Analysis of Special Cases in the Study of Bifurcations of Periodic Solutions of the Ikeda Equation

SynopsisThe existence of periodic solutions is proved for first order vector ordinary and functional differential equations when the right-hand side satisfies a one-sided growth restriction of Wintner type together with some conditions of asymptotic nature. Special cases in the line of Landesman-Lazer and of Winston are explicited.

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Periodic Solutions for n-Species Lotka-Volterra Competitive Systems with Pure Delays

Chinese Journal of Mathematics ◽

10.1155/2015/856959 ◽

2015 ◽

Vol 2015 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Ahmadjan Muhammadhaji ◽

Rouzimaimaiti Mahemuti ◽

Zhidong Teng

Keyword(s):

Periodic Solutions ◽

Lyapunov Functional ◽

Global Attractivity ◽

Sufficient Conditions ◽

Coincidence Degree ◽

Degree Theory ◽

Positive Periodic Solutions ◽

Competitive Systems ◽

Volterra Systems ◽

Special Cases

We study a class of periodic general n-species competitive Lotka-Volterra systems with pure delays. Based on the continuation theorem of the coincidence degree theory and Lyapunov functional, some new sufficient conditions on the existence and global attractivity of positive periodic solutions for the n-species competitive Lotka-Volterra systems are established. As an application, we also examine some special cases of the system, which have been studied extensively in the literature.

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A Novel Approach to a Time-Dependent-Coefficient WBK System: Doubly Periodic Waves and Singular Nonlinear Dynamics

Complexity ◽

10.1155/2018/3158126 ◽

2018 ◽

Vol 2018 ◽

pp. 1-14 ◽

Cited By ~ 2

Author(s):

Bo Xu ◽

Sheng Zhang

Keyword(s):

Periodic Solutions ◽

Water Waves ◽

Boussinesq Equations ◽

Expansion Method ◽

Periodic Wave ◽

Time Dependent ◽

Hyperbolic Function ◽

Periodic Waves ◽

Novel Approach ◽

Special Cases

Under investigation in this paper is a more general time-dependent-coefficient Whitham-Broer-Kaup (tdcWBK) system, which includes some important models as special cases, such as the approximate equations for long water waves, the WBK equations in shallow water, the Boussinesq-Burgers equations, and the variant Boussinesq equations. To construct doubly periodic wave solutions, we extend the generalized F-expansion method for the first time to the tdcWBK system. As a result, many new Jacobi elliptic doubly periodic solutions are obtained; the limit forms of which are the hyperbolic function solutions and trigonometric function solutions. It is shown that the original F-expansion method cannot derive Jacobi elliptic doubly periodic solutions of the tdcWBK system, but the novel approach of this paper is valid. To gain more insight into the doubly periodic waves contained in the tdcWBK system, we simulate the dynamical evolutions of some obtained Jacobi elliptic doubly periodic solutions. The simulations show that the doubly periodic waves possess time-varying amplitudes and velocities as well as singularities in the process of propagations.

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Positive Periodic Solutions for Impulsive Functional Differential Equations with Infinite Delay and Two Parameters

Journal of Applied Mathematics ◽

10.1155/2014/751612 ◽

2014 ◽

Vol 2014 ◽

pp. 1-17 ◽

Cited By ~ 4

Author(s):

Zhenguo Luo ◽

Liping Luo ◽

Yunhui Zeng

Keyword(s):

Fixed Point ◽

Differential Equations ◽

Periodic Solutions ◽

Infinite Delay ◽

Functional Differential Equations ◽

Positive Periodic Solutions ◽

Krasnoselskii’S Fixed Point Theorem ◽

Special Cases ◽

Functional Differential ◽

Two Parameters

We apply the Krasnoselskii’s fixed point theorem to study the existence of multiple positive periodic solutions for a class of impulsive functional differential equations with infinite delay and two parameters. In particular, the presented criteria improve and generalize some related results in the literature. As an application, we study some special cases of systems, which have been studied extensively in the literature.

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Periodic Solutions for a Semi-Ratio-Dependent Predator-Prey System with Delays on Time Scales

Discrete Dynamics in Nature and Society ◽

10.1155/2012/928704 ◽

2012 ◽

Vol 2012 ◽

pp. 1-15 ◽

Cited By ~ 1

Author(s):

Xiaoquan Ding ◽

Gaifang Zhao

Keyword(s):

Periodic Solutions ◽

Time Scales ◽

Sufficient Conditions ◽

Coincidence Degree ◽

Functional Responses ◽

Predator Prey ◽

Prey System ◽

Special Cases ◽

Existence Of Periodic Solutions ◽

Ratio Dependent

This paper is devoted to the existence of periodic solutions for a semi-ratio-dependent predator-prey system with time delays on time scales. With the help of a continuation theorem based on coincidence degree theory, we establish necessary and sufficient conditions for the existence of periodic solutions. Our results show that for the most monotonic prey growth such as the logistic, the Gilpin, and the Smith growth, and the most celebrated functional responses such as the Holling type, the sigmoidal type, the Ivlev type, the Monod-Haldane type, and the Beddington-DeAngelis type, the system always has at least one periodic solution. Some known results are shown to be special cases of the present paper.

