Special Periodic Solutions Research Articles

The Three-Body and n-Body Problems

This project explores the multi-body problem according to Newton's Laws of Gravitation in two dimensions. First, I will derive and numerically solve the differential equations which govern the motion of the three-body problem. Then, I will explore the chaotic nature of the system through a variety of demonstrations, plots, animations, and Explore windows. After, I will look at a few special periodic solutions to the three-body problem. Next, I will apply this system to astronomy, exploring the gravitational orbits that comprise our solar system. Finally, I will to generalize the three-body problem into a higher order n-body system (specifically, five bodies).

Solitons, lump and interactional solutions of the (3+1)-dimensional BLMP equation in incompressible fluid

Based on the Hirota bilinear method, we systematically investigate the (3+1)-dimensional Boiti-Leon-Manana-Pempinelli (BLMP) equation in incompressible fluids and main results include: (1) the formulas of N-kink-soliton solutions and the bound states of multi solitons are all presented, (2) the lump solution is derived by the positive quadratic function method, (3) the interactional solutions are given, i.e., one lump interacts with one- and two-kink-soliton, (4) some special periodic solutions are discussed, i.e., lump-periodic solutions and homoclinic breather solutions.

A new approach to mathematical modeling of chemical synapses

The purpose of this work is to study a new mathematical model of a ring neural network with unidirectional chemical connections, which is a singularly perturbed system of differential-difference equations with delay. Methods. A combination of analytical and numerical methods is used to study the existence and stability of special periodic solutions in this system, the so-called traveling waves. Results. The proposed methods make it possible to show that the ring system under study allows the number of stable traveling waves to increase with the number of oscillators in the network. Conclusion. In this article, we rethink and refine the previously proposed method of mathematical modeling of chemical synapses. On the one hand, it was possible to fully take into account the requirement of the Volterra structure of the corresponding equations and, on the other hand, the hypothesis of saturating conductivity. This makes it possible to observe the principle of uniformity: the new mathematical model is based on the same principles as the previously proposed model of electrical synapses.

Spatial structure of the non-integrable discrete defocusing Hirota equation.

In this paper, we investigate the spatial property of the non-integrable discrete defocusing Hirota equation utilizing a planar nonlinear discrete dynamical map method. We construct the periodic orbit solutions of the stationary discrete defocusing Hirota equation. The behavior of the orbits in the vicinity of the special periodic solution is analyzed by taking advantage of the named residue. We characterize the effects of the parameters on the aperiodic orbits with the aid of numerical simulations. A comparison with the non-integrable discrete defocusing nonlinear Schrödinger equation case reveals that the non-integrable discrete defocusing Hirota equation has more abundant spatial properties. Rather an interesting and novel thing is that for any initial value, there exists triperiodic solutions for a reduced map.

Strict Nonlinear Normal Modes of Systems Characterized by Scalar Functions on Riemannian Manifolds

For the study of highly nonlinear, conservative dynamic systems, finding special periodic solutions which can be seen as generalization of the well-known normal modes of linear systems is very attractive. However, the study of low-dimensional invariant manifolds in the form of nonlinear normal modes is rather a niche topic, treated mainly in the context of structural mechanics for systems with Euclidean metrics, i.e., for point masses connected by nonlinear springs. In our previous research [1], [16], [17] we recognized, however, that a very rich structure of periodic and low-dimensional solutions exist also within nonlinear systems such as elastic multi-body systems encountered in the biomechanics of humans and animals or of humanoid and quadruped robots, which are characterized by a non-constant metric tensor. This letter briefly discusses different generalizations of linear oscillation modes to nonlinear systems and proposes a definition of strict nonlinear normal modes, which matches most of the relevant properties of the linear modes. The main contributions are a theorem providing necessary and sufficient conditions for the existence of strict oscillation modes on systems endowed with a Riemannian metric and a potential field as well as a constructive example of designing such modes in the case of an elastic double pendulum.

Analytical solution of the Colombo top problem

The Colombo top is a basic model in the rotation dynamics of a celestial body moving on a precessing orbit and perturbed by a gravitational torque. The paper presents a detailed study of analytical solution to this problem. By solving algebraic equations of degree 4, we provide the expressions for the extreme points of trajectories as functions of their energy. The location of stationary points (known as the Cassini states) is found as the function of the two parameters of the problem. Analytical solution in terms of the Weierstrass and the Jacobi elliptic functions is given for regular trajectories. Some trajectories are expressible through elementary functions: not only the homoclinic orbits, as expected, but also a special periodic solution whose energy is equal to that of the first Cassini state (unnoticed in previous studies).

