Small Amplitude Periodic Solutions Research Articles

Small amplitude periodic solution of Hopf Bifurcation Theorem for fractional differential equations of balance point in group competitive martial arts

Abstract Differential equation modelling was earlier used to discover better and understand various biological phenomena and social problems. We hope to understand the stability of the system and the Hopf bifurcation based on the characteristic roots of the linear system. Because group competitive sports require participants to have certain competitive skills, those who do not have sports skills but want to develop into activities must receive training and specific training. Therefore, based on the research background, the article proposes a time-lag group competitive martial arts activity model with a time lag effect. Through delay differential equation theory and Hopf bifurcation theory, the stability of the equilibrium point and the existence of periodic solutions generated by the Hopf bifurcation caused by the ‘instability’ of the equilibrium point are discussed. Finally, the theoretical results are simulated and verified with the help of MATLAB software.

Research on Small Amplitude Periodic Solutions of Hopf Bifurcation Theorem of Fractional Differential Equation of Equilibrium Point of Group Competitive Wushu Activity

Research on Small Amplitude Periodic Solutions of Hopf Bifurcation Theorem of Fractional Differential Equation of Equilibrium Point of Group Competitive Wushu Activity

Integrability and zero-Hopf bifurcation in the Sprott A system

The first objective of this paper is to study the Darboux integrability of the polynomial differential systemx˙=y,y˙=−x−yz,z˙=y2−a and the second one is to show that for a>0 sufficiently small this model exhibits one small amplitude periodic solution that bifurcates from the origin of coordinates when a=0. This model was introduced by Hoover as the first example of a differential equation with a hidden attractor and it was used by Sprott to illustrate a differential equation having a chaotic behavior without equilibrium points, and now this system is known as the Sprott A system.

Oscillations induced by quiescent adult female in a model of wild aedes aegypti mosquitoes

Aedes aegypti (Ae. aegypti: mosquito) is a known vector of several viruses including yellow fever, dengue, chikungunya and zika. In the current paper, we present a delayed mathematical model describing the dynamics of Ae. aegypti. Our model is governed by a system of three delay differential equations modeling the interactions between three compartments of the Ae. aegypti life cycle (females, eggs and pupae). By using time delay as a parameter of bifurcation, we prove stability/switch stability of the possible equilibrium points and the existence of bifurcating branch of small amplitude periodic solutions when time delay crosses some critical value. We establish an algorithm determining the direction of bifurcation and stability of bifurcating periodic solutions. In the end, some numerical simulations are carried out to support theoretical results..

Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system

In this paper, we consider a diffusive Leslie-Gower predator-prey system with prey subject to Allee effect. First, taking into account the diffusion of both species, we obtain the existence of traveling wave solution connecting predator-free constant steady state and coexistence steady state by using the upper and lower solutions method. However, due to the singularity in the predator equation, we need construct a positive suitable lower solution for the prey density. Such a traveling wave solution can model the spatial-temporal process where the predator invades the territory of the prey and they eventually coexist. Second, taking into account two cases: the diffusion of both species and the diffusion of prey-only, we prove the existence of small amplitude periodic traveling wave train solutions by using the Hopf bifurcation theory. Such traveling wave solutions show that the predator invasion leads to the periodic population densities in the coexistence domain.

Predator interference in a Leslie–Gower intraguild predation model

The aim of this paper is to analyze general dynamics features of a new Intraguild Predation (IGP) model where the top predator feeds only on the mesopredator and also affects its consumption rate. Important dynamical aspects of the model are described. Specifically, we prove that the trajectories of the associated system are bounded and defined for all positive time; there is a trapping domain; there are open subsets of parameters, such that the system in the first octant has at most five equilibrium solutions and at most three of them are of co-existence. Here we characterize the existence of Hopf bifurcations and we prove that this model exhibits either one, two or three small amplitude periodic solutions which arise from a zero-Hopf bifurcation. We prove the existence of alternative stable states. Finally, some numerical computations have been given in order to support our analytical results. The importance of some parameters of the model is discussed, in particular the role of the interference rate. The numerical exploration suggests that the equilibrium biomass of each of the three species grows as the level of interference grows.

