Periodic Hyperbolic Solutions Research Articles

New exact solitary waves for the Sasa-Satsuma model with variable coefficients

In this paper, we investigate the variable coefficients Sasa-Satsuma model, which can describe the propagation of a light pulse in a cylindrical fiber. We study this model and obtain rich solutions using two separate methods. We obtain analytical Weierstrass elliptic function solutions using the Weierstrass elliptic function expansion method. Some Jacobi elliptic function solutions are obtained using the modified Jacobi elliptic function expansion method. When the Jacobi elliptic function degenerates, we obtain the corresponding trigonometric, hyperbolic function solutions and periodic solutions. We also try to take the coefficients of the equation as some functions and obtain some more complicated exact solutions, which have not appeared in previous studies. Finally, we simulate some waveform diagrams of the solutions using the computer software Mathematica and obtain periodic waves, bright and dark soliton, double solitons and some complex periodic waves. With these waveform diagrams, we can observe the dynamical behavior of the solutions more clearly.

Numerical dynamics of integrodifference equations

Integrodifference equations are versatile models in theoretical ecology for the spatial dispersal of species evolving in non-overlapping generations. The dynamics of these infinite-dimensional discrete dynamical systems is often illustrated using computational simulations. This paper studies the effect of Nyström discretization to the local dynamics of periodic integrodifference equations having Hölder continuous functions over a compact domain as state space. We prove persistence and convergence for hyperbolic periodic solutions and their associated stable and unstable manifolds respecting the convergence order of the quadrature/cubature method.

New generalized ϕ6-model expansion method and its applications to the (3+1) dimensional resonant nonlinear Schrödinger equation with parabolic law nonlinearity

In this paper, a new generalized ϕ6-model expansion method is proposed for the first time to construct optical solitons and other solutions of the (3+1) dimensional resonant nonlinear Schrödinger equation with parabolic law nonlinearity, which describes the propagation of optical pulse in nonlinear optical fibers. The generalized Jacobi elliptic function solutions and Weierstrass elliptic function solutions have been found. In particular, hyperbolic function solutions and periodic solutions are also obtained.

Invariant Manifolds in the Hamiltonian–Hopf Bifurcation

We study the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter $$\nu $$. The eigenvalues of the linearized system are pure imaginary for $$\nu 0$$ (these are the same basic assumptions as found in the Hamiltonian–Hopf bifurcation theorem of the authors). For $$ \nu > 0 $$ the equilibrium has a two-dimensional stable manifold and a two-dimensional unstable manifold, but for $$ \nu < 0 $$ there are no longer stable and unstable manifolds attached to the equilibrium. We study the evolution of these manifolds as the parameter is varied. If the sign of a certain term in the normal form is positive then for small positive $$\nu $$ the stable and unstable manifolds of the system are either identical or must have transverse intersection. Thus, either the system is totally degenerate or the system admits a suspended Smale horseshoe as an invariant set. This happens at the Lagrange equilibrium point $${\mathcal {L}}_4$$ of the restricted three-body problem at the Routh critical value $$\mu _1$$. On the other hand if the sign of this term in the normal form is negative then for $$\nu =0$$ the stable and unstable manifolds persists and then as $$\nu $$ decreases from zero they detach from the equilibrium to follow a hyperbolic periodic solution.

Stability of periodic solutions of periodic systems of differential equations with a heteroclinic contour

A two-dimensional periodic system of differential equations with two hyperbolic periodic solutions is considered. It is assumed that heteroclinic solutions lie at the intersection of stable and unstable manifolds of fixed points; more precisely, the existence of a heteroclinic contour is assumed. I study the case in which stable and unstable manifolds intersect nontransversally at the points of at least one heteroclinic solution. There are various ways of nontransversally intersecting a stable manifold with an unstable manifold at the points of a heteroclinic solution. Earlier, in the works of L.P. Shilnikov, S.V. Gonchenko, B.F. Ivanov, et al., it was suggested that, at the points of nontransversal intersection of a stable and an unstable manifold, there is a tangency of no more than finite order. It follows from the works of these authors that there exist systems in which there are stable periodic solutions in the neighborhood of the heteroclinic contour. In this paper, heteroclinic contours are studied under the assumption that, at the points of nontransversal intersection of a stable and an unstable manifold at the points of the heteroclinic solution, the tangency is not a tangency of finite order. It is shown that a countable set of periodic solutions is situated in the neighborhood of such a heteroclinic contour the characteristic exponents of which are separated from zero.

