Planar Dynamical Systems Research Articles

Kowalevskaya exponents and normal forms of planar dynamical systems. Application to integrability problem"

Kowalevskaya exponents and normal forms of planar dynamical systems. Application to integrability problem"

Bifurcation analysis of Van der Pol–Mathieu equation for the dust grain charge with regularized (κ)-distributed electron-ion

This paper presents a novel viewpoint on nonlinear dust-acoustic waves (DAWs) in an unmagnetized collisionless plasma containing regularized [Formula: see text] distributed electrons and ions (ei) and negative dust grain. The nonlinear oscillatory system based on hybridization of the Van der Pol–Mathieu equation (VdPME) is derived by a new technique. By bifurcation analysis of the planar dynamical system (DS), the effects of parameters with the assistance of phase planes and time series of VdPME are studied. After analyzing the equation to identify the resonance region, a fourth-order Runge–Kutta method is used to solve it numerically. We explained the behavior of DA periodic, stable limit cycle, and chaotic limit cycle wave solutions with different parameters. These types of numerical solutions are illustrated in two-dimensional and three-dimensional graphics by changing the rate at which charged dust grain is produced [Formula: see text], as well as waste [Formula: see text] and comparing the results with those of earlier research. A novel bifurcation analysis of VdPE and VdPME is obtained with the effects of the cut-off factor [Formula: see text] of regularized [Formula: see text]-distribution (RKD) distributed ei, the superthermality of ei particles [Formula: see text], the ratio of ion to electron temperature [Formula: see text], and the ratio of dust to electron density [Formula: see text] illustrated. It is noticed that the DAW shows promotion in width and amplitude as the frequency [Formula: see text] increases and it becomes rarefaction as the cut-off parameter [Formula: see text] increases. On the contrary, it becomes compressive under the impact of superthermality [Formula: see text] and [Formula: see text]. The obtained conclusions may help explain and comprehend a variety of applications in experimental plasma containing highly energy regularized [Formula: see text] dispersed ei, as well as in the interstellar medium.

Bifurcation, chaotic behaviors and solitary wave solutions for the fractional Twin-Core couplers with Kerr law non-linearity

The main purpose of this article is to analyze the bifurcation, chaotic behaviors, and solitary wave solutions of the fractional Twin-Core couplers with Kerr law non-linearity by using the planar dynamical system method. This equation has profound physical significance and application value in the areas of optics and optical communication. Firstly, the traveling wave transformation is applied to convert the beta-derivative Twin-Core couplers with Kerr law non-linearity into the ordinary differential equations. Secondly, phase portraits and Poincaré sections of two-dimensional dynamical system and its perturbation system are plotted by using mathematical software. For different initial values, the planar phase diagram and three-dimensional phase diagram in red and blue are plotted, respectively. Finally, the solitary wave solutions of the fractional Twin-Core couplers with Kerr law non-linearity are obtained by using theory of planar dynamical system. In addition, three-dimensional graphs, two-dimensional graphs, and the contour graphs of the solitary wave solutions are drawn.

Optical solitons, dynamics of bifurcation, and chaos in the generalized integrable (2+1)-dimensional nonlinear conformable Schrödinger equations using a new Kudryashov technique

In the present paper, the new Kudryashov approach is utilized to construct several novel optical soliton solutions for the generalized integrable (2 + 1)-dimensional nonlinear Schrödinger system with conformable derivative. Additionally, the dynamics of bifurcation behavior and chaos analysis in this system are investigated. We applied bifurcation and chaos theories to enhance our understanding of the planar dynamical system derived from the current model while we obtained and illustrated the chaotic solutions for the perturbed dynamical system using graphs. The study yields a class of new optical soliton solutions, including bell-shaped, wave, dark, dark-bright, dark, multi-dark, and singular soliton solutions. Three-dimensional, two-dimensional, and contour plots are presented to visually demonstrate the physical implications and dynamic characteristics of the current conformable equation system. Further, an analysis is discussed on how the conformable derivative parameter and the parameter of time impact the present optical solutions, demonstrating the system’s importance. It is believed that the solutions analyzed in this study are entirely new and have not been previously reported. These discoveries have the potential to significantly enhance our understanding of nonlinear physical phenomena, especially in nonlinear optics and traffic signaling effects with optical dromion transmission.