Generalized multi-hump wave solutions of Kdv-Kdv system of Boussinesq equations

The KdV-KdV system of Boussinesq equations belongs to the class of Boussinesq equations modeling two-way propagation of small-amplitude long waves on the surface of an ideal fluid. It has been numerically shown that this system possesses solutions with two humps which tend to a periodic solution with much smaller amplitude at infinity (called generalized two-hump wave solutions). This paper presents the first rigorous proof. The traveling form of this system can be formulated into a dynamical system with dimension 4. The classical dynamical system approach provides the existence of a solution with an exponentially decaying part and an oscillatory part (small-amplitude periodic solution) at positive infinity, which has a single hump at the origin and is reversible near negative infinity if some free constants, such as the amplitude and the phase shit of the periodic solution, are activated. This eventually yields a generalized two-hump wave solution. The method here can be applied to obtain generalized $2^k$-hump wave solutions for any positive integer $k$.

Selective Pyragas control of Hamiltonian systems

We consider a Newton system which has a branch (surface) of neutrally stable periodic orbits. We discuss sufficient conditions which allow arbitrarily small delayed Pyragas control to make one selected cycle asymptotically stable. In the case of small amplitude periodic solutions we give conditions in terms of the asymptotic expansion of the right hand side, while in the case of larger cycles we frame the conditions in terms of the Floquet modes of the target orbit as a solution of the uncontrolled system.

Dihedral molecular configurations interacting by Lennard-Jones and Coulomb forces

In this paper, we investigate nonlinear periodic vibrations of a group of particles with a planar dihedral configuration governed by the Lennard-Jones and Coulomb forces. Using the gradient equivariant degree, we provide a full topological classification of the periodic solutions with both temporal and spatial symmetries. In the process, we provide general formulae for the spectrum of the linearized system of equations describing the above configuration, which allows us to obtain the critical frequencies of the particles' motions. The obtained frequencies represent the set of all critical periods for small amplitude periodic solutions emerging from a given stationary symmetric orbit of solutions.

Invariant Algebraic Surfaces and Hopf Bifurcation of a Finance Model

Recently there are several works studying the finance model [Formula: see text] where [Formula: see text] and [Formula: see text] are positive parameters. The first objective of this paper is to show that this model exhibits one small-amplitude periodic solution emerging from a Hopf bifurcation at the equilibrium point [Formula: see text] and in the second one we show that this system does not have invariant algebraic surfaces for any value of the parameters.

Small-amplitude nonlinear periodic oscillations in a suspension bridge system

This paper deals with a coupled nonlinear beam-wave system, proposed by Lazer and McKenna, modeling a suspension bridge. By constructing a suitable Nash–Moser-type iteration scheme, the existence of small-amplitude nonlinear periodic oscillations, i.e., the existence of small-amplitude periodic solutions for the suspension bridge system, is obtained, and so is their local uniqueness.

Unique stationary distribution and ergodicity of a stochastic Logistic model with distributed delay

Dynamics of a stochastic Logistic model with distributed delay are considered. We first transfer a scalar stochastic Logistic model with strong kernel or weak kernel into an equivalent stochastic system through the linear chain technique. Then we obtain the sufficient and necessary conditions for extinction and persistence of the species with probability one. Moreover, in the case of persistence, we prove that there exists a unique stationary distribution by the Markov semigroups theory. The results show that, the stronger white noise results in the extinction of the species and the weaker white noise guarantees the existence of a unique stationary distribution, though for the deterministic model with strong kernel or weak kernel, the average delay may induce the existence of a group of small amplitude periodic solutions.

Periodic dynamics of a derivative nonlinear Schrödinger equation with variable coefficients

ABSTRACTWe study the existence and multiplicity of periodic waves with a nontrivial phase on the derivative nonlinear Schrödinger equation with a periodic coefficient. The existence of infinitely many periodic solutions with a nontrivial phase is proved by using the Poincaré–Birkhoff twist theorem and the method of averaging. The sequence of rotation numbers for large amplitude periodic solutions tends to infinity, while the one for small amplitude periodic solutions tends to a certain constant. Additionally, exact expressions of small amplitude periodic solutions are obtained by introducing a small parameter.

High-frequency instabilities of small-amplitude solutions of Hamiltonian PDEs

Generalizing ideas of MacKay, and MacKay and Saffman, a necessary condition for the presence of high-frequency (i.e., not modulational) instabilities of small-amplitude periodic solutions of Hamiltonian partial differential equations is presented, entirely in terms of the Hamiltonian of the linearized problem. With the exception of a Krein signature calculation, the theory is completely phrased in terms of the dispersion relation of the linear problem. The general theory changes as the Poisson structure of the Hamiltonian partial differential equation is changed. Two important cases of such Poisson structures are worked out in full generality. An example not fitting these two important cases is presented as well, using a candidate Boussinesq-Whitham equation.

