Bifurcation Theory Of Dynamical Systems Research Articles

Bifurcations and Exact Traveling Wave Solutions of the Celebrated Green–Naghdi Equations

In this paper, we study the bifurcations and exact traveling wave solutions of the celebrated Green–Naghdi equations by using the qualitative theory of differential equations and the bifurcation theory of dynamical systems. We obtain all possible phase portraits of bifurcations of the system under various conditions about the parameters associated with the planar dynamical system. Then we show the existence of traveling wave solutions including solitary wave solutions, blow-up solutions, periodic wave solutions and periodic blow-up solutions, and give their exact explicit expressions. These results can help to understand the dynamical behavior of the traveling wave solutions of the system.

SINGULAR PERIODIC WAVES OF AN INTEGRABLE EQUATION FROM SHORT CAPILLARY-GRAVITY WAVES

The effects of parabola singular curves in the integrable nonlinear wave equation are studied by using the bifurcation theory of dynamical system. We find new singular periodic waves for a nonlinear wave equation from short capillary-gravity waves. The new periodic waves possess weaker singularity than the periodic peakon. It is shown that the second derivatives of the new singular periodic wave solutions do not exist in countable number of points but the first derivatives exist. We show that there exist close connection between the new singular periodic waves and parabola singular curve in phase plane of traveling wave system for the first time.

Exact Solutions and Bifurcations in Invariant Manifolds for a Nonic Derivative Nonlinear Schrödinger Equation

Propagating modes in a class of nonic derivative nonlinear Schrödinger equations incorporating ninth order nonlinearity are investigated by the method of dynamical systems. Because the functions [Formula: see text] and [Formula: see text] in the solutions [Formula: see text], [Formula: see text] satisfy a four-dimensional integral system having two first integrals (i.e. the invariants of motion), a planar dynamical system for the squared wave amplitude [Formula: see text] can be derived in the invariant manifold of the four-dimensional integrable system. By using the bifurcation theory of dynamical systems, under different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions for this planar dynamical system can be given. Therefore, under some parameter conditions, solutions [Formula: see text] and [Formula: see text] can be exactly obtained. Thirty six exact explicit solutions of equation are derived.

Three kinds of periodic wave solutions and their limit forms for a modified KdV-type equation

A modified KdV-type equation is studied by using the bifurcation theory of dynamical system. By investigating the dynamical behavior with phase space analysis, all possible explicit exact traveling wave solutions including peakon solutions, kink and anti-kink wave solutions, blow-up wave solutions, smooth periodic wave solutions, periodic cusp wave solutions, and periodic blow-up wave solutions are obtained. When the first integral varies, we also show the convergence of the periodic wave solutions, such as the smooth periodic wave solutions converge to the kink and anti-kink wave solutions, the periodic cusp wave solutions converge to the peakon solution, the periodic blow-up wave solutions converge to the blow-up wave solution, the blow-up wave solutions converge to the blow-up wave solution, and the periodic blow-up wave solutions converge to the periodic blow-up wave solution.

Research on a Dynamic Master-Slave Cournot Triopoly Game Model with Bounded Rational Rule and Its Control

The oligopoly market is modelled by a new dynamic master-slave Cournot triopoly game model with bounded rational rule. The local stabile conditions and the stable region are got by the dynamical systems bifurcation theory. The dynamics characteristics of the system with the changes of the adjustment speed parameters are analyzed by means of bifurcation diagram, largest Lyapunov exponents, phase portrait, and sensitive dependence on initial conditions. Furthermore, the parameters adjustment method is used to control the complex dynamical behaviors of the systems. The derived results have some important theoretical and practical meanings for the oligopoly market.

Exact solutions for the quadratic mixed-parity Helmholtz–Duffing oscillator by bifurcation theory of dynamical systems

The dynamical behavior and exact solutions of the quadratic mixed-parity Helmholtz–Duffing oscillator are studied by using bifurcation theory of dynamical systems. As a result, all possible phase portraits in the parametric space are obtained. All possible explicit parametric representations of the bounded solutions (soliton solutions, kink and anti-kink solutions and periodic solutions ) are given. When parameters are varied, under different parametric conditions, various sufficient conditions guarantee the existence of the above solutions are given.

