Planar Dynamical Systems Research Articles

Algebraic Traveling Wave Solutions to Nonlinear Evolution Equations

In this paper, we employ planar dynamical systems and invariant algebraic curves to characterize all algebraic traveling wave solutions to nonlinear evolution equations. In order to demonstrate the applicability and efficiency of the method, we apply the approach to four (2+1)-dimensional integrable extensions of the Kadomtsev–Petviashvili equation. The numerical simulations are also plotted for better understanding the physical phenomena.

Characteristic of the algebraic traveling wave solutions for two extended (2 + 1)-dimensional Kadomtsev–Petviashvili equations

With the aid of the planar dynamical systems and invariant algebraic cure, all algebraic traveling wave solutions for two extended (2 + 1)-dimensional Kadomtsev–Petviashvili equations, which can be used to model shallow water waves with weakly nonlinear restoring forces and to describe waves in ferromagnetic media, were obtained. Meanwhile, some new rational solutions are also yielded through an invariant algebraic cure with two different traveling wave transformations for the first time. These results are an effective complement to existing knowledge. It can help us understand the mechanism of shallow water waves more deeply.

Oblique plane waves with bifurcation behaviors and chaotic motion for resonant nonlinear Schrodinger equations having fractional temporal evolution

This study deal with the oblique plane wave solutions with dynamical behaviours for (2 + 1)-dimensional resonant nonlinear Schrodinger equations having Bhom’s quantum potential with distinct law of nonlinearities (Kerr and parabolic law) and fractional temporal evolution. The considered equations are converted to solvable form by assuming conformable Khalil’s fractional derivatives. The bifurcation behaviors and chaotic motion for the existence of traveling waves are investigated by forming the planar dynamical system from the considered equations. The novel auxiliary ordinary differential equation method is therefore used to divulge several forms of plane wave solutions of these equations. It is investigated that the widths of resonant wave dynamics are significantly modified with the influence of obliqueness. The evaluated results may be very useful for better examining the resonant optical solitons in nonlinear dynamics because of obliqueness are existed in various nonlinear systems, specifically in optical bullets, Madelung fluids, etc.

Poincaré Maps of “ <”-Shape Planar Piecewise Linear Dynamical Systems with a Saddle

It is important in the study of limit cycles to investigate the properties of Poincaré maps of discontinuous dynamical systems. In this paper, we focus on a class of planar piecewise linear dynamical systems with “[Formula: see text]”-shape regions and prove that the Poincaré map of a subsystem with a saddle has at most one inflection point which can be reached. Furthermore, we show that one class of such systems with a saddle-center has at least three limit cycles; a class of such systems with saddle and center in the normal form has at most one limit cycle which can be reached; and a class of such systems with saddle and center at the origin has at most three limit cycles with a lower bound of two. We try to reveal the reasons for the increase of the number of limit cycles when the discontinuity happens to a system.

Kink soliton solutions and bifurcation for a nonlinear space-fractional Kolmogorov–Petrovskii–Piskunov equation in circuitry, chemistry or biology

Circuitry and chemistry are applied in such fields as communication engineering and automatic control, environmental protection and material/medicine sciences, respectively. Biology works as the basis of agriculture and medicine. Studied in this paper is a nonlinear space-fractional Kolmogorov–Petrovskii–Piskunov equation for the electronic circuitry, chemical kinetics, population dynamics, neurophysiology, population genetics, mutant gene propagation, nerve impulses transmission or molecular crossbridge property in living muscles. Kink soliton solutions are obtained via the fractional sub-equation method. Change of the fractional order does not affect the amplitudes of the kink solitons. Via the traveling transformation, the original equation is transformed into the ordinary differential equation, while we obtain two equivalent two-dimensional planar dynamic systems of that ordinary differential equation. According to the bifurcation and qualitative considerations of the planar dynamic systems, we display the corresponding phase portraits when the traveling-wave velocity is nonzero or zero. Nonlinear periodic waves of the original equation are obtained when the traveling-wave velocity is zero.

