Two-dimensional Dynamical System Research Articles

Qualitative analysis and modulation instability for the extended (3+1)-dimensional nonlinear Schrödinger equation with conformable derivative

The focus of the article studies the qualitative analysis and modulation instability for the extended (3+1)-dimensional nonlinear Schrödinger equation with conformable derivative. Firstly, the extended (3+1)-dimensional nonlinear Schrödinger equation with conformable derivative is transformed into two-dimensional planar dynamic system through complex exponential wave transformation. Secondly, phase portrait and its orbit of the system are analyzed through the theory of dynamical system. By using Maple software the phase portrait, Poincaré section and sensitivity analysis are compared and displayed. Finally, parameter conditions for stable systems are provided through modulation instability.

Dynamical behavior analysis and soliton solutions of the generalized Whitham–Broer–Kaup–Boussineq–Kupershmidt equations

This article investigates the dynamical behavior analysis and soliton solutions of the generalized Whitham–Broer–Kaup–Boussineq–Kupershmidt equations in fluid flow dynamics and plasma waves. Firstly, the generalized Whitham–Broer–Kaup–Boussineq–Kupershmidt equations are simplified into the ordinary differential equations. Secondly, two-dimensional planar dynamical system and its phase portraits are given by utilizing the theory of planar dynamical system analysis. Finally, the soliton solutions of the generalized Whitham–Broer–Kaup–Boussineq–Kupershmidt equations are presented by using the fourth-order complete discriminant system. In order to better reveal the soliton propagation phenomenon of the generalized Whitham–Broer–Kaup–Boussineq–Kupershmidt equations, some three-dimensional, two-dimensional, and contour graphs of the solutions were drawn.

An evolutive model of a boundedly rational consumer with changing preferences and reference group consumption

AbstractIn this paper, two dynamic models, recently proposed to describe the adaptive repeated choices of a boundedly rational consumer, are joined together. One considers a consumer adjusting the consumption level of a given good over time according to the observed discrepancy between expected and realized utility gain and modifies the utility function according to past consumption experience, also including saturation effects when past consumption is excessive. The other one considers the same adjustment mechanism with constant preferences but with a behavioral effect that introduces a tendency (or bias) to imitate a reference group of consumers. Merging these two models, a two-dimensional nonlinear dynamical system is obtained which describes consumers that decide their next period consumption of a given good by following two different (sometimes contrasting) criteria: their own utility maximization on the one side and imitation of a reference group of consumers on the other side. This leads to a greater uncertainty with respect to the model without the behavioral bias. Such uncertainty is studied through a numerical exploration of the long-run dynamics, guided by some global dynamical features of the nonlinear model, such as the folding action of the critical curves that characterize the behavior of the iterated noninvertible map and the singularities related to the presence of a vanishing denominator, namely focal points and prefocal curves. So, the aim of the paper is twofold: on the one side, it tries to contribute to the literature on the economic theory of boundedly rational consumers represented by evolutionary and behavioral approaches; On the other side, it tries to contribute to the recent literature about the global analysis of discrete dynamical systems characterized by contact bifurcations leading to the creation of complex topological structures of the attractors and their basins of attraction.

The dynamical behavior analysis of the fractional perturbed Gerdjikov–Ivanov equation

In this article, the fractional perturbed Gerdjikov–Ivanov equation is investigated. Firstly, the fractional perturbed Gerdjikov–Ivanov equation is transformed into an ordinary differential equation through traveling wave transformation. Secondly, using the trial method of rank homogeneous equation polynomials and the principle of homogeneous equilibrium, a two-dimensional planar dynamic system is presented and its bifurcation behavior is studied. Then, its two-dimensional phase portraits are drawn by using Maple software. Finally, disturbance factors are introduced into the planar dynamical system to study its chaotic behavior, and some two-dimensional phase portraits, three-dimensional phase portraits, Poincaré sections, and sensitivity analysis graphs of the perturbed system are plotted by using Maple software. The novelty lies in studying the dynamic behavior of the objective equation without the need for solving.

Qualitative analysis and optical solitons for the (1+1)-dimensional Biswas-Milovic equation with parabolic law and nonlocal nonlinearity

In this paper, the (1+1)-dimensional Biswas-Milovic equation with parabolic law and nonlocal nonlinearity is studied. Firstly, the Biswas-Milovic equation with parabolic law and nonlocal nonlinearity is transformed into ordinary differential equation through traveling wave transformation. Secondly, two-dimensional planar dynamic system is given by using the trial method of polynomial for rank homogeneous equations together with the principle of homogeneous balance. Moreover, the phase portrait of the dynamical system is given. Meanwhile, the phase portrait, Poincaré section and sensitivity analysis of its perturbation system are drawn by mathematical software. Finally, the optical soliton solutions of the Biswas-Milovic equation with parabolic law and nonlocal nonlinearity are constructed by using the complete discrimination system method.

