Planar Dynamical Systems Research Articles

Bifurcation analysis, chaotic analysis and diverse optical soliton solutions of time-fractional (2+1) dimensional generalized Camassa-Holm Kadomtsev-Petviashvili equation arising in shallow water waves

This paper presents a study on (2+1) generalized Camassa-Holm Kadomtsev-Petviashvili equation, which is used to describe the behavior of shallow water waves in nonlinear media. The considered equation provides a more accurate description of wave behavior compared to linear wave equations and can account for wave breaking and other nonlinear effects. This model can be used to describe and study the behavior of nonlinear waves such as rogue waves in complex fluid dynamics scenarios. This includes the behavior of waves in stratified fluids, nonlinear dispersive media and wave interactions in fluid flows with varying velocities and densities. The bifurcation analysis of the governing equation has been performed using the planar dynamical system method. The chaotic behavior of the dynamical system has been examined by utilizing various techniques such as time series analysis and the construction of 2D and 3D phase space trajectories. Furthermore, the introduction of a perturbed term has resulted in the observation of chaotic and quasi-periodic behaviors across a range of parameter values. The considered equation has been reduced to ordinary differential equation by performing symmetry reduction. The Kudryashov method has been used to obtain the exact solution of reduced equation. The single soliton solution of governed equation has been obtained by using Hirota method and impact of fractional parameter on the obtained solution has been studied using graphical representation. The extended sinh-Gordon equation expansion method and modified generalized exponential rational function method have been exploited to obtain dark, bright and singular soliton solutions of considered equation. The motivation for this study arises from the need to understand and analyze the complex dynamics of shallow water waves in nonlinear media with a particular focus on the (2+1) generalized Camassa-Holm Kadomtsev-Petviashvili equation. By performing symmetry reduction and applying various analytical methods, we aim to unravel the intricate behavior and soliton solutions of considered equation, contributing to the broader understanding of nonlinear wave phenomena.

Analysis of bifurcation and chaos in the traveling wave solution in optical fibers using the Radhakrishnan–Kundu–Lakshmanan equation

The study examines the propagation of solitary waves in optical fibers controlled by the Radhakrishnan–Kundu–Lakshmanan model (RKL) equation, which includes a cubic–quintic non-Kerr refractive index. The first step in this study involves utilizing the extended algebraic method to derive solitonic solutions for the given equation. These solutions are subsequently graphically represented for a variety of parameter values. The proposed strategy allows for the discovery of a range of solutions.The controlling model is incorporated into a planar dynamical system via the Galilean transformation, resulting in the transformation of the governing model. The effects of both the quintic nonlinearity and self-phase modulation on the perturbed and unperturbed systems are explored using the effective potential and related phase portrait. The governing model is thus transformed into a planar dynamical system. Furthermore, this study harnessed a variety of numerical techniques. These methods encompassed time series analysis, phase portraits examination, and meticulous sensitivity inspections. Additionally, we employed a Poincaré section to assess the equation’s responsiveness to external perturbations. The outcomes reveal a fascinating observation: the perturbed system’s behavior shifts from a quasi-periodic state to a chaotic one, contingent upon the amplitude and frequency of the external perturbation. Moreover, we conducted multistability analysis for specific sets of physical parameters, uncovering the equation’s tendency to exhibit multistability in such scenarios.

Nonlinear dynamics of porous fin temperature profile: The extended simplest equation approach

This paper discusses the extended simplest equation method for analyzing the behavior of a porous fin while taking into account the porous and convection parameters. This method helps in obtaining exact solutions and understanding the temperature distribution and heat transfer characteristics of the fin. Furthermore, the equation is transformed into a planar dynamical system using the Galilean transformation. Under dynamic and quasiperiodic conditions, the behavior of the equation is extensively discussed. The bifurcation of the planar system of dynamics described by the transformed equation is studied using bifurcation theory and phase portrait analysis. Additionally, a perturbed force is introduced to the dynamical system, and the RK4 method is employed to identify periodic or quasiperiodic patterns in the system. Sensitivity and multistability analysis are applied to the porous fin equation, considering different initial conditions to investigate its quasiperiodic and periodic behavior. The chaotic structure of the perturbed dynamical system using Lyapunov exponent analysis has also been discussed. Through a comparison between numerical solutions and graphical representations, the validity and applicability of the exact technique are demonstrated. The reported findings contribute novel insights into understanding the behavior of nonlinear equations encountered in mathematical physics.