Periodic Wave Solutions and Their Limit Forms of the Modified Novikov Equation

The modified Novikov equationut-utxx+(b+1)u2ux=buuxuxx+u2uxxxis studied by using the bifurcation theory of dynamical system and the method of phase portraits analysis. The existences, dynamic properties, and limit forms of periodic wave solutions forbbeing a negative even are investigated. All possible exact parametric representations of the different kinds of nonlinear waves also are presented.

Explicit Wave Solutions and Qualitative Analysis of the (1 + 2)-Dimensional Nonlinear Schrödinger Equation with Dual-Power Law Nonlinearity

The (1 + 2)-dimensional nonlinear Schrödinger equation with dual-power law nonlinearity is studied using the factorization technique, bifurcation theory of dynamical system, and phase portraits analysis. From a dynamic point of view, the existence of smooth solitary wave, and kink and antikink waves is proved and all possible explicit parametric representations of these waves are presented.

PEAKON, PSEUDO-PEAKON, LOOP, AND PERIODIC CUSP WAVE SOLUTIONS OF A THREE-DIMENSIONAL 3DKP(2, 2) EQUATION WITH NONLINEAR DISPERSION

In this paper, we study the three-dimensional Kadomtsev-Petviashvili equation (3DKP(m, n)) with nonlinear dispersion for <i>m</i>=<i>n</i>=2. By using the bifurcation theory of dynamical systems, we study the dynamical behavior and obtain peakon, pseudo-peakon, loop and periodic cusp wave solutions of the three-dimensional 3DKP(2, 2) equation. The parameter expressions of peakon, pseudo-peakon, loop and periodic cusp wave solutions are obtained and numerical graph are provided for those peakon, pseudo-peakon, loop and periodic cusp wave solutions.

Bifurcations and exact traveling wave solutions for a class of nonlinear fourth-order partial differential equations

In this paper, we study the bifurcations and exact traveling wave solutions for a class of nonlinear fourth-order partial differential equations. Based on the method of bifurcation theory of dynamical systems, the nonlinear fourth-order partial differential equations are shown the existence and implicit expressions of peakon, loop, periodic cusp wave, unbounded wave and periodic wave solutions. In particular, the numerical simulations of the bounded wave solutions are given to show the correctness of our results.

Effects of Quadratic Singular Curves in Integrable Equations

In this paper, the effects of quadratic singular curves in integrable wave equations are studied by using the bifurcation theory of dynamical system. Some new singular solitary waves (pseudo‐cuspons) and periodic waves are found more weak than regular singular traveling waves such as peaked soliton (peakon), cusp soliton (cuspon), cusp periodic wave, etc. We show that while the first‐order derivatives of the new singular solitary wave and periodic waves exist, their second‐order derivatives are discontinuous at finite number of points for the solitary waves or at infinitely countable points for the periodic wave. Moreover, an intrinsic connection is constructed between the singular traveling waves and quadratic singular curves in the phase plane of traveling wave system. The new singular periodic waves, pseudo‐cuspons, and compactons emerge if corresponding periodic orbits or homoclinic orbits are tangent to a hyperbola, ellipse, and parabola. In particular, pseudo‐cuspon is proposed for the first time. Finally, we study the qualitative behavior of the new singular solitary wave and periodic wave solutions through theoretical analysis and numerical simulation.

Complexity analysis of the dual-channel supply chain model with delay decision

In the oligopoly e-commerce market, the oligarch retailers sell products through traditional channel, while others through both network and traditional channel in order to obtain greater profits. Instead of discussing classic Bertrand game model, which past studies have done, we considered dual-channel retailer who makes price decision through both in network channel and traditional channel. This paper used the bifurcation theory of dynamical system, considering dual-channel retailer who makes delay decision. We performed a numerical simulation on system with different conditions, and some complex phenomenons occured, such as bifurcation and chaos. The results showed that adopting price delay decision in tradition channel would make the system more stable. While, adopting price delay decision in network channel makes the system less stable. When the market is in chaotic state, the using of delay decision would have an effect on the system stability in either traditional or network channels. The system become stable from chaos and would return to chaotic again with the increasing of weight in past period. Some interesting phenomenons happened when dual-channel retailer adopted delay decision in both channels. The superposition of delay decision would make the system more complex. At last, we measured the system’s performance by using profit index. We analyzed the profits of different oligarchs when the system is in different states. When the system is in chaos, the total profits of the oligarchs are obviously less than that in a stable state. Adopting delay decision is a way to avoid profit loss when system is in chaotic period, but this requires the retailer has rich operational experience. That is because adopting delayed decision may not always enhance the competitive strength of oligarchs.