Chaotic pattern and traveling wave solution of the perturbed stochastic nonlinear Schrödinger equation with generalized anti-cubic law nonlinearity and spatio-temporal dispersion

The main object of this paper is to study the bifurcation, chaotic pattern and traveling wave solution of the perturbed stochastic nonlinear Schrödinger equation with generalized anti-cubic law nonlinearity and spatio-temporal dispersion. A traveling wave transformation is used to simplified the perturbed stochastic nonlinear Schrödinger equation into ordinary differential equation. The dynamic behavior of two-dimensional planar dynamical systems and their perturbed systems are studied, and bifurcation, phase portrait, and Poincaré section are presented. Furthermore, traveling wave solutions included Jacobian function solutions, trigonometric function solutions and hyperbolic function solutions are constructed.

Differential recycling strategies and government intervention in a closed-loop supply chain

Previous literature has discussed recycling strategy and government intervention in a closed-loop supply chain; however, differential recycling strategies considering the bounded rationality of decision-makers are unexplored. This study establishes evolutionary game models with a two-dimensional dynamical system to analyze the long-term recycling behavior of enterprises in a supply chain, which consists of manufacturers and retailers. The evolutionarily stable strategy (ESS) of enterprises is derived, and the government intervention is investigated. The enterprises choose from two recycling strategies: (1) ordinary recycling strategy, i.e., only recycling high-quality waste electrical and electronic equipment (WEEE); (2) differentiated recycling strategy, i.e., recycling high- and low-quality WEEE. The results show that when the recycling difficulty of enterprises is at different thresholds, the system will evolve into a different ESS. Moreover, the government's intervention is effective. If government subsidies and penalties meet a certain range, the enterprises without recycling qualifications will choose the ordinary recycling strategy, while the enterprises with recycling qualifications will choose the differentiated recycling strategy. Finally, when enterprises make decisions on price and recycling rate, the recycling difficulty and consumer environmental awareness will affect the decisions and profits of enterprises. However, only government intervention can effectively affect the ESS.

Planar Bistable Structures Detection via the Conley Index and Applications to Biological Systems

Bistability is a ubiquitous phenomenon in life sciences. In this paper, two kinds of bistable structures in two-dimensional dynamical systems are studied: one is two one-point attractors, another is a one-point attractor accompanied by a cycle attractor. By the Conley index theory, we prove that there exist other isolated invariant sets besides the two attractors, and also obtain the possible components and their configuration. Moreover, we find that there is always a separatrix or cycle separatrix, which separates the two attractors. Finally, the biological meanings and implications of these structures are given and discussed.

Soliton solutions and dynamics analysis of fractional Radhakrishnan–Kundu–Lakshmanan equation with multiplicative noise in the Stratonovich sense

The main objective of this paper is to study the soliton solutions and dynamics analysis of the fractional Radhakrishnan–Kundu–Lakshmanan equation with multiplicative noise in the Stratonovich sense. Firstly, the wave transformation is used to obtain the nonlinear ordinary differential equation, and then the nonlinear ordinary differential equation is transformed into a two-dimensional plane dynamic system with a Hamiltonian system. Secondly, the phase portrait and sensitivity of the plane dynamic system and its perturbed system are studied using Maple software. Thirdly, the soliton solutions of the stochastic fractional Radhakrishnan–Kundu–Lakshmanan equation can be constructed, and the Jacobian function solutions and hyperbolic function solutions are obtained. Finally, some three-dimensional and two-dimensional diagrams of the obtained solutions are also drawn. Moreover, the modulation stability of the equation under consideration is also given.

Bifurcation, phase portrait and traveling wave solution of time-fractional thin-film ferroelectric material equation with beta fractional derivative

In this paper, the bifurcation, phase portrait and traveling wave solution of time-fractional thin-film ferroelectric material equation with beta fractional derivative are studied, which are also the biggest innovations of this article compared with existing literature. Firstly, the time-fractional thin-film ferroelectric material equation with beta fractional derivative is converted into the nonlinear ordinary differential equation by using the traveling wave transformation. Secondly, the nonlinear ordinary differential equation is regarded as two-dimensional planar dynamical system. What's more, the bifurcation and phase portraits of two-dimensional planar dynamical system are plotted by using the Maple software. Finally, the traveling wave solutions of time-fractional thin-film ferroelectric material equation with beta fractional derivative are constructed by using the planar dynamical system method.