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Transition Curves for Nonhomogeneous Mathieu Equation

Volume 1: 20th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C ◽

10.1115/detc2005-85170 ◽

2005 ◽

Author(s):

D. Younesian ◽

E. Esmailzadeh ◽

R. Sedaghati

Keyword(s):

Differential Equation ◽

Numerical Simulation ◽

Periodic Solutions ◽

Dynamic Behavior ◽

Analytical Solutions ◽

Mathieu Equation ◽

Special Cases ◽

Transition Curves ◽

Support Motion

The generalized form of the non-homogeneous Mathieu differential equation is analyzed in this paper. This type of differential equation arises from dynamic behavior of a pendulum subjected to the butterfly support motion. The Lindstedt-Poincare’s technique is considered in order to obtain the analytical solutions. The transition curves in some special cases are presented and their related periodic solutions with periods of 2π and 4π are obtained. Numerical simulation is carried out for some typical points in ε-δ plane.

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On nearly-commensurable periods in the restricted problem of three bodies, with calculations of the long-period variations in the interior 2:1 case

Symposium - International Astronomical Union ◽

10.1017/s0074180900105480 ◽

1966 ◽

Vol 25 ◽

pp. 197-222 ◽

Cited By ~ 4

Author(s):

P. J. Message

Keyword(s):

Periodic Solution ◽

Periodic Solutions ◽

Large Amplitude ◽

Restricted Problem ◽

Small Amplitude ◽

Minor Planets ◽

Long Period ◽

Numerical Integrations ◽

Period Variations ◽

The Mean

An analytical discussion of that case of motion in the restricted problem, in which the mean motions of the infinitesimal, and smaller-massed, bodies about the larger one are nearly in the ratio of two small integers displays the existence of a series of periodic solutions which, for commensurabilities of the typep+ 1:p, includes solutions of Poincaré'sdeuxième sortewhen the commensurability is very close, and of thepremière sortewhen it is less close. A linear treatment of the long-period variations of the elements, valid for motions in which the elements remain close to a particular periodic solution of this type, shows the continuity of near-commensurable motion with other motion, and some of the properties of long-period librations of small amplitude.To extend the investigation to other types of motion near commensurability, numerical integrations of the equations for the long-period variations of the elements were carried out for the 2:1 interior case (of which the planet 108 “Hecuba” is an example) to survey those motions in which the eccentricity takes values less than 0·1. An investigation of the effect of the large amplitude perturbations near commensurability on a distribution of minor planets, which is originally uniform over mean motion, shows a “draining off” effect from the vicinity of exact commensurability of a magnitude large enough to account for the observed gap in the distribution at the 2:1 commensurability.

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Electron Beam Damage of Biomolecules Assessed by Energy Loss Spectroscopy

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100069855 ◽

1978 ◽

Vol 36 (3) ◽

pp. 61-69

Author(s):

M. Isaacson ◽

M.L. Collins ◽

M. Listvan

Keyword(s):

Electron Microscopy ◽

Radiation Damage ◽

Structural Integrity ◽

Analytical Electron Microscopy ◽

High Dose ◽

Dose Rates ◽

Special Cases ◽

Analytical Electron ◽

Atom Migration ◽

Beam Damage

Over the past five years it has become evident that radiation damage provides the fundamental limit to the study of blomolecular structure by electron microscopy. In some special cases structural determinations at very low doses can be achieved through superposition techniques to study periodic (Unwin & Henderson, 1975) and nonperiodic (Saxton & Frank, 1977) specimens. In addition, protection methods such as glucose embedding (Unwin & Henderson, 1975) and maintenance of specimen hydration at low temperatures (Taylor & Glaeser, 1976) have also shown promise. Despite these successes, the basic nature of radiation damage in the electron microscope is far from clear. In general we cannot predict exactly how different structures will behave during electron Irradiation at high dose rates. Moreover, with the rapid rise of analytical electron microscopy over the last few years, nvicroscopists are becoming concerned with questions of compositional as well as structural integrity. It is important to measure changes in elemental composition arising from atom migration in or loss from the specimen as a result of electron bombardment.

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Decoration technique and surface physics

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s042482010010932x ◽

1978 ◽

Vol 36 (1) ◽

pp. 436-437 ◽

Cited By ~ 1

Author(s):

H. Bethge

Keyword(s):

Diffusion Path ◽

Special Importance ◽

Surface Physics ◽

Step Structure ◽

Initial Stage ◽

Special Cases ◽

Atomic Surface ◽

The Mean ◽

Bulk Structure ◽

Decoration Technique

Besides the atomic surface structure, diverging in special cases with respect to the bulk structure, the real structure of a surface Is determined by the step structure. Using the decoration technique /1/ it is possible to image step structures having step heights down to a single lattice plane distance electron-microscopically. For a number of problems the knowledge of the monatomic step structures is important, because numerous problems of surface physics are directly connected with processes taking place at these steps, e.g. crystal growth or evaporation, sorption and nucleatlon as initial stage of overgrowth of thin films.To demonstrate the decoration technique by means of evaporation of heavy metals Fig. 1 from our former investigations shows the monatomic step structure of an evaporated NaCI crystal. of special Importance Is the detection of the movement of steps during the growth or evaporation of a crystal. From the velocity of a step fundamental quantities for the molecular processes can be determined, e.g. the mean free diffusion path of molecules.

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