Bifurcation Of Solutions Research Articles

Disconnected Stationary Solutions in 3D Kolmogorov Flow and Their Relation to Chaotic Dynamics

This paper aims to investigate the nonlinear transition to turbulence in generalized 3D Kolmogorov flow. The difference between this and classical Kolmogorov flow is that the forcing term in the x direction sin(y) is replaced with sin(y)cos(z). This drastically complicates the problem. First, a stability analysis is performed by deriving the analog of the Orr–Sommerfeld equation. It is shown that for infinite stretching, the flow is stable, contrary to classical forcing. Next, a neutral curve is constructed, and the stability of the main solution is analyzed. It is shown that for the cubic domain, the main solution is linearly stable, at least for 0<R≤100. Next, we turn our attention to the numerical investigation of the solutions in the cubic domain. The main feature of this problem is that it is spatially periodic, allowing one to apply a relatively simple pseudo-spectral numerical method for its investigation. We apply the method of deflation to find distinct solutions in the discrete system and the method of arc length continuation to trace the bifurcation solution branches. Such solutions are called disconnected solutions if these are solutions not connected to the branch of the main solution. We investigate the influence of disconnected solutions on the dynamics of the system. It is demonstrated that when disconnected solutions are formed, the nonlinear transition to turbulence is possible, and dangerous initial conditions are these disconnected solutions.

Effects of extra resource and harvesting on the pattern formation for a predation system

We deal with a reaction–diffusion predation system with extra resource provided to predator and harvesting on prey. We first discuss the long-time behaviors of parabolic system, including the dissipativeness and persistence of positive solutions. It is shown that under certain constraints of harvesting, the quality of extra resource, the predation and transition rates of predator, the dissipativeness is compatible with persistence. Secondly, some properties of steady-state system are investigated, mainly including the existence of non-constant positive solutions, Turing and steady-state bifurcation phenomena. It is found that the extra resource, prey harvesting and diffusion have significant impacts on the pattern formations. Furthermore, some numerical simulations on Turing patterns and steady-state bifurcation solutions are performed to illustrate the theoretical analysis. We observe that when the quantity of extra resources is low, changes in the quality of extra resources can lead to significant changes in the spatial distribution of species, which is in sharp contrast to the case of high quantity of extra resource. Additionally, we conclude that different diffusion rates of predator can lead to different spatial patterns for the system.

The bifurcation, chaotic behavior and exact solutions of the fractional stochastic Jimbo–Miwa equations

The Jimbo–Miwa equation is the second equation in the KP hierarchy of integrable systems. In this paper, this equation is extended and introduced with the stochastic process and fractional derivatives. Firstly, the phase portrait of the Hamiltonian system generated by it is studied to understand its bifurcation behavior. Additionally, non-periodic and periodic perturbation terms are added to this system. Different values are assigned to the parameters in the perturbation terms to analyze its sensitivity and the resulting chaos is obtained. Finally, through integration techniques, the expression of the solution of this equation is obtained. These solutions are related to rational functions, trigonometric functions, exponential functions and Jacobi elliptic functions. To observe the form of the solutions more intuitively, 3D and 2D numerical simulations are conducted on the solutions and the solution images of the stochastic fractional differential equation are given by Matlab software. Compared with the existing literature, the research on the stochastic fractional equation of this equation is relatively rare and the analysis of the phase portrait is even scarcer. Our solution method is quite different from that in the previous literature. Therefore, this paper is novel. The conclusion of this paper will be of great help for the practical application of this equation.

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.