Calculating Floquet Multipliers for Periodic Solution of Non-smooth Dynamical System

In this paper, we will discuss how to calculate Floquet multipliers of non-smooth dynamical systems which are continuous or discontinuous. The saltation matrix of discontinuous system plays an important role when computing the Floquet multipliers. In addition, a special periodic solution, semi-limit cycle, will be presented. We find that the stability of this periodic solution cannot be analyzed by the classical Floquet theory. More investigation is needed to enrich the theory of Floquet multipliers to non-smooth systems.

New periodic wave solutions of a (3+1)-dimensional Jimbo–Miwa equation

A new three-wave method is efficient and well-developed approach to solve nonlinear partial differential equation. In this paper, a (3+1)-dimensional Jimbo–Miwa equation is investigated by using this approach. Some periodic wave solutions and kink solutions are obtained through the Hirota bilinear form. Furthermore, figures of some special periodic wave solutions and kink solutions are presented to illustrate the dynamical features of these solutions.

The Impulse-Refractive Mode in a Neural Network with Ring Synaptic Interaction

This paper considers a mathematical model of a ring neural network with an even number of synaptically interacting elements. The model is a system of scalar nonlinear differential-difference equations, the right-hand sides of which depend on a large parameter. The unknown functions being contained in the system characterize membrane potentials of neurons. It is of interest to search within the system for special periodic solutions, so-called impulse-refractory modes. Functions with odd numbers of the impulse-refraction cycle have an asymptotically large burst and the functions with even numbers are asymptotically small. For this purpose, two substitutions are made sequentially, making it possible to study of a two-dimensional system of nonlinear differential-difference singularly perturbed equations with two delays instead of the initial system. Further, as the large parameter tends to infinity, the limiting object is defined, which is a relay system of equations with two delays. Using the step-by-step method, we prove that the solution of a relay system with an initial function from a suitable class is a periodic function with required properties. Then, using the Poincare operator and the Schauder principle, the existence of a relaxation periodic solution of a two-dimensional singularly perturbed system is proven. To do this, we construct asymptotics of this solution, and then prove its closeness to the solution of the relay system. The exponential estimate of the Frechet derivative of the Poincare operator implies uniqueness of the solution of a two-dimensional differential-difference system of equations with two delays in the constructed class of functions and is used to justify the exponential orbital stability of this solution. Furthermore, with the help of reverse replacement, the proven result is transferred to the original system.

Solitary and periodic wave solutions of higher-dimensional conformable time-fractional differential equations using the ( frac{G'}{G},frac{1}{G} ) -expansion method

In this paper, the two variables ( frac{G'}{G},frac{1}{G} ) -expansion method is applied to obtain new exact solutions with parameters of higher-dimensional nonlinear time-fractional differential equations (NTFDEs) in the sense of the conformable fractional derivative. To clarify the veracity of this method, it is implemented in nonlinear (2+1)-dimensional time-fractional biological population (BP) model and nonlinear (3+1)-dimensional KdV–Zakharov–Kuznetsov (KdV–ZK) equation with time-fractional derivative. When the parameters take some special values, the solitary and periodic solutions are obtained from the hyperbolic and trigonometric function solutions.

Solitons on a Periodic Wave Background of the Modified KdV-Sine-Gordon Equation**Supported by the Natural Science Foundation of Zhejiang Province under Grant No. LZ15A050001, the National Natural Science Foundation of China under Grant No. 11675146, and Talent Fund and K. C. Wong Magna Fund in Ningbo University

The Bäcklund transformation(BT) of the mKdV-sG equation is constructed by introducing a new transformation. Infinitely many nonlocal symmetries are obtained in terms of its BT. The soliton-periodic wave interaction solutions are explicitly derived by the classical Lie-group reduction method. Particularly, some special concrete soliton and periodic wave interaction solutions and their behaviours are discussed both in analytical and graphical ways.

Explicit integration of the system of invariant relations for the case of M. Adler and P. van Moerbeke

For the general integrability case of M. Adler and P. van Moerbeke, invariant relations are obtained in which the rank of the momentum map is 1. Thereby, special periodic solutions generating the edges of the bifurcation diagram are defined. All phase variables are expressed in terms of a set of constants and one auxiliary variable, for which a differential equation integrable in elliptic functions time is given. An explicit expression for the characteristic exponent determining the type of special periodic solutions is presented, which makes it possible to study the character of stability of the obtained solution.

