptotically away from these limiting cases. A reduction in the complexity of the PDE for modes oscillating at the inertial frequency is exploited. A variational formulation is constructed in which the stratification parameter is treated as an eigenvalue of the system for fixed wave frequency. The spectral information is again extended asymptotically away from these ‘accessible’ points. Although the PDE reduces to Laplace's tidal equations (LTE) only under stringent parameter restrictions, it is observed that aspects of the behaviour of low frequency LTE modes are reproduced in the general model.

Quasi‐adjoint third order difference equations: oscillatory and asymptotic behavior

In this paper, asymptotic properties of solutions of urn:x-wiley:01611712:media:ijmm806976:ijmm806976-math-0001 are investigated via the quasi‐adjoint equation urn:x-wiley:01611712:media:ijmm806976:ijmm806976-math-0002 A necessary and sufficient condition for the existence of oscillatory solutions of (E+) is given. An example showing that it is possible for (E+) to have only nonoscillatory solutions is also given.Corrigendum: Quasi-adjoint third order difference equations: oscillatory and asymptotic behaviordx.doi.org/10.1155/S0161171287000747

Oscillation, nonoscillation, and asymptotic behavior for third order nonlinear differential equations

This paper discusses the behavior of real-valued solutions to the equation $$y''' + p(t)y'' + q(t)y' + r(t)y^\mu = 0$$ (1.1) where p, q, r are continuous and real-valued on some half-line (t0, + ∞) and μ is the quotient of odd positive integers. Oriteria are obtained for the existence of nonoscillatory solutions, several stability theorems are proved, and the existence of oscillatory solutions is shown. Of primary concern are the two cases q(t) ≤ 0, r(t) > 0, and q(t)>0, r(t)>0. Some of the main techniques used involve comparison theorems for linear equations and results in the theory of second order nonlinear oscillations.

Existence of oscillatory solutions and asymptotic behavior for a class of third order linear differential equations

Existence of oscillatory solutions and asymptotic behavior for a class of third order linear differential equations

On the existence of oscillatory solutions in negative feedback cellular control processes

It is shown that the differential equation $$\frac{{d^3 Z}}{{dt^3 }} + (\alpha + \beta + \gamma )\frac{{d^2 Z}}{{dt^2 }} + (\alpha \beta + \beta \gamma + \gamma \alpha )\frac{{dZ}}{{dt}} + \alpha \beta \gamma Z = (1 + Z^m )^{ - 1}$$ has at least one periodic solution past the instability of the stationary state solution, Z=Z0, the unique real positive root of $$\beta \gamma Z = (1 + Z^m )^{ - 1}$$

The Existence of Oscillatory Solutions in the Field–Noyes Model for the Belousov–Zhabotinskii Reaction

The cerium ion catalyzed oxidation of malonic acid by bromate in a sulphuric acid solution is essentially the Belousov–Zhabotinskii reaction. Belousov [1] observed temporal oscillations in the concentrations of the cerium ions Ce(III) and Ce(IV) when the reagent was stirred. The quantitative model for the chemical mechanism suggested by Field and Noyes [3] is examined here and it is shown that the solutions of the model equations are oscillatory, of finite amplitude and must possess at least one periodic solution.

Concerning solutions of third order self-adjoint differential equations

This work is concerned with the behavior of solutions of third order self-adjoint differential equations. It is well known that products of solutions of a second order self-adjoint equation are solutions of the third order equation. A converse of this fact is established. Existence of oscillatory solutions and asymptotic behavior of nonoscillatory solutions is established. Separation and comparison theorems are obtained for such equations.

The Existence of Oscillatory Solutions for the Equation d 2 y/dt 2 + q(t)y r = 0, 0 < r < 1

The Existence of Oscillatory Solutions for the Equation d 2 y/dt 2 + q(t)y r = 0, 0 < r < 1