Interesting Bifurcation Phenomena Research Articles

Tricritical state and quasi-periodicity triggered by the non-linear elasticity in an Upper Convected Maxwell fluid confined between two co-oscillating cylinders about zero-mean

The effects of harmonically co-oscillating the inner and outer cylinders about zero mean rotation in a Taylor–Couette flow are examined numerically using Floquet theory, for the case where the fluid confined between the cylinders obeys the upper convected Maxwell model. Although stability diagrams and mode competition involved in the system were clearly elucidated recently by Hayani Choujaa et al. (2021) in weakly elastic fluids, attention is focused, in this paper, on the dynamic of the system at higher elasticity with emphasis on the nature of the primary bifurcation. In this framework, we are dealing with pure inertio-elastic parametric resonant instabilities where the elastic and inertial mechanisms are considered of the same order of magnitude. It turns out, on the one hand, that the fluid elasticity gives rise, at the onset of instability, to the appearance of a family of new harmonic modes having different axial wavelengths and breaking the spatio-temporal symmetry of the base flow: invariance in the axial direction generating the O(2) symmetry group and a half-period-reflection symmetry in the azimuthal direction generating a spatio-temporal Z2 symmetry group. On the other hand, new quasi-periodic flow emerging in the high frequency limit and other interesting bifurcation phenomena including bi and tricritical states are also among the features induced by the fluid elasticity. Lastly, and in comparison with the Newtonian configuration of this system, the fluid elasticity leads to a total suppression of the non-reversing flow besides emergence of instabilities with lower wavelengths. Such a comparison provides insights into the dynamics of elastic hoop stresses in altering the flow reversal in modulated Taylor–Couette flow.

New Traveling Wave Solutions and Interesting Bifurcation Phenomena of Generalized KdV-mKdV-Like Equation

Using the bifurcation method of dynamical systems, we investigate the nonlinear waves and their limit properties for the generalized KdV-mKdV-like equation. We obtain the following results: (i) three types of new explicit expressions of nonlinear waves are obtained. (ii) Under different parameter conditions, we point out these expressions represent different waves, such as the solitary waves, the 1-blow-up waves, and the 2-blow-up waves. (iii) We revealed a kind of new interesting bifurcation phenomenon. The phenomenon is that the 1-blow-up waves can be bifurcated from 2-blow-up waves. Also, we gain other interesting bifurcation phenomena. We also show that our expressions include existing results.

Bifurcations of Nonlinear Waves in the Generalized KdV–mKdV-Like Equation

Using bifurcation analytic method of dynamical systems, we investigate the nonlinear waves and their bifurcations of the generalized KdV–mKdV-like equation. We obtain the following results : (i) Three types of new explicit expressions of nonlinear waves are obtained. They are trigonometric expressions, exp-function expressions, and hyperbolic expressions. (ii) Under different parameteric conditions, these expressions represent different waves, such as solitary waves, kink waves, 1-blow-up waves, 2-blow-up waves, smooth periodic waves and periodic blow-up waves. (iii) Two kinds of new interesting bifurcation phenomena are revealed. The first phenomenon is that the single-sided periodic blow-up waves can bifurcate from double-sided periodic blow-up waves. The second phenomenon is that the double-sided 1-blow-up waves can bifurcate from 2-blow-up waves. Furthermore, we show that the new expressions encompass many existing results.

Families of reversing and non-reversing Taylor vortex flows between two co-oscillating cylinders with different amplitudes

This paper deals with the centrifugal instability of time-modulated Taylor-Couette flow for the case in which the inner and outer cylinders are co-oscillating around zero mean with the angular velocities Ωin = Ω0 cos(ωt) and Ωout = εΩ0 cos(ωt), respectively (Ω0, ω, and ε denote, respectively, the amplitude, the frequency of the modulated rotation, and the amplitudes ratio). The small-gap equations for the stability of this flow with respect to axisymmetric disturbances are derived and solved on the basis of Floquet theory. We recover in the case ε = 0 where the outer cylinder is stationary while the inner is modulated the two well-known reversing and non-reversing Taylor vortex flows. Attention is focused on the evolution of these time-dependent flows when one allows the oscillation of the outer cylinder. It turns out that an increase in the parameter ε leads to the discovery of families of reversing and non-reversing flows and other interesting bifurcation phenomena including codimension-two bifurcation points. In addition, a proper tuning of this parameter ε provides a control of the onset of instability as well as the nature of the primary bifurcation. Moreover, it is shown that when ε > 1, the instability is suppressed in low frequencies and the flow is always stable in good agreement to what is obtained by a quasi-steady approach where transient instability is detected. This latter is attributed to the fluid inertia taking place when the cylinders are reversing their rotation’s direction. However, no effect of the parameter ε is observed in high frequencies where the instability develops in thin boundary Stokes layers near the oscillating cylinders.