Moduli of stability for heteroclinic cycles of periodic solutions

We consider $ C^2 $ vector fields in the three dimensional sphere with an attracting heteroclinic cycle between two periodic hyperbolic solutions with real Floquet multipliers. The proper basin of this attracting set exhibits historic behavior and from the asymptotic properties of its orbits we obtain a complete set of invariants under topological conjugacy in a neighborhood of the cycle. As expected, this set contains the periods of the orbits involved in the cycle, a combination of their angular speeds, the rates of expansion and contraction in linearizing neighborhoods of them, besides information regarding the transition maps and the transition times between these neighborhoods. We conclude with an application of this result to a class of cycles obtained by the lifting of an example of R. Bowen.

Stable Periodic Solutions of Periodic Systems of Differential Equations

An infinitely differentiable periodic two-dimensional system of differential equations is considered. It is assumed that there is a hyperbolic periodic solution and there exists a homoclinic solution to the periodic solution. It is shown that, for a certain type of tangency of the stable manifold and unstable manifold, any neighborhood of the nontransversal homoclinic solution contains a countable set of stable periodic solutions such that their characteristic exponents are separated from zero.

Boundary crisis for degenerate singular cycles

The term boundary crisis refers to the destruction or creation of a chaotic attractor when parameters vary. The locus of a boundary crisis may contain regions of positive Lebesgue measure marking the transition from regular dynamics to the chaotic regime. This article investigates the dynamics occurring near a heteroclinic cycle involving a hyperbolic equilibrium point E and a hyperbolic periodic solution P, such that the connection from E to P is of codimension one and the connection from P to E occurs at a quadratic tangency (also of codimension one). We study these cycles as organizing centers of two-parameter bifurcation scenarios and, depending on properties of the transition maps, we find different types of shift dynamics that appear near the cycle. Breaking one or both of the connections we further explore the bifurcation diagrams previously begun by other authors. In particular, we identify the region of crisis near the cycle, by giving information on multipulse homoclinic solutions to E and P as well as multipulse heteroclinic tangencies from P to E, and bifurcating periodic solutions, giving partial answers to the problems (Q1)–(Q3) of Knobloch (2008 Nonlinearity 21 45–60). Throughout our analysis, we focus on the case where E has real eigenvalues and P has positive Floquet multipliers.

New Exact Solutions of the System of Equations for the Ion Sound and Langmuir Waves by ETEM

This manuscript focuses attention on new exact solutions of the system of equations for the ion sound wave under the action of the ponderomotive force due to high-frequency field and for the Langmuir wave. The extended trial equation method (ETEM), which is one of the analytical methods, has been handled for finding exact solutions of the system of equations for the ion sound wave and the Langmuir wave. By using this method, exact solutions including the rational function solution, traveling wave solution, soliton solution, Jacobi elliptic function solution, hyperbolic function solution and periodic wave solution of this system of equations have been obtained. In addition, by using Mathematica Release 9, some graphical simulations were done to see the behavior of these solutions.

New solitary wave solutions of Maccari system

In this paper, we consider exact solutions of Maccari system. We tackle extended trial equation method and generalized Kudryashov method to find exact solutions of this system. Initially, we obtain some exact solutions such as rational function solution, travelling wave solution, soliton solution, Jacobi elliptic function solution, hyperbolic function solution and periodic wave solution of this system by using extended trial equation method. Also, we attain dark soliton solutions of this system by using generalized Kudryashov method. Finally, for suitable parameter values, we plot two and three dimensional graphics of imaginary and real values of some exact solutions that we found by using both methods.

Transversality for time-periodic competitive-cooperative tridiagonal systems

Transversality of the stable and unstable manifolds of hyperbolic periodic solutions is proved for tridiagonal competitive-cooperative time-periodic systems. We further show that such systems admit the Morse-Smale property provided that all the fixed points (of the corresponding Poincare map) are hyperbolic. The main tools used here are the integer-valued Lyapunov function, as well as the Floquet theory developed in [1] for general time-dependent tridiagonal linear systems.

Computational study of some nonlinear shallow water equations

Abstract The shallow water equations have wide applications in ocean, atmospheric modeling and hydraulic engineering, also they can be used to model flows in rivers and coastal areas. In this article we obtained exact solutions of three equations of shallow water by using $\frac{{G'}} {G} $-expansion method. Hyperbolic and triangular periodic solutions can be obtained from the $\frac{{G'}} {G} $-expansion method.