Periodic solutions of inhomogeneous Schrödinger flows into 2-sphere

In this paper,we consider the so called generalized inhomogeneousSchrödinger flows from a closed Riemann surface $M$ into thestandard 2-sphere $S^2$ associated with the energy functional givenby\begin{align*}E_{f,P}(u)=\int_M\left(\frac{1}{2}f|\nabla u|^2+P(u_3)\right)dV_g.\end{align*}We showed the existence of special periodic solutions to thegeneralized inhomogeneous Schrödinger flows from $M$ withconvolution symmetry (especially $M = S^2$) into $S^2$ when thefunction $f$ and $P$ satisfy certain conditions respectively.Especially, we show that the inhomogeneous Heisenberg spin chainsystem from a closed Riemann surface with convolution symmetryadmits some special periodic solutions if the couplingfunction $f$ satisfies some suitable conditions. We also prove thatthere exist an infinite number of special periodic solutions to theLandau-Lifshitz system with an external magnetic field from $S^2$into $S^2$.

Periodic solutions of travelling-wave type in circular gene networks

We consider circular chains of unidirectionally coupled ordinary differential equations which are mathematical models of artificial gene networks. We study the problems of the existence and stability of special periodic solutions, the so-called travelling waves, in these chains. We establish that the number of such periodic solutions grows unboundedly as the number of links in the chain grows. However, at most one of these travelling waves can be stable. We give an explicit algorithm for choosing the stable cycle.

Exact Solutions for the Nonlinear (2+1)-Dimensional Davey-Stewartson Equation Using the Generalized $(\frac{G'}{G})$-Expansion Method

In this article, we construct the exact traveling wave solutions of the nonlinear (2+1)-dimensional Davey-Stewartson equation (D-S) using the generalized $(\frac{G'}{G})$-expansion method which play an important role in mathematical physics. As a result, hyperbolic, trigonometric and rational function solutions with parameters are obtained. When these parameters are taken special values, the solitary and periodic solutions are derived from the hyperbolic and trigonometric function solutions respectively. New complex type traveling wave solutions to the nonlinear (2+1)-dimensional Davey-Stewartson equation were obtained with Liu's theorem.

Cell cycle dynamics: clustering is universal in negative feedback systems.

We study a model of cell cycle ensemble dynamics with cell-cell feedback in which cells in one fixed phase of the cycle S (Signaling) produce chemical agents that affect the growth and development rate of cells that are in another phase R (Responsive). For this type of system there are special periodic solutions that we call k-cyclic or clustered. Biologically, a k-cyclic solution represents k cohorts of synchronized cells spaced nearly evenly around the cell cycle. We show, under very general nonlinear feedback, that for a fixed k the stability of the k-cyclic solutions can be characterized completely in parameter space, a 2 dimensional triangle T. We show that T is naturally partitioned into k(2) sub-triangles on each of which the k-cyclic solutions all have the same stability type. For negative feedback we observe that while the synchronous solution (k = 1) is unstable, regions of stability of k ≥ 2 clustered solutions seem to occupy all of T. We also observe bi-stability or multi-stability for many parameter values in negative feedback systems. Thus in systems with negative feedback we should expect to observe cyclic solutions for some k. This is in contrast to the case of positive feedback, where we observe that the only asymptotically stable periodic orbit is the synchronous solution.

Formula omitted]-expansion method for the fifth-order forms of KdV–Sawada–Kotera equation

In this work we establish the traveling wave solution with parameters of the KdV–Sawada–Kotera equation by using the G′G-expansion method. This method is effectively employed. When the parameters are taken as special values, solitary, periodic and rational solutions are obtained.

Coexistence of stable ECM solutions in the Lang–Kobayashi system

The Lang‐Kobayashi system of delay differential equations describes the behavior of the complex electric field E and inversion N inside an external cavity semiconductor laser. This system has a family of special periodic solutions known as external cavity modes (ECMs). It is well known that these ECM solutions appear through saddle-node bifurcations, then lose stability through a Hopf bifurcation before new ECM solutions are born through a secondary saddle-node bifurcation. Employing analytical and numerical techniques, we show that for certain short external cavity lasers the loss of stability happens only after the secondary saddle-node bifurcations, which means that stable ECM solutions can coexist in these systems. We also investigate the basins of these ECM attractors.

Constructing of exact solutions to the KdV and Burgers equations with power-law nonlinearity by the extended [formula omitted]-expansion method

The extended G ′ G -expansion method is introduced to construct more general exact traveling wave solutions to the KdV and Burgers equations with any order nonlinear terms. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. When the parameters take up special values, new solitary, periodic and complex solutions are also derived from the traveling wave solutions.

Exact solutions for a class of integrable Hénon–Heiles-type systems

We study the exact solutions of a class of integrable Hénon–Heiles-type systems (according to the analysis of Bountis et al. [Phys. Rev. A 25, 1257 (1982)]). These solutions are expressed in terms of two-dimensional Kleinian functions. Special periodic solutions are expressed in terms of the well-known Weierstrass function. We extend some of our results to a generalized Hénon–Heiles-type system with (n+1) degrees of freedom.