Dynamical Analysis and Exact Solutions of a New (2+1)-Dimensional Generalized Boussinesq Model Equation for Nonlinear Rossby Waves**Supported by the National Natural Science Foundation of China under Grant Nos. 11562014, 11762011, 11671101, 71471020, 51839002; the Natural Science Foundation of Inner Mongolia under Grant No. 2017MS0108; Hunan Provincial Natural Science Foundation of China under Grant No. 2016JJ2061; the Scientific Research Fund of

In this paper, we study the higher dimensional nonlinear Rossby waves under the generalized beta effect. Using methods of the multiple scales and weak nonlinear perturbation expansions [Q. S. Liu, et al., Phys. Lett. A 383 (2019) 514], we derive a new (2+1)-dimensional generalized Boussinesq equation from the barotropic potential vorticity equation. Based on bifurcation theory of planar dynamical systems and the qualitative theory of ordinary differential equations, the dynamical analysis and exact traveling wave solutions of the new generalized Boussinesq equation are obtained. Moreover, we provide the numerical simulations of these exact solutions under some conditions of all parameters. The numerical results show that these traveling wave solutions are all the Rossby solitary waves.

Bifurcations and Exact Solutions of Generalized Two-Component Peakon Type Dual Systems

This paper investigates two generalized two-component peakon type dual systems, which can be reduced to the same planar dynamical systems via the dynamical system approach and the theory of singular traveling wave systems, where one of them contains the two-component Camassa–Holm system. By bifurcation analysis on the corresponding traveling wave system, we obtain the phase portraits and derive possible exact traveling wave solutions that include solitary wave solution, peakon and anti-peakon, pseudo-peakon, periodic peakon, compacton and periodic wave solution. Our results are also applicable to the two-component Camassa–Holm equation.

Dynamics of ion-acoustic waves in Thomas-Fermi plasmas with source term

Dynamics of ion acoustic waves are studied in Thomas-Fermi plasmas with source term consisting of electrons, positrons and positive ions, where electrons and positrons follow zero-temperature Fermi-gas statistics, but ions behave as classical fluid. Using RPT (reductive perturbation technique) the forced KdV and modified KdV equations are obtained. Employing the concept of planar dynamical systems, periodic, quasiperiodic and chaotic features of ion-acoustic waves are studied in Thomas-Fermi plasmas with space debris depending on positron concentration (α), speed (vd) of the space debris and strength (f0) of source perturbation. The results of this work may have relevance in the Thomas-Fermi plasma environments with space debris.

Isochronous cosmological solutions of the Friedmann–Robertson–Walker model

In this paper, the periodic solutions of the equation of Friedmann–Robertson–Walker cosmology with a cosmological constant are investigated. Using variable transformation, the original second-order ordinary differential equation is converted to a planar dynamical system with cosmic time t. Numerical simulations indicate that period function T(h) of this dynamical system is monotonically increasing. However, a new planar dynamical system could be deduced by using conformal time variable [Formula: see text]. We prove that the new planar dynamical system has two isochronous centers under certain parameter conditions by using Picard–Fuchs equation. Explicitly, we find that there exist two families of periodic solutions with equal period for the new planar dynamical system which is derived from the Friedmann–Robertson–Walker model.

Limit Cycles of a Class of Discontinuous Planar Piecewise Linear Systems with Three Regions of Y-Type

In this paper, we investigate the crossing limit cycles of a class of discontinuous planar piecewise linear dynamical systems with three regions of Y-type. We want to find the reason that cause the increase of the number of such limit cycles of the piecewise systems. We obtain the canonical form of such systems with focus–focus–focus type, and prove that the graph of Poincare map of one of three subsystems has one inflection point. Naturally, we prove that the lower bound of the number of crossing limit cycles for this class systems is 3. We also give an example to simulate the four real crossing limit cycles.