Bifurcation, chaotic pattern and optical soliton solutions of generalized nonlinear Schrödinger equation

This article studies the generalized nonlinear Schrödinger equation, which is used to simulate the propagation model of optical pulses in Non-Kerr medium. Building upon the traveling wave transformation, the generalized nonlinear Schrödinger equation is simplified to an ordinary differential equation. By employing the two-dimensional planar dynamic system to analyze, the bifurcation, phase portraits and chaotic behaviors of the system is presented. Furthermore, 2D and 3D phase portraits of the dynamic system with perturbation term are plotted with the Maple software. The optical soliton solutions of the generalized nonlinear Schrödinger equation are constructed by using the polynomial complete discriminant method.

Qualitative analysis of stochastic Schrödinger–Hirota equation in birefringent fibers with spatiotemporal dispersion and parabolic law nonlinearity

In this paper, the stochastic Schrödinger–Hirota equation in birefringent fibers with spatiotemporal dispersion and parabolic law nonlinearity is studied, which is usually used to describe the mathematical model of optical soliton propagation in dispersive optical fibers from the field of nonlinear optics. Firstly, the stochastic Schrödinger–Hirota equation is converted into two-dimensional planar dynamic system by using the traveling wave transformation. Secondly, the qualitative analysis of the stochastic Schrödinger–Hirota equation is discussed by using the theory of planar dynamical systems. Finally, the chaos pattern of the stochastic Schrödinger–Hirota equation with perturbed system is considered. What is more, phase portraits and sensitivity analysis of the perturbed system are plotted by using the Maple software.

Phase Portraits and Traveling Wave Solutions of Fokas System in Monomode Optical Fibers

The main purpose of this paper is to investigate the bifurcation and traveling wave solution of the Fokas system in monomode optical fibers by using the method of planar dynamical system. Firstly, the Fokas systems are reduced to two-dimensional planar dynamic system by using the traveling wave transformation. Secondly, by selecting fixed parameters, the phase portraits are drawn by using the Maple software. Finally, the Jacobi elliptic function solutions, the hyperbolic function solutions, and trigonometric function solutions of the Fokas system are obtained.

Chaotic pattern, bifurcation, sensitivity and traveling wave solution of the coupled Kundu–Mukherjee–Naskar equation

In this paper, the coupled Kundu–Mukherjee–Naskar equation is considered, which is usually used to describe the oceanic rogue waves, hole waves and Bragg grating fibers. Firstly, the traveling wave transformation is applied to convert the coupled Kundu–Mukherjee–Naskar equation into two-dimensional planar dynamic system. Secondly, bifurcation and two-dimensional phase portrait of the coupled Kundu–Mukherjee–Naskar equation are discussed by using the plane dynamics system. What is more, after adding periodic perturbation term, two-dimensional phase portrait, three-dimensional phase portrait, Poincaré section and sensitivity of its perturbed system are plotted by the Maple software. Finally, the optical soliton solutions of the coupled Kundu–Mukherjee–Naskar equation are obtained by using planar dynamical system method.

Chaotic pattern, phase portrait, sensitivity and optical soliton solutions of coupled conformable fractional Fokas-Lenells equation with spatio-temporal dispersion in birefringent fibers

In this paper, coupled conformable fractional Fokas-Lenells equation with spatio-temporal dispersion is introduced, which is usually used to simulate the propagation model of pulses’ temporal evolution in birefringent fibers. Firstly, the traveling wave transformation is applied to convert the conformable time–space fractional Fokas-Lenells model into two-dimensional planar dynamic system. Secondly, the qualitative analysis of the coupled conformable fractional Fokas-Lenells equation with spatio-temporal dispersion is discussed by using the plane dynamics system. Moreover, two-dimensional phase portraits, three-dimensional phase portraits, Poincaré sections and sensitivity analysis of its perturbed system are drawn by using the Maple mathematical software. Finally, the optical soliton solutions of the coupled conformable fractional Fokas-Lenells equation with spatio-temporal dispersion are obtained by using planar dynamical system method.

Bifurcation, sensitivity analysis and exact traveling wave solutions for the stochastic fractional Hirota–Maccari system

The stochastic fraction Hirota–Maccari system is used as a governing model to check the wave propagation in nonlinear optics, plasma physics and hydrodynamics. Firstly, the random fractional order Hirota–Maccari system is transformed into a two-dimensional plane dynamic system by using traveling wave transformation. Secondly, the bifurcation is discussed by using the theory of plane dynamic system, and the sensitivity to the strength and frequency of perturbation is analyzed. In addition, the exact traveling wave solution of the stochastic fractional Hirota–Maccari system is constructed by using the analytical method of the plane dynamic system. Finally, the results of comparison analysis indicated that the fractional derivatives and noises will greatly influence the solution behavior.