Bifurcation, quasi‐periodic, chaotic pattern, and soliton solutions for a time‐fractional dynamical system of ion sound and Langmuir waves

This paper strives to investigate the time fractional system that characterizes the ion sound wave influenced by the ponderomotive force induced by a high‐frequency field, as well as the Langmuir wave in plasma. Initially, based on the qualitative theory for planar integrable systems, four‐phase portraits are found in the phase plane under certain conditions on the physical parameters. These conditions are used to prove analytically the existence of solitary, kink (anti‐kink), periodic, super‐periodic, and unbounded wave solutions. The correspondence between the energy levels, phase orbits, and consequently the type of the solution is announced. We derived the bounded wave solutions associated with the phase orbits, which are shown to be consistent with the qualitative analysis of the types of solutions. Moreover, we studied the consistency between the obtained solutions by investigating the degeneracy of the solutions through the transmission between the phase orbits, or equivalently, through the dependence on the initial conditions. With the presence of perturbed periodic terms, the quasi‐periodic behavior and chaotic patterns are investigated.

Bifurcation and dynamics of periodic solutions of MEMS model with squeeze film damping

In this paper, we study the oscillations of an idealized mass–spring model of micro-electro-mechanical system (MEMS) with squeeze film damping. The model consists of two parallel electrodes separated by a gap d: one of them is fixed, and another one is movable and attached to a linear spring with stiffness coefficient k>0. The oscillation, under the influence of AC–DC voltage V(t)=vdc+vaccos2πTt, is ruled by the following singular differential equation my′′+[A(d−y)3+Ad−y]y′+ky=θ0A2V2(t)(d−y)2.Here, y is the vertical displacement of the moving plate (y is always assumed to be less than d), m>0 is its mass, A>0 is the electrode area, and θ0>0 is the absolute dielectric constant of vacuum. Taking d as the parameter, we show the existence of saddle–node bifurcation of T-periodic solutions to the equation in the parameter space. This answers, from certain point of view, the open problem proposed by Torres in his monograph, see Torres (2015, Open Problem 2.1, p. 18). Further, we prove that the equation has exactly two classes of T-periodic solutions: as d tends to +∞, one of them uniformly tends to +∞ at the rate of d, while the minimum values of the second class tend to, or cross, 0.

Dynamics of a size-structured predator–prey model with chemotaxis mechanism

This paper is concerned with a size-structured diffusive predator–prey model with chemotaxis mechanism. The existence, linearized stability and monotonicity with respect to the growth rates of boundary steady-state solutions are analyzed. Moreover, the global stability of trivial steady-state solution under certain conditions is proved by constructing Lyapunov functional. We investigate the local and global bifurcations of positive steady-state solutions that emanate from semi-trivial steady-state solutions using Lyapunov–Schmidt reduction and bifurcation techniques when the fertility intensity of a predator or prey is used as a bifurcation parameter. It is shown that the nonlinear nonlocal chemotaxis term can lead to the emergence of Allee effect.

Periodic Solutions of Wave Propagation in a Strongly Nonlinear Monatomic Chain and Their Novel Stability and Bifurcation Analyses

Abstract A modified incremental harmonic balance (IHB) method is used to determine periodic solutions of wave propagation in discrete, strongly nonlinear, periodic structures, and solutions are found to be in a two-dimensional hyperplane. A novel method based on the Hill’s method is developed to analyze stability and bifurcations of periodic solutions. A simplified model of wave propagation in a strongly nonlinear monatomic chain is examined in detail. The study reveals the amplitude-dependent property of nonlinear wave propagation in the structure and relationships among the frequency, the amplitude, the propagation constant, and the nonlinear stiffness. Numerous bifurcations are identified for the strongly nonlinear chain. Attenuation zones for wave propagation that are determined using an analysis of results from the modified IHB method and directly using the modified IHB method are in excellent agreement. Two frequency formulae for weakly and strongly nonlinear monatomic chains are obtained by a fitting method for results from the modified IHB method, and the one for a weakly nonlinear monatomic chain is consistent with the result from a perturbation method in the literature.