Small amplitude periodic solutions of Klein–Gordon equations

We consider a class of nonlinear Klein–Gordon equations utt=uxx−u+f(u) and obtain a family of small amplitude periodic solutions, where the temporal and spatial period have different scales. The proof is based on a combination of Lyapunov–Schmidt reduction, averaging and Nash–Moser iteration.

Stability of small-amplitude periodic solutions near Hopf bifurcations in time-delayed fully-connected PLL networks

In this paper we investigate stability conditions for small-amplitude periodic solutions emerging near symmetry-preserving Hopf bifurcations in a time-delayed fully-connected N-node PLL network. The study of this type of systems which includes the time delay between connections has attracted much attention among researchers mainly because the delayed coupling between nodes emerges almost naturally in mathematical modeling in many areas of science such as neurobiology, population dynamics, physiology and engineering. In a previous work it has been shown that symmetry breaking and symmetry preserving Hopf bifurcations can emerge in the parameter space. We analyze the stability along branches of periodic solutions near fully-synchronized Hopf bifurcations in the fixed-point space, based on the reduction of the infinite-dimensional space onto a two-dimensional center manifold in normal form. Numerical results are also presented in order to confirm our analytical results.

Impulsive Steering Between Coexisting Stable Periodic Solutions With an Application to Vibrating Plates

Single-degree-of-freedom (single-DOF) nonlinear mechanical systems under periodic excitation may possess multiple coexisting stable periodic solutions. Depending on the application, one of these stable periodic solutions is desired. In energy-harvesting applications, the large-amplitude periodic solutions are preferred, and in vibration reduction problems, the small-amplitude periodic solutions are desired. We propose a method to design an impulsive force that will bring the system from an undesired to a desired stable periodic solution, which requires only limited information about the applied force. We illustrate our method for a single-degree-of-freedom model of a rectangular plate with geometric nonlinearity, which takes the form of a monostable forced Duffing equation with hardening nonlinearity.

Dynamics at Infinity, Degenerate Hopf and Zero-Hopf Bifurcation for Kingni–Jafari System with Hidden Attractors

To understand the complex dynamics of Kingni–Jafari system with hidden attractors, the first objective of this paper is to study the global dynamics, and give a complete description of the dynamics of Kingni–Jafari system at infinity by using the Poincaré compactification of a polynomial vector field in [Formula: see text]. The second objective of this paper is to prove the existence of periodic solutions in the Kingni–Jafari system by classical Hopf bifurcation and degenerate Hopf bifurcation. Moreover, based on averaging theory, a small amplitude periodic solution that bifurcates from a zero-Hopf equilibrium was derived in the Kingni–Jafari system. The theoretical analysis and simulations demonstrate the rich dynamics of the system.

Small amplitude periodic solutions in time for one-dimensional nonlinear wave equations

This paper is devoted to the construction of solutions for one-dimensional wave equations with Dirichlet or Neumann boundary conditions by means of a Nash-Moser iteration scheme, for a large set of frequencies.

Global dynamical aspects of a generalized Chen–Wang differential system

We study theoretically the global chaotic behavior of the generalized Chen–Wang differential system $$\begin{aligned} \dot{x} = y, \quad \dot{y} = z,\quad \dot{z} = -y - b x^2 - x z + 3 y^2 + a, \end{aligned}$$ where \(a,b \in {\mathbb {R}}\) are parameters and \(b\ne 0\). This polynomial differential system is relevant because is the first polynomial differential system in \({\mathbb {R}}^3\) with two parameters exhibiting chaotic motion without having equilibria. We first show that for \(a>0\) sufficiently small it can exhibit up to three small amplitude periodic solutions that bifurcate from a zero-Hopf equilibrium point located at the origin of coordinates when \(a=0\). We also show that the system exhibits two limit cycles emerging from two classical Hopf bifurcations at the equilibrium points \(({\pm }\sqrt{2a}, 0,0)\), for \(a>0, b=1/2\). We also give a complete description of its dynamics on the Poincare sphere at infinity by using the Poincare compactification of a polynomial vector field in \({\mathbb {R}}^3\), and we show that it has no first integrals neither in the class of analytic functions nor in the class of Darboux functions.