Lie symmetries, bifurcation analysis, and Jacobi elliptic function solutions to the nonlinear Kodama equation

The present article formally studies the propagation of optical pulses in a nonlinear medium governed by the nonlinear Kodama equation. For this purpose, the Lie symmetries group is first adopted for similarity reduction and constructing some exact solutions of the governing equation. After deriving the dynamical system of the nonlinear Kodama equation, its bifurcation analysis is accomplished using the idea of the planar dynamical system. Through perturbing the resultant dynamical system using a trigonometric function, chaotic characteristics of the governing model are analyzed by serving several two- and three-dimensional phase portraits. A sensitivity analysis of the dynamical system is performed using the Runge–Kutta method to ensure that small changes in initial conditions have little impact on solution stability. Finally, using the technique of the planar dynamical system, a number of Jacobi elliptic function solutions (in special cases, bright and dark solitons) are constructed for the nonlinear Kodama equation. It has been shown that bright and dark solitons can be controlled for their width and height effectively by the achievements of the current paper.

Bifurcation and Exact Traveling Wave Solutions for the Nonlinear Klein-Gordon Equation

This paper focuses on the construction of traveling wave solutions to the nonlinear Klein--Gordon equation by employing the qualitative theory of planar dynamical systems. Based on this theory, we analytically study the existence of periodic, kink (anti-kink), and solitary wave solutions. We then attempt to construct such solutions. For this purpose, we apply a well-known traveling wave solution to convert the nonlinear Klein--Gordon equation into an ordinary differential equation that can be written as a one-dimensional Hamiltonian system. The qualitative theory is applied to investigate and describe phase portraits of the Hamiltonian system. Based on the bifurcation constraints on the system parameters, we integrate the conserved quantities to build new wave solutions that can be classified into periodic, kink (anti-kink), and solitary wave solutions. Some of the obtained solutions are clarified graphically and their connection with the phase orbits is derived.

The only limit cycle that appears in damped harmonic oscillators affected by state-dependent impulses

This study is devoted to planar hybrid dynamical systems with state-dependent impulses to investigate phenomena related to sudden changes in the velocity of an object. In particular, the effect of the state-dependent impulses on the appearance of limit cycles is addressed, which has rarely been studied to date. When a toddler is placed on a swing and pushed gently from behind, the discontinuous changes in the direction and speed of movement of both the toddler and the swing can be described using such a hybrid dynamical system. The main theorem presented in this study reveals that, regardless of the amount of state-dependent impulses, they yield only one stable limit cycle within the region that is identified by the considered system. Moreover, state-dependent impulses lead to destabilization of the equilibrium point. In the proof of the main theorem, a Poincaré mapping is derived by examining the behavior of positive semi-orbits, and the characteristics of the sequence that is obtained from the Poincaré mapping are investigated in detail. Several simulations are performed to demonstrate the main theorem.

Dynamical features and sensitivity visualization of thin-film Polarisation equation

The present investigation describes the dynamical behavior, multi-stability, and traveling wave solutions of thin-film polarisation equations (TFPE) which describes the propagation of waves in thin-film ferroelectric materials. The extended direct algebraic technique is used to construct the traveling wave patterns. Visual representations of a few randomly selected solutions are provided for physical comprehension. The ordinary differential equation can be expressed in the planar dynamical system using the Galilean transformation. Using various initial conditions for the unperturbed dynamical system, phase portraits with various sorts of trajectories are created. Additionally, the Runge-Kutta method is used to plot nonlinear periodic waves and super nonlinear waves. Additionally, the Hamiltonian function for this undisturbed dynamical system is computed and shown. It also included the source term with amplitude and frequency parameters for the chaotic and quasi-periodic behaviors, and the system is also stated in the non-autonomous form. For the dynamical system under investigation, multi-stability is also thoroughly described. Furthermore, a full inspection of the sensitivity of the perturbed dynamical structure under various initial conditions has been conducted.

The traveling wave solution and dynamics analysis of the parabolic law nonlinear stochastic dispersive Schrödinger-Hirota equation with multiplicative white noise

This article investigates the traveling wave solution of the parabolic law nonlinear stochastic dispersive Schrödinger-Hirota equation with multiplicative white noise, and its conclusions can be applied to optical communication. Firstly, the equation is transformed into a nonlinear ordinary differential equation through traveling wave transformation, and its planar dynamic system is studied. The phase portraits of the equation in various cases are drawn by Maple software. Secondly, the traveling wave solutions of this equation are obtained by using the complete discriminant system theory of cubic polynomials. Finally, we draw two-dimensional graphics, three-dimensional graphics, density plots and contour plots for the module length of some solutions under specific parameters. In addition, the above research results can help readers have a more comprehensive and clear understanding of this equation and its application.