Traveling Wave Solutions for a Coupled KdV Equations

In this paper, we use a method in order to find exact explicit traveling solutions in the subspace of the phase space for coupled KdV equations. The key idea is removing a coupled relation for the given system so that the new systems can be solved. The existence of solitary wave solutions is obtained. It is shown that bifurcation theory of dynamical systems provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.

Dynamics of Bose-Einstein Condensates: Exact Representation and Topological Classification of Coherent Matter Waves

By using the bifurcation theory of dynamical systems, we present the exact representation and topological classification of coherent matter waves in Bose-Einstein condensates (BECs), such as solitary waves and modulate amplitude waves (MAWs). The existence and multiplicity of such waves are determined by the parameter regions selected. The results show that the characteristic of coherent matter waves can be determined by the “angular momentum” in attractive BECs while for repulsive BECs; the waves of the coherent form are all MAWs. All exact explicit parametric representations of the above waves are exhibited and numerical simulations support the result.

New Exact Solutions for High Dispersive Cubic-Quintic Nonlinear Schrödinger Equation

We study a class of high dispersive cubic-quintic nonlinear Schrödinger equations, which describes the propagation of femtosecond light pulses in a medium that exhibits a parabolic nonlinearity law. Applying bifurcation theory of dynamical systems and the Fan sub-equations method, more types of exact solutions, particularly solitary wave solutions, are obtained for the first time.

Bifurcations of Travelling Wave Solutions for the B(m,n) Equation

Using the bifurcation theory of dynamical systems to a class of nonlinear fourth order analogue of the B(m,n) equation, the existence of solitary wave solutions, periodic cusp wave solutions, compactons solutions, and uncountably infinite many smooth wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.

Bifurcation and exact travelling wave solutions for Gardner–KP equation

By using the bifurcation theory of dynamical systems, this paper researches the bifurcation and exact travelling wave solutions for Gardner–KP equation. As a result, exact parametric representations of all wave solutions, including solitary wave solution, periodic wave solution, kink (anti-kink) wave solution and breaking wave solution, are given.

BIFURCATION BEHAVIORS OF THE TWO-STATE VARIABLE FRICTION LAW OF A ROCK MASS SYSTEM

Based on the stability and bifurcation theory of dynamical systems, the bifurcation behaviors and chaotic motions of the two-state variable friction law of a rock mass system are investigated by the bifurcation diagrams based on the continuation method and the Poincaré maps. The stick-slip of the rock mass is formulated as an initial values problem for an autonomous system of three coupled nonlinear ordinary differential equations (ODEs) of first order. The results of linear stability analysis indicate that there is an equilibrium position in the rock mass system. Furthermore, numerical results of nonlinear analysis indicate that the equilibrium position loses its stability from a sup-critical Hopf bifurcation point, and then the bifurcating periodic motion evolves into chaotic motion through a series of period-doubling bifurcations with the decreasing of the control parameter. The stick-slip and chaotic motions evolve into infinity in the end with some unstable periodic motions.

Bifurcation analysis and dimension reduction of a predator-prey model for the L-H transition

The L-H transition denotes a shift to an improved confinement state of a toroidal plasma in a fusion reactor. A model of the L-H transition is required to simulate the time dependence of tokamak discharges that include the L-H transition. A 3-ODE predator-prey type model of the L-H transition is investigated with bifurcation theory of dynamical systems. The analysis shows that the model contains three types of transitions: an oscillating transition, a sharp transition with hysteresis, and a smooth transition. The model is recognized as a slow-fast system. A reduced 2-ODE model consisting of the full model restricted to the flow on the critical manifold is found to contain all the same dynamics as the full model. This means that all the dynamics in the system is essentially 2-dimensional, and a minimal model of the L-H transition could be a 2-ODE model.

Lie symmetries and exact solutions for a short-wave model

In this paper, the Lie symmetry analysis and generalized symmetry method are performed for a short-wave model. The symmetries for this equation are given, and the phase portraits of the traveling wave systems are analyzed using the bifurcation theory of dynamical systems. The exact parametric representations of four types of traveling wave solutions are obtained.