Dynamics of Bifurcation, Chaos, Sensitivity and Diverse Soliton Solution to the Drinfeld-Sokolov-Wilson Equations Arise in Mathematical Physics

Dynamics of Bifurcation, Chaos, Sensitivity and Diverse Soliton Solution to the Drinfeld-Sokolov-Wilson Equations Arise in Mathematical Physics

Dynamics of quasi-periodic, bifurcation, sensitivity and three-wave solutions for (n + 1)-dimensional generalized Kadomtsev-Petviashvili equation.

This study endeavors to examine the dynamics of the generalized Kadomtsev-Petviashvili (gKP) equation in (n + 1) dimensions. Based on the comprehensive three-wave methodology and the Hirota's bilinear technique, the gKP equation is meticulously examined. By means of symbolic computation, a number of three-wave solutions are derived. Applying the Lie symmetry approach to the governing equation enables the determination of symmetry reduction, which aids in the reduction of the dimensionality of the said equation. Using symmetry reduction, we obtain the second order differential equation. By means of applying symmetry reduction, the second order differential equation is derived. The second order differential equation undergoes Galilean transformation to obtain a system of first order differential equations. The present study presents an analysis of bifurcation and sensitivity for a given dynamical system. Additionally, when an external force impacts the underlying dynamic system, its behavior resembles quasi-periodic phenomena. The presence of quasi-periodic patterns are identified using chaos detecting tools. These findings represent a novel contribution to the studied equation and significantly advance our understanding of dynamics in nonlinear wave models.

Complex Dynamics of a Simple Tumor-Immune Model with Tumor Malignancy

One main feature of a malignant tumor is its uncontrolled growth. In this paper, we propose a simple tumor-immune model to study the progressive characteristics of malignant and benign tumors, where the anti-tumor immunity can be described by the Michaelis–Menten function or the mass action law. The model includes only two state variables for the tumor cells and the effector cells representing the immune system. Three quantities with clear biological meanings are given to determine the asymptotic states of the tumor progression. Moreover, differences in asymptotic states between the two anti-tumor immunity descriptions are drawn. Differently from existing simple models, on the one hand, the model exhibits rich dynamical behaviors including super-critical and sub-critical Bogdanov–Takens bifurcations (consisting of Hopf bifurcation, saddle–node bifurcation, and homoclinic bifurcation) and saddle–node bifurcation of nonconstant periodic solutions (leading to the appearance of two periodic orbits) as the parameters vary; on the other hand, the malignant feature, dormancy, and immune escape of the tumor are revealed with numerical simulations. Furthermore, from the perspective of qualitative analysis and numerical simulations, how the obtained results can be applied to the treatment and control of tumors is illustrated.

Investigation of the complex dynamical structure of bifurcation and dark soliton solutions to fractional generalized double [formula omitted]-Gordon equation

In the current research work, the classical wave equation is combined with a nonlinear sinh source term in the sinh-Gordon equation. It has been used in several scientific domains such as differential geometry theory, integrable quantum field theory, kink dynamics, and statistical mechanics. It makes more comprehensible the dynamics of strings and multi-strings in the constant curvature space. The current study has three main objectives. Examine the governing model to get novel solutions by employing the modified Kudryashov technique. Then compare it with the numerical technique of modified variational iteration method (MVIM) to calculate the error terms. Furthermore, employing bifurcation theory to produce a dynamical system. Additionally, use the dynamical system’s sensitivity analysis to investigate the model’s sensitivity. At last, for the validation of acquired results, the cryptography technique of novel image encryption and decryption is used. The research is greatly enhanced by the presentation of thorough 2D and 3D phase portraits. The field of mathematics and other sciences will benefit from these discoveries.

Effect of density-dependent diffusion on a diffusive predator–prey model in spatially heterogeneous environment

The paper presents a class of predator–prey model with density-dependent diffusion in the spatially heterogeneous environment. We first provide the global existence and boundedness of the solution for the model. Then, by taking a variable transformation, the difficulty brought by the cross-diffusion can be overcome, and the existence, stability and local bifurcation of semi-trivial steady-state solutions for the equivalent system are further studied. Finally, the existence of positive solutions of the system is also given by using the Leray–Schauder degree theory and the method of principle eigenvalue, especially for the limit cases when the diffusion coefficient tends to zero or infinite.