Exact Solutions and Dynamics in Schrödinger–Hirota Model Having Multiplicative White Noise via Itô Calculus

For the Schrödinger–Hirota model having multiplicative white noise described via Itô calculus, to find exact explicit solutions, the corresponding differential system of the amplitude component is formulated, which is a planar dynamical system with a singular straight line. In this paper, by using the techniques from dynamical systems to analyze the parameter conditions of the associated system and to find the corresponding phase portraits, the dynamical behavior of the amplitude component can be derived. Under a special parameter condition, exact explicit homoclinic solutions, periodic solution families as well as compacton solutions can be found.

Dynamical profile and multi-stability of ion-acoustic waves with soliton solution arising in plasma physics

The present exploration studies the dynamical profile, multistability and soliton solution of ion-acoustic waves (IAWs) in plasma physics. The new auxiliary equation approach is implemented to compute a wide range of solitary wave structures of different types. The dynamical profile is studied by using the bifurcation theory. This system is converted into a planar dynamical system via the Galilean transformation. Taking different cases of its coefficients, the phase plane analysis of this resultant dynamical system is performed. The phase diagrams show the different characteristics of the equilibrium points of the planar dynamical system. The quasi-periodic behaviors, multistability of this planar dynamical system are done by adding the perturbation term. The sensitivity assessment of this perturbed dynamical system is performed by taking different initial constraints.

Qualitative analysis and optical solitons of the coupled Biswas–Arshed equation in single mode polarization-preserving birefringence fibers

In this article, the topological phase portrait and dynamical behavior of the coupled Biswas–Arshed equation in single mode polarization-preserving birefringence fibers are discussed via qualitative analysis of planar dynamic system. When assigning values to the parameters of the system with perturbation terms, the system exhibits chaotic behavior. By the polynomial complete discrimination method, the optical solitons are obtained such as dark solitons, bright solitons, dark-bright solitons, and bright-dark solitons. Moreover, the physical properties of optical solitons are studied through two-dimensional and three-dimensional diagrams, density maps and contour maps of optical solitons. Finally, the modulation instability of the system is analyzed via standard linear stability.

The non-local Lotka–Volterra system with a top hat kernel—Part 1: dynamics and steady states with small diffusivity

We study the dynamics of the non-local Lotka–Volterra system u t = D u u x x + u ( 1 − ϕ ∗ u − α v ) , v t = D v v x x + v ( 1 − ϕ ∗ v − β u ) , where a star denotes the spatial convolution and the kernel ϕ is a top hat function. We initially focus on the case of small, equal diffusivities ( D = D u = D v ≪ 1 ) together with weak interspecies interaction ( α , β ≪ 1 ), and specifically α , β ≪ D . This can then be extended to consider small, but unequal, diffusivities and weak interactions, with now α , β ≪ D u , D v ≪ 1 . Finally, we are able to develop the theory for the situation when the diffusivities remain small, but the interactions become stronger. In each case, we find that u and v independently develop into periodic spatial patterns that consist of separated humps on an O ( 1 ) time scale, and that these patterns become quasi-steady on a time scale proportional to the inverse diffusivity. These then interact on a longer time scale proportional to the inverse interaction scale, and approach a meta-stable state. Finally, a stable steady state is achieved on a much longer timescale, which is exponentially large relative to the preceding time scales. We are able to quantify this interaction process by determining a planar dynamical system that governs the temporal evolution of the separation between the two periodic arrays of humps on these sequentially algebraically and then exponentially long time scales. We find that, once the humps no longer overlap, the subsequent dynamics lead to a symmetric disposition of the humps, occurring on the exponentially long time scale. Numerical solutions of the full evolution problem cannot access the behaviour on this final extreme time scale, but it can be fully explored through the dynamical system.

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.