Classical and nonclassical Lie symmetries, bifurcation analysis, and Jacobi elliptic function solutions to a 3D-modified nonlinear wave equation in liquid involving gas bubbles

The current paper undertakes an in-depth exploration of the dynamics of nonlinear waves governed by a 3D-modified nonlinear wave equation, a significant model in the study of complex wave phenomena. To this end, the study employs both classical and nonclassical Lie symmetries for rigorously deriving invariant solutions of the governing equation. These symmetries enable the formal construction of exact solutions, which are crucial for understanding the complex behavior of the model. Furthermore, the research extends into the realm of bifurcation analysis through the application of planar dynamical system theory. Such an analysis reveals the conditions under which the 3D-modified nonlinear wave equation admits Jacobi elliptic function solutions. The study also delves into the impact of the nonlinear parameter on the physical characteristics of bright and kink solitary waves as well as continuous periodic waves using Maple. Overall, the comprehensive analysis presented not only enhances the understanding of complex nonlinear wave dynamics but also sets the stage for future advancements in vast areas of fluid dynamics and plasma physics.

Impact of Ion Pressure Anisotropy in Collisional Quantum Magneto-Plasma with Heavy and Light Ions

We have examined collisional degenerate plasma composed of charged state of heavy positive ion and light positive as well as negative ion. Employing the reductive perturbation method, we derived the damped Korteweg-de Vries-Burgers (dKdV-B) equation and by using its standard solution we analyze the characteristics of the solitary-shock profile under varying parameters. Furthermore, with the application of planar dynamical systems bifurcation theory, the phase portraits have been analyzed. This dynamical system analysis allowed us to extract important information on the stability of these structures as represented by the dKdV-B equation.

Qualitative analysis and optical soliton solutions galore: scrutinizing the (2+1)-dimensional complex modified Korteweg–de Vries system

This investigation discusses the (2+1)-dimensional complex modified Korteweg–de Vries (cmKdV) system. The cmKdV system describes the nontrivial dynamics of water particles from the surface to the bottom of a water layer, providing a more comprehensive understanding of wave behavior. The cmKdV system finds applications in various fields of physics and engineering, including fluid dynamics, nonlinear optics, plasma physics, and condensed matter physics. Understanding the behavior predicted by the cmKdV system can lead to insights into the underlying physical processes in these systems and potentially inform the design of novel technologies. A new version of the generalized exponential rational function method (nGERFM) is utilized to discover diverse soliton solutions. This method uncovers analytical solutions, including exponential function, singular periodic wave, combo trigonometric, shock wave, singular soliton, and hyperbolic solutions in mixed form. Moreover, the planar dynamical system of the concerned equation is created, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. To gain a deeper understanding of the dynamic behavior of the solutions, analytical results are supplemented with numerical simulations. These obtained outcomes provide a foundation for further investigation, making the solutions useful, manageable, and trustworthy for the future development of intricate nonlinear issues. This study’s methodology is reliable, strong, effective, and applicable to various nonlinear partial differential equations (NLPDEs). As far as we know, this type of research has never been conducted to such an extent for this equation before. The Maple software application is used to verify the correctness of all obtained solutions.

Investigating the invariant solutions of (1+1)-dimensional Sawada–Kotera model using Lie symmetries analysis

Here attention is focused on the (1+1)-dimensional Sawada–Kotera (SK) model that is prominent in mathematical physics and engineering to analyze plasmas and coherent systems for communication. Our techniques offer fresh perspectives on the model’s attributes and structure, deepening our comprehension of the underlying dynamics. First, the SK model is reduced to ordinary differential equations by constructing the Lie symmetries and using the associated transformation. Graphs are used to build and display invariant solutions. This strategy has caused the revelation of novel constant solutions that have not been found in the previous works. We offer new understandings of nature and changes observed during the SK derivation by taking advantage of the Lie symmetries powerful tools. Next, the fluctuating layout of proposed framework is examined from several perspectives such as sensitivity and bifurcation analysis. We examined the bifurcation analysis of planar dynamical system by using bifurcation theory. We also include an external periodic perturbation term that breaks regular patterns in the perturbed dynamical system. Graphical structures are provided to display the invariant solutions. The sensitivity of the SK model is determined to be strong after sensitivity analysis under different initial conditions. These results are fascinating, fresh, and conceptually useful for understanding the suggested framework. In mathematics and the applied sciences, forecasting and learning about new technologies are greatly aided by the dynamic aspect of system processing.