Bifurcations of phase portraits and exact solutions of the ($$2+1$$)-dimensional integro-differential Jaulent–Miodek equation

Bifurcations of phase portraits and exact solutions of the ($$2+1$$)-dimensional integro-differential Jaulent–Miodek equation

Analyzing bifurcation, stability, and wave solutions in nonlinear telecommunications models using transmission lines, Hamiltonian and Jacobian techniques

This study presents a comprehensive analysis of a nonlinear telecommunications model, exploring bifurcation, stability, and wave solutions using Hamiltonian and Jacobian techniques. The investigation begins with a thorough examination of bifurcation behavior, identifying critical points and their stability characteristics, leading to the discovery of diverse bifurcation scenarios. The stability of critical points is further assessed through graphical and numerical methods, highlighting the sensitivity to parameter variations. The study delves into the derivation of both numerical and analytical wave solutions, aligning them with energy orbits depicted in phase portraits, revealing a spectrum of wave behaviors. Additionally, the analysis extends to traveling wave solutions, providing insights into wave propagation dynamics. Notably, the study underscores the efficacy of the planar dynamical approach in capturing system behavior in harmony with phase portrait orbits. The findings have significant implications for telecommunications engineers and researchers, offering insights into system behavior, stability, and signal propagation, ultimately advancing our understanding of complex nonlinear dynamics in telecommunications networks.

Stability, bifurcation and spikes of stationary solutions in a chemotaxis system with singular sensitivity and logistic source

In this paper, we study stability, bifurcation and spikes of positive stationary solutions of the following parabolic–elliptic chemotaxis system with singular sensitivity and logistic source: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are positive constants. Among others, we prove there are [Formula: see text] and [Formula: see text] ([Formula: see text]) such that the constant solution [Formula: see text] of system is locally stable when [Formula: see text] and is unstable when [Formula: see text], and under some generic condition, for each [Formula: see text], a (local) branch of nonconstant stationary solutions of system bifurcates from [Formula: see text] when [Formula: see text] passes through [Formula: see text], and global extension of the local bifurcation branch is obtained. We also prove that any sequence of nonconstant positive stationary solutions [Formula: see text] of system with [Formula: see text] develops spikes at any [Formula: see text] satisfying [Formula: see text]. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from [Formula: see text] when [Formula: see text] passes through [Formula: see text] can be extended to [Formula: see text] and the stationary solutions on this global bifurcation extension are locally stable when [Formula: see text] and develop spikes as [Formula: see text].

The Nonlinear Flow Characteristics within Two-Dimensional and Three-Dimensional Counterflow Models within Symmetrical Structures

In this paper, we investigate the nonlinear characteristics of the flow in a two-dimensional and a three-dimensional counterflow model with symmetrical structures. Through numerical simulations, we obtain the velocity fields of the fluid flow in these models for different Re. The numerical results are analyzed to understand the nonlinear characteristics and differences between the two-dimensional and three-dimensional models. The findings indicate that, when Re varies, both the two-dimensional and three-dimensional models exhibit solution bifurcations and nonlinear phenomena such as symmetry breaking, self-sustained oscillations, and chaos. As Re increases, the two-dimensional counterflow model displays a unique solution, an asymmetric solution, and an oscillating solution. Specifically, when Re < 4320, both the laminar and turbulent models show a unique, symmetric, and steady-state velocity distribution. For 4652 < Re < 8639, the two-dimensional model solutions are not unique, presenting a pair of antisymmetric, asymmetric solutions that nevertheless remain steady-state. When Re > 8639, the solution becomes oscillatory and unsteady. The three-dimensional counterflow model exhibits a two-dimensional solution independent of the Z-axis. At Re = 4652, both the three-dimensional and two-dimensional models produce the same unique, symmetric, and steady-state velocity distribution with no three-dimensional flow (W = 0). At Re = 8639, the three-dimensional model solutions are not unique, showing a pair of antisymmetric, asymmetric solutions, while still being steady and time-independent. At Re = 87,627, the three-dimensional model solution becomes oscillatory and unsteady. By elucidating the flow characteristics and nonlinear features of both models, this study compares the differences between the two-dimensional and three-dimensional flows, thereby laying the groundwork for simplification of the problem and further theoretical research.