Hyperbolic tori in Hamiltonian systems with slowly varying parameter

This paper looks at a Hamiltonian system which depends periodically on a parameter. For each value of the parameter the system is assumed to have a hyperbolic periodic solution. Using the methods in KAM-theory it is proved that if the Hamiltonian is perturbed so that the value of the parameter varies with constant small frequency, then the nonautonomous system will have hyperbolic 2-tori in the extended phase space.Bibliography: 12 titles.

The [formula omitted]-expansion method for the nonlinear lattice equations

In this paper, we extended the G′G-expansion method to three well-known nonlinear lattice equations. With the aid of symbolic computation, we choose nonlinear lattice equations to illustrate the validity and advantages of the algorithm. This method could give many kinds of exact solutions including soliton solutions expressed by hyperbolic functions and periodic solutions expressed by trigonometric functions in a uniform way if solutions of these kinds exist. It is shown that the proposed algorithm is effective and can be used for many other nonlinear lattice equations in mathematical physics and applied mathematics.

Exact solitary solution of Schamel equation in plasmas with negative ions

A theoretical investigation is carried out for understanding the properties of the solitary solution in plasmas with negative ions. Schamel equation for a plasma consisting of electron, positive ions and negative ions has been derived by using the reductive perturbation method. The effects of negative ions and the density on the properties of the solitary solution is discussed. We make use of the extended mapping method and auxiliary equation to obtain the solution of Schamel equation. This solution includes the Jacobi elliptic function solutions, hyperbolic function solutions, rational solutions, and periodic wave solutions. Furthermore, we show that the incorporate negative ion effects in the nonlinear propagation of ion acoustic waves that are controlled by trapped electrons and the results of the solitary solution in plasmas with negative ions model are discussed.

Smoothness of Asymptotic Phase Revisited

Abstract It is well-known that solutions on the stable manifold of a hyperbolic periodic solution of an autonomous system of ordinary differential equations have an asymptotic phase which has the same order of smoothness as the vector field. In this paper we show if the system depends on a parameter that, in general, the asymptotic phase loses one order of smoothness in the parameter.

Network periodic solutions: full oscillation and rigid synchrony

We prove two results about hyperbolic periodic solutions in networks of systems of ODEs. First, we show that generically hyperbolic periodic solutions of network admissible systems of differential equations oscillate in each node if and only if the network is transitive. We can associate a polydiagonal Δ(Z(t)) with each hyperbolic periodic solution Z(t) as follows. The cell coordinates of a point in Δ(Z(t)) are equal if the corresponding cell coordinates of Z(t) are equal for all t; that is, the outputs from the two cells are synchronous. Second, we prove that Δ(Z(t)) is rigid (unchanged by small admissible perturbations) if and only if it is flow-invariant for all admissible vector fields.

New Exact Solutions of Two Nonlinear Physical Models

Abundant new exact solutions of the Schamel–Korteweg-de Vries (S-KdV) equation and modified Zakharov–Kuznetsov equation arising in plasma and dust plasma are presented by using the extended mapping method and the availability of symbolic computation. These solutions include the Jacobi elliptic function solutions, hyperbolic function solutions, rational solutions, and periodic wave solutions. In the limiting cases, the solitary wave solutions are obtained and some known solutions are also recovered.

Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems

We consider a singularly perturbed system with a normally hyperbolic centre manifold. Assuming the existence of a fast homoclinic orbit to a point of the centre manifold belonging to a hyperbolic periodic solution for the slow system, we prove an old and a new result concerning the existence of solutions of the singularly perturbed system that are homoclinic to a periodic solution of the system on the centre manifold. We also give examples in dimensions greater than three of Sil'nikov [16] periodic-to-periodic homoclinic orbits.

Exact solitary-wave solutions and periodic wave solutions for generalized modified Boussinesq equation and the effect of wave velocity on wave shape

By means of the undetermined assumption method, we obtain some new exact solitary-wave solutions with hyperbolic secant function fractional form and periodic wave solutions with cosine function form for the generalized modified Boussinesq equation. We also discuss the boundedness of these solutions. More over, we study the correlative characteristic of the solitary-wave solutions and the periodic wave solutions along with the travelling wave velocity’s variation.