Inquisition of optical soliton structure and qualitative analysis for the complex-coupled Kuralay system

This study seeks to explore the integrable dynamics of induced curves through the utilization of the complex-coupled Kuralay system. The importance of the coupled complex Kuralay equation lies in its role as a fundamental model that contributes to the understanding of intricate physical and mathematical concepts, making it a valuable tool in scientific research and applications. The soliton solutions originating from the Kuralay equations are believed to encapsulate cutting-edge research in various important domains such as optical fibers, nonlinear optics, and ferromagnetic materials. Analytical techniques are employed to derive traveling wave solutions for this model, given that the Cauchy problem cannot be resolved using the inverse scattering transform. In the quest for solitary wave solutions, the extended modified auxiliary equation mapping (EMAEM) method is employed. We derive several novel families of precise traveling wave solutions, encompassing trigonometric, hyperbolic, and exponential forms. Moreover, the planar dynamical system of the concerned equation is created, all probable phase portraits are given, and sensitive inspection is applied to check the sensitivity of the considered equation. Furthermore, after adding a perturbed term, chaotic and quasi-periodic behaviors have been observed for different values of parameters, and multistability is reported at the end. Numerical simulations of the solutions are incorporated alongside the analytical results to enhance comprehension of the dynamic characteristics of the solutions obtained. This study’s outcomes can offer valuable insights for addressing other nonlinear partial differential equations (NLPDEs). The soliton solutions obtained in this study offer important insights into the intricate nonlinear equation being examined.

The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model

Abstract In this article, the phase portraits, chaotic patterns, and traveling wave solutions of the conformable extended Korteweg–de-Vries (KdV) model are investigated. First, the conformal fractional order extended KdV model is transformed into ordinary differential equation through traveling wave transformation. Second, two-dimensional (2D) planar dynamical system is presented and its chaotic behavior is studied by using the planar dynamical system method. Moreover, some three-dimensional (3D), 2D phase portraits and the Lyapunov exponent diagram are drawn. Finally, many meaningful solutions are constructed by using the complete discriminant system method, which include rational, trigonometric, hyperbolic, and Jacobi elliptic function solutions. In order to facilitate readers to see the impact of fractional order changes more intuitively, Maple software is used to draw 2D graphics, 3D graphics, density plots, contour plots, and comparison charts of some obtained solutions.

Investigating optical soliton pattern and dynamical analysis of Lonngren wave equation via phase portraits

This study focuses on finding effective solutions to a mathematical equation known as the Lonngren wave equation. The solutions from the Lonngren wave equation can be used to evaluate electromagnetic signals in cable lines and sound waves in stochastic systems. The main equation is transformed into an ordinary differential equation by utilizing a suitable wave transformation, allowing for the exploration of mathematical models by using the modified Khater technique to detect the exact solution of a solitary wave. We use the provided method to derive the trigonometric, rational, and hyperbolic solutions. To illustrate the model’s physical behavior, we also present graphical plots of selected solutions to illustrate the physical behavior of the model. By choosing appropriate values for arbitrary factors, the visual representation enhances the understanding of the dynamical system. Furthermore, the system is transformed into a planar dynamical system, and phase portrait analysis is conducted. Additionally, the sensitivity analysis of the dynamical system confirms that slight changes in the initial conditions will have minimal impact on the stability of the solution. The existence of chaotic dynamics in the Lonngren wave equation is explored by introducing a perturbed term in the dynamical system. Two and three-dimensional phase portraits will be used to demonstrate these chaotic behaviors.

Analyzing the Kuralay-II equation: bifurcation, chaos, and sensitivity insights through conformable derivative and Jacobi elliptic function expansion

This study explores the intricate dynamics of the Kuralay-II equation by employing the conformable derivative. Using the Galilean transformation, we can establish a dynamical system related to the equation. We investigate bifurcation methods in this derived system using planar dynamical systems theory. By introducing a perturbed term, we thoroughly investigate the possibility of chaotic behaviors in the Kuralay-II equation using comprehensive two-phase portraiture. Through careful analysis, we have determined that even small changes in the initial conditions have little impact on the stability of the solution, which has been confirmed by employing the Runge–Kutta method. In addition, we obtain exact solutions for the Kuralay-II equation by the Jacobi elliptic function expansion method. Graphical results of some solutions are showcased, offering a comprehensive evaluation using MATLAB across various dimensions. This study has yielded significant findings, such as the discovery of bifurcation points, the determination of conditions for chaos, and the analysis of stability under perturbations. These results have enhanced our understanding of the behavior of the Kuralay-II equation.