Analyzing the bifurcation, chaos and soliton solutions to (3+1)-dimensional nonlinear hyperbolic Schrödinger equation

Water wave development, soliton dynamics and electromagnetic field propagation are just a few phenomena that are studied using the (3+1)-dimensional nonlinear hyperbolic Schrödinger model (NLHSM). This work extracts unique soliton solutions such as solitary, dark periodic, combination of dark and bright solutions, and rational wave solutions by analyzing the NLHSM using the modified Sardar sub-equation technique (MSSET). Also, the chaotic structure for the governing model is examined with and without disturbance using chaos theory and bifurcation theory. The innovations of our research are in discovering hitherto unexplored bifurcation situations and chaotic behaviors within the governing equation. The focus of these investigations is on the solutions of nonlinear dynamics, which are illustrated using density plots, 3-D plots, 2-D curves, and pertinent physical property descriptions. Outcomes demonstrate the value of NLHSM for generating soliton solutions and evaluating them in nonlinear models, offering useful numerical tools for applied mathematics.

Study of self-oscillations of a distributed rotor from a material with hereditarily elastic properties

Self-oscillatory solutions of dynamic equations of transverse vibrations of a distributed rotating ideally balanced rotor (shaft) on an isotropic support made of a material with nonlinear hereditary properties are studied. The derivation of dynamic equations (mathematical model), which are a system of two nonlinear partial differential equations with an infinite (integral) delay of the time argument, is given. A mathematical formulation of the initial–boundary value problem is formulated, its solvability and correctness are proven. The stability of the zero solution (stable rotation), the mechanisms of loss of stability, and the self-oscillatory solutions bifurcating in this case are studied. The possibility of bifurcation of periodic solutions (direct asynchronous precessions) and invariant two-dimensional tori (beat modes) is shown. The stability of self-oscillating solutions is determined by the parameters of the problem under consideration. The method of central manifolds of distributed systems and the theory of bifurcations were used as a research method. Asymptotic formulas are constructed for self-oscillating solutions.

Symmetry and complexity: a Lie symmetry method to bifurcation, chaos, multistability and soliton solutions of the nonlinear generalized advection-diffusion-reaction equation

This paper deals with the complexities of nonlinear dynamics within the nonlinear generalized advection-diffusion-reaction equation, which describes intricate transport phenomena involving advection, diffusion, and reaction processes occurring simultaneously. Through the utilization of the Lie symmetry approach, we thoroughly examine this proposed model, transforming the partial differential equation into an ordinary differential equation using similarity reduction techniques to facilitate a more comprehensive analysis. Exact solutions for this transformed equation are derived employing the extended simplest equation method and the new extended direct algebraic method. To enhance understanding, contour plots along with 2D and 3D visualizations of solutions are employed. Additionally, we explore bifurcation and chaotic behaviors through a qualitative analysis of the model. Phase portraits are meticulously scrutinized across various parameter values, offering insights into system behavior. The introduction of an external periodic strength allows us to utilize various tools including time series, 3D, and 2D phase patterns to discern chaotic and quasi-periodic behaviors. Furthermore, a multistability analysis is conducted to examine the impacts of diverse initial conditions. These findings underscore the efficacy and practicality of the proposed methodologies in evaluating soliton solutions and elucidating phase dynamics across a spectrum of nonlinear models, offering novel perspectives on intricate physical phenomena