Narrow Results

In this paper, we consider the existence of periodic solutions of Lagrangian systems of the form $\frac{d}{dt}(\nabla \mathrm{\Phi }(\dot{u}(t)))+V_u(t,u(t))+\lambda G_u(t,u(t))=0$, where $\mathrm{\Phi }:{ℝ}^n\to [0,\infty )$ is a ${𝒢}$-function in the sense of Trudinger, $V,G:ℝ\times {ℝ}^n\to ℝ$ are $C^1$-smooth, $T$-periodic in the time variable $t$ and $\lambda$ is a real parameter. We prove the existence of a $T$-periodic solution $u_{\lambda }:ℝ\to {ℝ}^n$ for any sufficiently small $\left|\lambda \right|$, and show that the found solutions converge to a $T$-periodic solution of the unperturbed system if $\lambda$ tends to $0$. Let us stress that our theorem only makes a natural assumption on the potential $V$ and no additional assumption on $G$ except that it is continuously differentiable. Its proof requires to work in a rather unusual (mixed) Orlicz–Sobolev space setting, which bears several challenges.

Our main interest in this study is to obtain approximate time-dependent solutions of the Lotka–Volterra predator–prey model given the populations of both species at a certain time. We achieve this by considering an earlier scheme by Shinohara and Yamamoto and modifying it so as to address the initial conditions. An important feature of this scheme is that it treats the system’s period as an unknown variable, making it possible to calculate it as a side product. In this method, the Galerkin procedure is utilized to convert the problem to a system of nonlinear algebraic equations, where sine and cosine waves are chosen as the basis functions in order to ensure the periodicity of solutions. The resulting overdetermined system is then solved in a least squares sense, which yields the coefficients making up the Galerkin approximations. After a detailed presentation of the numerical scheme, we apply it to two example problems and evaluate the results according to several accuracy criteria.

Sufficient conditions are obtained for the existence and global exponential stability of a unique periodic solution of a class of impulsive tow-neuron networks with variable and unbounded delays. The approaches are based on Mawhin's continuation theorem of coincidence degree theory and Lyapunov functions.

In this paper, the modified Jacobi elliptic function method is applied for Salerno equation which describes the nonlinear discrete electrical lattice in the forbidden bandgaps. Dark and bright solitons are obtained. Also, periodic solutions and periodic Jacobi elliptic function solutions are reported. Moreover, for the physical illustration of the obtained solutions, three-dimensional and two-dimensional graphs are presented.

This paper gives an analytical study of dynamic behavior of the exact solutions of nonlinear Korteweg–de Vries equation with space–time local fractional derivatives. By using the improved $(\frac{G^{\prime }}{G})$-expansion method, the explicit traveling wave solutions including periodic solutions, dark soliton solutions, soliton solutions and soliton-like solutions, are obtained for the first time. They can better help us further understand the physical phenomena and provide a strong basis. Meanwhile, some solutions are presented through 3D-graphs.

In recent years, neutral-type differential-difference equations have been applied extensively in the field of engineering, and their dynamical behaviors are more complex than that of the delay differential-difference equations. In this paper, the equations used to describe a neutral-type neural network system of differential difference equation with two delays are studied (i.e. neutral-type differential equations). Firstly, by selecting ${\tau }_1$, ${\tau }_2$ respectively as a parameter, we provide an analysis about the local stability of the zero equilibrium point of the equations, and sufficient conditions of asymptotic stability for the system are derived. Secondly, by using the theory of normal form and applying the theorem of center manifold introduced by Hassard et al., the Hopf bifurcation is found and some formulas for deciding the stability of periodic solutions and the direction of Hopf bifurcation are given. Moreover, by applying the theorem of global Hopf bifurcation, the existence of global periodic solution of the system is studied. Finally, an example is given, and some computer numerical simulations are taken to demonstrate and certify the correctness of the presented results.

In this paper, based on proposed Riccati mapping technique, we investigated the soliton solutions of fourth-order dispersive nonlinear Schrödinger equation with nonlinearity of dual-power law. The various types of solitons solutions involving some parameters are constructed. These soliton solutions can be useful for understanding the physical nature of the waves spread in the dispersive medium. Furthermore, the Modulation Instability (MI) is discussed by standard linear-stability analysis that shows all achieved results are exact and stable. The movements of some achieved results were presented graphically by giving suitable values to parameters that provide easy understanding to the physical phenomenon of this dynamical model. The obtained results show the simplicity and efficiency of the current used approach.

In this study, several novel solutions of strain wave equation for micro-structured solids are obtained by using the powerful modified extended direct algebraic method. In micro-structured solids, the strain wave dynamical equation is utilized to model the wave propagation in micro-structured materials. We have profitably obtained exact traveling wave solutions, including solitons, solitary waves, kink and anti-kink solitary waves, periodic and rational solutions have been found. This approach can not only produce the same solution, but also find new solutions that we think other researchers have missed. These attained solutions help researchers for knowing the physical phenomena of this dynamical equation. The three-dimensional maps of some solutions obtained in this study are plotted through giving appropriate values of the parameters engaged in the solutions for knowing the physical interpretation. The computational work and obtained results demonstrate that the new extended technique, gives an efficient and direct mathematical tools for resolving the nonlinear models in engineering and mathematical physics.

The periodic solutions for the $(2+1)$-dimensional complex modified Korteweg–de Vries (cmKdV) equation are obtained by using the Darboux transformation (DT). Starting with a periodic seed, we get two types of periodic solutions that are respectively, induced by different properties of the parameter $l_2$, namely, the periodic line wave solutions for $l_2=\alpha \in ℝ$ and the breather solutions for $l_2=\alpha {\lambda }_1$. Especially, for each type, two different kinds of solutions can be obtained via choosing different constrains, namely, $c-2\gamma \ge 0$ or $c-2\gamma <0$. The detailed dynamical characteristics of these solutions are also analyzed, including period, velocity, extreme values, amputations and the rules of temporal evolutions.