Dynamical analysis and the soliton solutions of (2+1)-dimensional Heisenberg ferro-magnetic spin chains model with beta fractional derivative

This paper investigates the soliton solutions and dynamical analysis of (2+1)-dimensional Heisenberg ferro-magnetic spin chains model with beta fractional derivative, which is transformed into the ordinary differential equation. By using the second-order complete discriminant system, the soliton solutions are presented. By utilizing the theory of planar dynamical system, the phase portraits of the dynamical system and its disturbance system are drawn. Moreover, three-dimensional, two-dimensional, and contour plots of soliton solutions for (2+1)-dimensional Heisenberg ferro-magnetic spin chains model with beta fractional derivative have also been plotted.

Chaotic Pattern and Solitary Solutions for the (21)-Dimensional Beta-Fractional Double-Chain DNA System

The dynamical behavior of the double-chain deoxyribonucleic acid (DNA) system holds significant implications for advancing the understanding of DNA transmission laws in the realms of biology and medicine. This study delves into the investigation of chaos patterns and solitary wave solutions for the (2+1) Beta-fractional double-chain DNA system, employing the theory of planar dynamical systems and the method of complete discrimination system for polynomials (CDSP). The results demonstrate a diverse spectrum of solitary wave solutions, sensitivity to perturbations, and manifestations of chaotic behavior within the system. Through the utilization of the complete discrimination system for polynomials, a multitude of novel solitary wave solutions, encompassing periodic, solitary wave, and Jacobian elliptic function solutions, were systematically constructed. The influence of Beta derivatives on the solutions was elucidated through parameter comparison analysis, emphasizing the innovative nature of this study. These findings underscore the potential of this system in unraveling various biologically significant DNA transmission mechanisms.

Symmetry-Optimized Dynamical Analysis of Optical Soliton Patterns in the Flexibly Supported Euler–Bernoulli Beam Equation: A Semi-Analytical Solution Approach

This study employs spatial optimization principles to investigate the nonlinear vibration of a flexibly supported Euler–Bernoulli beam, a (1 + 1)-dimensional system subjected to axial loads. The modified Khater method, a crucial tool in mechanical engineering, is utilized to analyze analytical solutions, which include a symmetric spatial representation of the waveform as an integral part of each solution. Notably, periodic soliton solutions for the nonlinear model closely align with numerical and approximate analytical solutions, demonstrating the accuracy of our modeling approach. Density diagrams, contour diagrams, and Poincaré maps depicting the obtained analytical solutions are presented to elucidate their accuracy and provide visual confirmation of the optimized engineering model’s physical significance. The planar dynamical system is derived through the Galilean transformation by employing mathematical models and appropriate parameter values, thereby further refining problem understanding. Sensitivity analysis is conducted, and phase portraits with equilibrium points are illustrated by analyzing a special case of the investigated dynamical system, emphasizing its symmetrical properties. Lastly, we perform a global analysis to identify periodic, quasi-periodic, and chaotic behaviors, with an extra weak forcing term confirmed by Poincaré maps and a two-dimensional symmetric basin of the largest Lyapunov exponent.

Bifurcation analysis and soliton structures of the truncated [formula omitted]-fractional Kuralay-II equation with two analytical techniques

The truncated M-fractional Kuralay (TMFK)-II equation is prevalent in the exploration of specific complex nonlinear wave phenomena. Such types of wave phenomena are more applicable in science and engineering. These equations could potentially provide insights into understanding the intricate dynamics of optical phenomena, encompassing solitons, nonlinear effects, and wave interactions. This study aims to uncover a diverse range of soliton solutions to the model, spanning trigonometric, hyperbolic, exponential, and rational expressions. These solutions are unveiled through the application of extended hyperbolic functions and improved F-expansion techniques, representing the primary objective of this research. The three-dimensional (3D) and two-dimensional (2D) combined charts are plotted for some of these solutions. The impact of the fractional parameters and time variations is also illustrated. Moreover, the models are converted into a planar dynamical system using a Galilean transformation, and the analysis of bifurcation is examined. This research underscores the versatility of the aforementioned techniques for exploring complex nonlinear phenomena across various engineering and scientific disciplines. Finally, the findings of this study hold significant implications for advancing our understanding and analysis of nonlinear wave dynamics in diverse physical systems.