Archipelagos, islands, necklaces, and other exotic structures in external force-driven chaotic dusty plasmas

In this paper, the modified Kadomtsev Petviashvili (mKP) equation is derived by considering the dynamics of the dust-acoustic waves (DAWs) with kappa-distributed hot and cold ions and Boltzmannian electrons. The reductive perturbation technique is applied for deriving mKP. The bifurcation theory of the planar dynamical system is used to obtain the phase portrait of the DAWs in the framework of mKP equation. The formation of dust-acoustic solitary waves (DASWs) is analyzed for different plasma parameters in the Saturn’s magnetosphere. In the current plasma model, the external periodic force is introduced to study the quasiperiodic and chaotic behavior of the DAWs. It is noted that periodic initial conditions lead to the emergence of many outlandish features in the system by comparison with the solitary initial conditions. The frequency of the external periodic force ω plays an important role in the transition from quasiperiodic to chaotic behavior. Moreover, the strength of the external periodic force, the value of the nonlinear coefficient of the mKP equation, and the nonlinear acoustic speed are also found to have a significant effect on the chaotic behavior of the system.

Optical soliton solutions and dynamic behavior analysis of generalized nonlinear fractional Tzitzéica-type equation

The application of fractional partial differential equations in the area of nonlinear optics has greatly facilitated the development of optical fiber communication. The primary objective of this article is to analyze the dynamic behavior of generalized nonlinear fractional Tzitzéica-type equation and obtain its optical soliton solutions. Using the planar dynamic system, phase portraits of the system are studied from a qualitative perspective, and then the polynomial complete discrimination method is utilized to obtain various optical soliton solutions of the equation. Furthermore, the relevant characteristics are examined through digital simulation and image analysis.

Solving the Regularized Schamel Equation by the Singular Planar Dynamical System Method and the Deep Learning Method

In this paper, two methods are applied to solve the regularized Schamel equation. Firstly, by using the singular planar dynamical system method, we discover its peakon structure which was not reported before. We also derive some new explicit traveling wave solutions of this equation, including various solitary wave solutions, periodic wave solutions and compactons. Especially, for the first time, we find the W-shape solitary wave solutions and W-shape periodic wave solutions of the equation. Then, in order to discover more wave phenomena, a deep learning framework is introduced to solve complicated initial boundary value problems of this equation. In comparison with the exact solutions given previously, our deep learning framework is reliable and highly accurate in capturing the dynamical behavior of the traveling wave solutions of the equation. As an application, we use it to solve a specific initial boundary value problem and obtain a new data-driven solution.

Further physical study about solution structures for the fractional equal width equation (EW) in shallow water hydrodynamics

In this article, with the help of Matlab R2021a software, the exact solutions of the EW are successfully examined by Sub-Equation method and the unified F-expansion method. The main novelty of this paper lies in the six aspects: (1) Singular periodic wave U-shaped solition is discovered by the sub equation method which did not happen in previous papers [35–37]. (2) More and more intensive multi-periodic M-shape wave solutions are also discovered by the unified F-expansion method which did not happen in previous papers [32,34–37]. (3) The dynamic wave behaviours about the got results and the effects of fractionality α are illustrated by Figs. 1–7. The fractional parameter α masters the appearance of the solitons and makes the involved alphabetic shaped solitions due to the increase in the fractionality number α in Figs. 1–7. (4) The effect about free parameters c,k can be seen from Figs. 8–11 in this paper. If only the increase of wave velocity parameter c or wave number parameter k, the energy trend of the wave is always positive and intensive. This also never happens in previous literatures [34–37]. (5) The Jumarie’s modified Riemann–Liouville and beta fractional and conformable derivatives are tested for accuracy in Figs. 12 and 13. We seem firstly compare the relations and distinctions among different fractional order derivatives in EW model. (6) In our best cognitive, the planar dynamical system about EW model whose phase portraits are also first studied. Furthermore, phase portraits for the space–time fractional equal width equation are plotted, and the corresponding sensitivity and chaotic behaviors are analyzed. The geometrical representations of the EW provide the corporal information to explain the physically phenomenons.

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.

Instability of waves in deep water — A discrete Hamiltonian approach

The stability of waves in deep water has classically been approached via linear stability analysis, with various model equations, such as the nonlinear Schrödinger equation, serving as points of departure. Some of the most well-studied instabilities involve the interaction of four waves – so called Type I instabilities – or five waves – Type II instabilities. A unified description of four and five wave interaction can be provided by the reduced Hamiltonian derived by Krasitskii (1994). Exploiting additional conservation laws, the discretised Hamiltonian may be used to shed light on these four and five wave instabilities without restrictions on spectral bandwidth. We derive equivalent autonomous, planar dynamical systems which allow for straightforward insight into the emergence of instability and the long time dynamics. They also yield new steady-state solutions, as well as discrete breathers associated with heteroclinic orbits in the phase space.