The coupled nonlinear Schrödinger (CNLS) system is often used to describe physical problems such as wave propagation in a birefringent optical fiber. Under investigation in this paper is the nonlocal ${𝒫}{𝒯}$-symmetric reverse-space type of CNLS system, through constructing two types of Darboux transformation (DT), we will derive a new series of nonlocal analytic solutions: (1) Single-periodic and double-periodic solutions; (2) Four different combinations about dark and anti-dark solitons, as well as the bright and dark breathers on a periodic background. Moreover, the fundamental properties and dynamical behaviors of those solutions will be discussed.

The nonlinear coupled Schrödinger equation in fiber Bragg gratings is studied in this paper. The existence of soliton solutions and periodic solutions are proved by qualitative analysis, and exact solutions are given, as well as the parameter condition of each solution is described. Then the modulation instability (MI) analysis is carried out and the linear stability criterion is given. In particular, external perturbation terms are introduced to prove that the equation exists chaotic behaviors.

We show how to compute families of periodic solutions of conservative systems with two-point boundary value problem continuation software. The computations include detection of bifurcations and corresponding branch switching. A simple example is used to illustrate the main idea. Thereafter we compute families of periodic solutions of the circular restricted 3-body problem. We also continue the figure-8 orbit recently discovered by Chenciner and Montgomery, and numerically computed by Simó, as the mass of one of the bodies is allowed to vary. In particular, we show how the invariances (phase-shift, scaling law, and x, y, z translations and rotations) can be dealt with. Our numerical results show, among other things, that there exists a continuous path of periodic solutions from the figure-8 orbit to a periodic solution of the restricted 3-body problem.

By developing topological shooting methods, the existence of single-humped periodic solutions and homoclinic orbit for a class of fourth-order ordinary differential equations is obtained, under some general conditions. Using these strict mathematical conclusions to a model of the deflection patterns of elastic struts resting on elastic foundations, the existence of single-humped periodic solutions which have been found by asymptotical and numerical methods is determined. An estimation of the half-period of the periodic solutions is also given.

A system that consists of a damped oscillator impacting two immovable fenders has been considered in this paper. In the first part a method of analytical determination of the existence of periodic solutions to the equations of motion and a method of analysis of the stability of these solutions have been presented. The results of the computations carried out by means of these methods have been illustrated by a few examples. In the second part of the paper, the results of some numerical investigations have been presented. The goal of these studies was to determine, in which regions of parameters characterizing the system, the motion of the oscillator is periodic and stable.

In this paper, we consider the following nonlinear differential equation

This paper concerns the computation and local stability analysis of periodic solutions to semi-explicit differential algebraic equations with time delays (delay DAEs) of index 1 and index 2. By presenting different formulations of delay DAEs, we motivate our choice of a direct treatment of these equations. Periodic solutions are computed by solving a periodic two-point boundary value problem, which is an infinite-dimensional problem for delay DAEs. We investigate two collocation methods based on piecewise polynomials: collocation at Radau IIA and Gauss–Legendre nodes. Using the obtained collocation equations, we compute an approximation to the Floquet multipliers which determine the local asymptotic stability of a periodic solution. Based on numerical experiments, we present orders of convergence for the computed solutions and Floquet multipliers and compare our results with known theoretical convergence results for initial value problems for delay DAEs. We end with examples on bifurcation analysis of delay DAEs.

In this paper we study the existence, uniqueness and stability of periodic solutions for a two-neuron network system with or without external inputs. The system consists of two identical neurons, each possessing nonlinear feedback and connected to the other neuron via a nonlinear sigmoidal activation function. In the absence of external inputs but with appropriate conditions on the feedback and connection strengths, we prove the existence, uniqueness and stability of periodic solutions by using the Poincaré–Bendixson theorem together with Dulac's criterion. On the other hand, for the system with periodic external inputs, combining the techniques of the Liapunov function with the contraction mapping theorem, we propose some sufficient conditions for establishing the existence, uniqueness and exponential stability of the periodic solutions. Some numerical results are also provided to demonstrate the theoretical analysis.

A perturbation-incremental (PI) method is presented for the computation, continuation and bifurcation analysis of periodic solutions of nonlinear systems of delay differential equations (DDEs). Periodic solutions can be calculated to any desired degree of accuracy and their stabilities are determined by the Floquet theory. Branch switching at a period-doubling bifurcation is made simple by the present scheme as a parameter is simply increased from zero to a small positive value so that a solution on the new branch is obtained. Subsequent continuation of an emanating branch is also discussed. The advantage of the PI method lies in its simplicity and ease of implementation.

This paper considers stabilization of unstable periodic solutions of nonlinear systems. Based on differential geometry method, a nonlinear recursive delayed feedback controller is designed. The concept of γ dynamics is introduced and the stability of the periodic solution of the closed-loop system is proved rigorously. The proposed control method does not have the odd number limitation. Simulation results are also presented for validating the effectiveness of the proposed method.

A host-parasite model is proposed that incorporates a nonlinear incidence rate. Under the influence of multiple infectious attacks, the model admits bistable regions such that the infection dies out if initial states lie in one region, and the population and parasites coexist if initial states lie in the other region. It is also found that parasites can drive the population to extinction for suitable parameters. It is verified that the model has a saddle-node bifurcation, Hopf bifurcations and a cusp of codimension 2 or higher codimension. Stable limit cycles and unstable limit cycles are examined as the infection-reduced reproduction rate varies. It is shown that the model goes through the change of stages of infection extinction, infection persistence, infection extinction, and the extinction of both parasites and the population as the contact coefficient increases.