Dynamical Behavior of Supernonlinear Positron-Acoustic Periodic Waves and Chaos in Nonextensive Electron-Positron-Ion Plasmas

Abstract Propagation of nonlinear and supernonlinear positron-acoustic periodic waves is examined in an electron-positron-ion plasma composed of static positive ions, mobile cold positrons, and q-nonextensive electrons and hot positrons. Employing the phase plane theory of planar dynamical systems, all qualitatively different phase portraits that include nonlinear positron-acoustic homoclinic orbit, nonlinear positron-acoustic periodic orbit, supernonlinear positron-acoustic homoclinic orbit, and supernonlinear positron-acoustic periodic orbit are demonstrated subjected to the parameters q , μ 1 , μ 2 , σ 1 , σ 2 $q,{\mu_{1}},{\mu_{2}},{\sigma_{1}},{\sigma_{2}}$ , and V. The nonlinear and supernonlinear positron-acoustic periodic wave solutions are reported for different situations through numerical computations. It is observed that the nonextensive parameter (q) acts as a controlling parameter in the dynamic motion of nonlinear and supernonlinear positron-acoustic periodic waves. The dynamic motions for the positron-acoustic traveling waves with the influence of an extrinsic periodic force are investigated through distinct qualitative approaches, such as phase portrait analysis, sensitivity analysis, time series analysis, and Poincaré section. The results of this paper may be applicable in understanding nonlinear, supernonlinear positron-acoustic periodic waves, and their chaotic motion in space plasma environments.

Exact Traveling Wave Solutions and Bifurcations of the Time-Fractional Differential Equations with Applications

This paper presents a method to investigate exact traveling wave solutions and bifurcations of the nonlinear time-fractional partial differential equations with the conformable fractional derivative proposed by [Khalil et al., 2014]. The method is based on employing the bifurcation theory of planar dynamical systems proposed by [Li, 2013]. For the fractional PDEs, up till now, there is no related paper to obtain the exact solutions by applying bifurcation theory. We show how to use this method with applications to two fractional PDEs: the fractional Klein–Gordon equation and the fractional generalized Hirota–Satsuma coupled KdV system, respectively. We find the new exact solutions including periodic wave solution, kink wave solution, anti-kink wave solution and solitary wave solution (bright and dark), which are different from previous works in the literature. This approach can also be extended to other nonlinear time-fractional differential equations with the conformable fractional derivative.

Chaotic motions for a perturbed nonlinear Schrödinger equation with the power-law nonlinearity in a nano optical fiber

Perturbed nonlinear Schrödinger (NLS) equation with the power-law nonlinearity in a nano optical fiber is studied with the help of its equivalent two-dimensional planar dynamic system and Hamiltonian. Via the bifurcation theory and qualitative theory, equilibrium points for the two-dimensional planar dynamic system are obtained. With the external perturbation taken into consideration, chaotic motions for the perturbed NLS equation with the power-law nonlinearity are derived based on the equilibrium points.

Modeling the Role of Feedback in the Adaptive Response of Bacterial Quorum Sensing.

Bacterial quorum sensing (QS) is a form of intercellular communication that relies on the production and detection of diffusive signaling molecules called autoinducers. Such a mechanism allows the bacteria to track their cell density in order to regulate group behavior, such as biofilm formation and bioluminescence. In a number of bacterial QS systems, including V. harveyi, multiple signaling pathways are integrated into a single phosphorylation-dephosphorylation cycle. In this paper, we propose a weight control mechanism, in which QS uses feedback loops to 'decode' the integrated signals by actively changing the sensitivity in different pathways. We first use a slow/fast analysis to reduce a single-cell model to a planar dynamical system involving the concentrations of phosphorylated signaling protein LuxU and a small non-coding RNA. In addition to identifying the weight control mechanism, we show that adding a feedback loop can lead to a bistable QS response in certain parameter regimes. We then combine the slow/fast analysis with a contraction mapping theorem in order to reduce a population model to an effective single-cell model, and show how the weight control mechanism allows bacteria to have a finer discrimination of their social and physical environment.

The effects of the singular lines on the traveling wave solutions of modified dispersive water wave equations

In this paper, the bounded traveling wave solutions of the modified water wave equations of which one dependent variable attains the singular value 2c in finite or infinite time are investigated by using the bifurcation theory of planar dynamical systems. The line V=2c is the so-called singular line of the associated dynamical system and the results of this paper show that the solutions possess singularity if and only if their corresponding phase orbits intersect with this singular line. There are two types of solutions corresponding to these orbits intersecting with the singular line: smooth classical solutions and compact solutions possessing compact support in Hloc1(R), which suggests that the existence of singular line breaks the uniqueness of solutions in Hloc1(R) space. There is a significant discovery from the investigation of the modified water wave equations that there are new type of solitary wave solutions approaching the singular value 2c as time tends to infinite that correspond to some specific orbits connecting with singular lines of the associated traveling wave system, which refreshes and enriches the knowledge of the effects of singular lines on the traveling wave solutions to nonlinear wave equations. The explicit bounded smooth traveling wave solutions and compact solutions of the modified water wave equations are presented and simulated numerically.