Bifurcations of phase portraits and chaotic behaviors of the (2+1)-dimensional double‐chain DNA system with beta derivative: A qualitative approach

Qualitative analysis in mathematical modeling has become an important research area within the broad domain of nonlinear sciences. In the realm of qualitative analysis, the bifurcation method is one of the significant approaches for studying the structure of orbits in nonlinear dynamical systems. To apply the bifurcation method to the (2 + 1)-dimensional double-chain Deoxyribonucleic Acid system with beta derivative, the bifurcations of phase portraits and chaotic behaviors, combined with sensitivity and multi-stability analysis of this system, are examined. Initially, the bifurcations of phase portraits are visually identified at the obtained equilibrium points of a planar dynamical system via both Hamiltonian and Jacobian algorithms. The obtained results indicate Jacobian algorithm is more efficient in identifying the stability of bifurcations than the Hamiltonian algorithm for this system. Subsequently, by introducing an external perturbation term into the planar dynamical system, the chaotic behavior is effectively identified by using a variety of tools, such as two- and three-dimensional phase portraits, time series, Lyapunov exponents, and Poincaré maps. The findings suggest that the perturbed dynamical system deviates from regular patterns and exhibits behavior ranging from periodic to quasi-periodic and from quasi-periodic to chaotic. Finally, the sensitivity and multi-stability of the system are examined using the Runge-Kutta method to assess the model's response to minor variations in initial conditions through numerical solutions, revealing that the model is sensitive and multi-stable. The outcomes of this study will enhance a relationship between applied mathematicians and experimental biologists, helping to explore hidden features of Deoxyribonucleic Acid through the studied model.

Nonlinear propagation of ion-acoustic waves along with their bifurcation analysis in a negative ion plasma in the presence of quartic nonlinearity

In this article, the propagation of ion-acoustic shock and periodic waves along with their dynamical analysis around the supercritical values is studied in a (1 + 1)-dimensional collisionless negative ions plasma system comprising of inertia SF 6− with mass m−i and temperature T−i, inertia K+ with mass m+i and temperature T+i and inertialess non-extensive distributed electrons. By considering the appropriate starching coordinates and expansion of perturbation quantities, the Burgers-type equation with quartic nonlinearity is derived. Using the traveling wave transformation, a planar dynamical system is formed. The phase portrait is drawn and the associated nonlinear waves are analyzed. The research presented could be beneficial for understanding and forecasting localized electrostatic disturbances in the F- and D-layers of Earth's ionosphere as well as for guiding future experimental investigations in plasma laboratories.

Exploring multi-soliton patterns, bifurcation analysis, and chaos in (2+1) dimensions: A study on nonlinear dynamics

This study explores the intrinsic characteristics of the (2+1)-dimensional Schwarz-Korteweg-de Vries equation used to describe shallow water waves. The multiple solitons are successfully constructed using a technique involving multiple exponential functions. Graphical representations of the results are provided in 3D, 2D, and density plots to assess the compatibility of the solutions. Also, the dynamic nature of studied equation is examined based on bifurcation and chaos theory for nonlinear systems. Bifurcation signifies how our dynamical system is affected by physical parameters in planar dynamical system. After that, we apply the external force on planar dynamical system to show the chaotic like behavior of the studied model. Such behavior is confirmed by utilizing different chaos detecting tools. The obtained results serve to explain the effectiveness and applicability of the utilized methodologies in comprehend the exact solution and qualitative behavior of nonlinear physical models.

Qualitative behavior and variant soliton profiles of the generalized [formula omitted]-type equation with its sensitivity visualization

This study delves into the exploration of the dynamics of (3+1)-dimensional Painlevé integrable generalized model from different perspectives, which delineates the evolution of nonlinear phenomena in three spatial dimensions and one temporal dimension, displaying the remarkable Painlevé integrability property. The ϕ6-expansion technique is applied to extract traveling wave solutions in the form of Jacobi elliptic functions. To give physical insights of obtained solutions, we present these solutions through graphs such as 2D, 3D and contour plots. Further, to understand the planar dynamical system, we employ the concepts of bifurcation, chaos theory and sensitivity analysis. Bifurcation analysis reveals the dependence on the solution of a planar dynamical system at critical points. Additionally, the detection of chaotic movements in the perturbed dynamical system is achieved by detecting tools. Also the sensitivity analysis of the model is investigate by three distinct initial conditions. The findings are innovative, valuable and captivating for the readers in exploring this model.