Identification of parameters of non-linear oscillators

Many applied control problems are characterized by a situation where some or all parameters of the initial dynamic system are unknown. In such cases, the problem of identification arises, which consists in determining the unknown parameters of the system based on information about its output - known information about movement. The ability to solve the problem of identification is an essential property of identifiability depends on the analytical structure of the right-hand sides of the dynamics equations and available information [1]. To solve the identification problem itself, this work uses the method of invariant relations [2], which was developed in analytical mechanics and is intended, in particular, for finding partial solutions (dependencies between variables) in problems of the dynamics of a rigid body with a fixed point. The modification of this method to the problems of the theory of control, observation made it possible to synthesize additional connections between the known and unknown quantities of the original system that arise during the movement of its extended model [3 - 5]. It is worth noting that a some more general approach, which forms a suitable method for solving observation problems for nonlinear dynamic systems due to the synthesis of an invariant manifold in the space of an extended system, was proposed in the works [6], [7] as a certain modification of the method stabilization of nonlinear systems I&I (Input and Invariance). The purpose of this work is to spread the method of synthesis of invariant relations in control problems to the problem of identifying parameters of pendulum systems. A general scheme for constructing asymptotically accurate estimates of the parmeters of a two-dimensional dynamical system is proposed. A relatively simple case of the identification problem will be considered, namely: 1) the output of the original system is the complete phase vector and 2) the system depends linearly on the unknown parameters. Generalizations to more general designs of input-output systems, including with the involvement of information about the output obtained on several trajectories, can be carried out using the approach described below and is the subject of a separate study. The computational experiment on the estimation of the parameters of the mathematical pendulum confirms the efficiency of the proposed identification scheme.

Bifurcation, phase portrait, chaotic pattern and optical soliton solutions of the conformable Fokas–Lenells model in optical fibers

In this paper, conformable Fokas–Lenells model is under consideration, this model is usually used to simulate the propagation model of femtosecond pulses in single-mode optical fibers. Firstly, the traveling wave transformation is applied to convert the conformable Fokas–Lenells model into two-dimensional planar dynamic system. Secondly, the bifurcation of the dynamics system of the conformable Fokas–Lenells model is discussed by using the theory of the plane dynamics system. What is more, two-dimensional phase portraits, three-dimensional phase portraits and Poincaré sections of the dynamics system with perturbation term are drawn with the help of the Maple mathematical software. Finally, the optical soliton solutions of the conformable Fokas–Lenells model are construct by using the analysis method of planar dynamical system.

Sqrt{Toverline{T}}-deformed oscillator inspired by ModMax

Inspired by a recently proposed Duality and Conformal invariant modification of Maxwell theory (ModMax), we construct a one-parameter family of two-dimensional dynamical systems in classical mechanics that share many features with the ModMax theory. It consists of a couple of sqrt{Toverline{T}}-deformed oscillators that nevertheless preserves duality (q rightarrow p,p rightarrow -q) and depends on a continuous parameter gamma, as in the ModMax case. Despite its nonlinear features, the system is integrable. Remarkably, it can be interpreted as a pair of two coupled oscillators whose frequencies depend on some basic invariants that correspond to the duality symmetry and rotational symmetry. Based on the properties of the model, we can construct a nonlinear map dependent on gamma that maps the oscillator in 2D to the nonlinear one, but with parameter 2gamma. The reason behind the existence of such map can be revealed through a construction of two Lax pairs associated with the system. The dynamics also shows the phenomenon of energy transfer and we calculate a Hannay angle associated to geometric phases and holonomies.

Jacobi and Lyapunov Stability Analysis of Circular Geodesics around a Spherically Symmetric Dilaton Black Hole

We analyze the stability of the geodesic curves in the geometry of the Gibbons–Maeda–Garfinkle–Horowitz–Strominger black hole, describing the space time of a charged black hole in the low energy limit of the string theory. The stability analysis is performed by using both the linear (Lyapunov) stability method, as well as the notion of Jacobi stability, based on the Kosambi–Cartan–Chern theory. Brief reviews of the two stability methods are also presented. After obtaining the geodesic equations in spherical symmetry, we reformulate them as a two-dimensional dynamic system. The Jacobi stability analysis of the geodesic equations is performed by considering the important geometric invariants that can be used for the description of this system (the nonlinear and the Berwald connections), as well as the deviation curvature tensor, respectively. The characteristic values of the deviation curvature tensor are specifically calculated, as given by the second derivative of effective potential of the geodesic motion. The Lyapunov stability analysis leads to the same results. Hence, we can conclude that, in the particular case of the geodesic motion on circular orbits in the Gibbons–Maeda–Garfinkle–Horowitz–Strominger, the Lyapunov and the Jacobi stability analysis gives equivalent results.