Orbital Stability of Peakons for a Generalized Camassa-Holm Equation

We investigate the orbital stability of the peakons for a generalized Camassa-Holm equation (gCH). Using variable transformation, a planar dynamical system is obtained from the gCH equation. It is shown that the planar system has two heteroclinic cycles which correspond two peakon solutions. We then prove that the peakons for the gCH equation are orbitally stable by using the method of Constantin and Strauss.

More on Bifurcations and Dynamics of Traveling Wave Solutions for a Higher-Order Shallow Water Wave Equation

This paper studies the dynamics of traveling wave solutions to a shallow water wave model with a large-amplitude regime in phase space. The corresponding traveling wave system is a singular planar dynamical system with two singular straight lines. By using the method of dynamical systems, bifurcation diagrams are obtained. The existence of solitary wave solutions, periodic wave solutions, peakon, pseudo-peakon solution, periodic peakon solutions and compacton solutions are determined under different parameter conditions.

EXACT SOLUTIONS AND DYNAMICS OF THE RAMAN SOLITON MODEL IN NANOSCALE OPTICAL WAVEGUIDES, WITH METAMATERIALS, HAVING PARABOLIC LAW NON-LINEARITY

This paper investigate the Raman soliton model in nanoscale optical waveguides, with metamaterials, having parabolic law non-linearity by using the method of dynamical systems. The functions $ q(x, t) = \phi(\xi)\exp(i(-kx+\omega t)) $ are solutions of the equation (1.1) that governs the propagation of Raman solitons through optical metamaterials, where $ \xi = x-vt $ and $ \phi(\xi) $ in the solutions satisfy a singular planar dynamical system (1.5) which has two singular straight lines. By using the bifurcation theory method of dynamical systems to the equation of $ \phi(\xi) $, bifurcations of phase portraits for this dynamical system are obtained under 28 different parameter conditions. Based on those phase portraits, 62 exact solutions of system (1.5) including periodic solutions, heteroclinic and homoclinic solutions, periodic peakons and peakons as well as compacton solutions are derived.

Dynamical Behavior of Traveling Wave Solutions of a Long Waves–Short Waves Resonance Model

In this paper, we consider the model of a long waves–short waves, which model the resonant interaction between the ocean wave, to investigate the qualitative behavior of traveling wave solutions of dynamical systems. To understand the underlying complex dynamics, we apply the bifurcation method, that serve as a mechanism for fascinating the physical behavior such as solitons and we obtain all possible phase portraits of the system under various conditions depending on the parameters associated with the planar dynamical system.

Existence and Dynamics of Bounded Traveling Wave Solutions to Getmanou Equation**Supported by the National Natural Science Foundation of China under Grant No. 11701191, Program for Innovative Research Team in Science and Technology in Fujian Province University, and Quanzhou High-Level Talents Support Plan under Grant No. 2017ZT012

In this paper, we study the existence and dynamics of bounded traveling wave solutions to Getmanou equations by using the qualitative theory of differential equations and the bifurcation method of dynamical systems. We show that the corresponding traveling wave system is a singular planar dynamical system with two singular straight lines, and obtain the bifurcations of phase portraits of the system under different parameters conditions. Through phase portraits, we show the existence and dynamics of several types of bounded traveling wave solutions including solitary wave solutions, periodic wave solutions, compactons, kink-like and antikink-like wave solutions. Moreover, the expressions of solitary wave solutions are given. Additionally, we confirm abundant dynamical behaviors of the traveling wave s olutions to the equation, which are summarized as follows: i) We confirm that two types of orbits give rise to solitary wave solutions, that is, the homoclinic orbit passing the singular point, and the composed homoclinic orbit which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of associated system. ii) We confirm that two types of orbits correspond to periodic wave solutions, that is, the periodic orbit surrounding a center, and the homoclinic orbit of associated system, which is tangent to the singular line at the singular point of associated system.