Various Bifurcation Phenomena in a Nonlinear Curved Beam Subjected to Base Harmonic Excitation

Various bifurcation phenomena in a nonlinear curved beam subjected to base harmonic excitation, which is governed by a coupled nonlinear equation with both quadratic and cubic nonlinearities, are investigated using the incremental harmonic balance (IHB) method. The nonlinear partial differential equation that governs the motion of the curved beam is given using Hamilton’s principle. A spatially discretized governing equation is derived using Galerkin’s method, yielding a set of second-order nonlinear ordinary different equations. A high-dimensional model that can take nonlinear model coupling into account is derived. Specific attention is paid to the different bifurcation phenomena of frequency responses and amplitude responses of the system without and with an anti-symmetric mode being excited. Numerical results reveal the rich and interesting diverse bifurcation phenomena that have not been presented in the existent literature on the nonlinear vibration of the curved beam system. Saddle-node, Hopf, and period-doubling bifurcations are observed without an anti-symmetric mode being excited. Besides, a symmetry-breaking bifurcation is observed with an anti-symmetric mode being excited. Furthermore, the phase portraits and bifurcation points obtained by the IHB method agree very well with those obtained by the numerical integration using the fourth-order Runge–Kutta method.

Bifurcation dynamics of the tempered fractional Langevin equation.

Tempered fractional processes offer a useful extension for turbulence to include low frequencies. In this paper, we investigate the stochastic phenomenological bifurcation, or stochastic P-bifurcation, of the Langevin equation perturbed by tempered fractional Brownian motion. However, most standard tools from the well-studied framework of random dynamical systems cannot be applied to systems driven by non-Markovian noise, so it is desirable to construct possible approaches in a non-Markovian framework. We first derive the spectral density function of the considered system based on the generalized Parseval's formula and the Wiener-Khinchin theorem. Then we show that it enjoys interesting and diverse bifurcation phenomena exchanging between or among explosive-like, unimodal, and bimodal kurtosis. Therefore, our procedures in this paper are not merely comparable in scope to the existing theory of Markovian systems but also provide a possible approach to discern P-bifurcation dynamics in the non-Markovian settings.

Soret Convection in a Shallow Porous Cavity under a Magnetic Field and Submitted to Uniform Fluxes of Heat and Mass

Combined effects of magnetic field and thermodiffusion (Soret effect) on natural convection within an electrically conducting binary mixture, confined in a horizontal sparsely packed porous enclosure subjected to uniform fluxes of heat and mass, is studied analytically and numerically. In the limit of a shallow enclosure, an analytical solution is derived using the parallel flow approximation. The approximate analytical solution is validated against the numerical solution of the full governing equations using a finite difference method. Interesting flow bifurcation phenomena are obtained herein and discussed. The linear stability theory and the parallel flow concept are used to determine explicitly the thresholds for the onset of stationary, subcritical and oscillatory convections as functions of the governing parameters. The obtained results showed the existence of different regions in the (N, Le) plane that correspond to different parallel flow behaviors. The number and the locations of these regions depend on the Soret parameter. The existence of a codimension-2 point is demonstrated. The effects of the Hartmann number on the fluid flow intensity and heat and mass transfer characteristics are also discussed.

Double Bifurcation of Nilpotent Focus

In this paper, an interesting bifurcation phenomenon is investigated — a 3-multiple nilpotent focus of the planar dynamical systems could be broken into two element focuses and an element saddle, and the limit cycles could bifurcate out from two element focuses. As an example, a class of cubic systems with 3-multiple nilpotent focus O(0, 0) is investigated, we prove that nine limit cycles with the scheme 7 ⊃ (1 ∪ 1) could bifurcate out from the origin when the origin is a weak focus of order 8. At the end of this paper, the double bifurcations of a class of Z2 equivalent cubic system with 3-multiple nilpotent focus or center O(0, 0) are investigated.

Single-point kinetic energy density functionals: A pointwise kinetic energy density analysis and numerical convergence investigation

We present a comprehensive study of single-point kinetic energy density functionals (KEDFs) to be used in orbital-free density functional theory (DFT) calculations. We first propose a form of KEDFs based on a pointwise Kohn-Sham (KS) kinetic energy density (KED) and electron localization function (ELF) analysis. We find that the ELF and modified enhancement factor have a very strong and transferable correlation with the reduced density in various bulk metals. The non-self-consistent kinetic energy errors predicted by our KEDF models are decreased greatly compared to previously reported generalized gradient approximation (GGA) KEDFs. Second, we perform self-consistent calculations with various single-point KEDFs and investigate their numerical convergence behavior. We find striking numerical instabilities for previous GGA KEDFs; most of the GGA KEDFs fail to converge and show unphysical densities during the optimization. In contrast, our KEDFs demonstrate stable convergence, and their self-consistent results of various bulk properties agree reasonably well with KSDFT. A further detailed KED analysis reveals an interesting bifurcation phenomenon in defective metals and alloys, which may shed light on directions for future KEDF development.

Some Interesting Bifurcations of Nonlinear Waves for the Generalized Drinfel’d-Sokolov System

We study the bifurcations of nonlinear waves for the generalized Drinfel’d-Sokolov systemut+(vm)x=0,vt+a(vn)xxx+buxv+cuvx=0calledD(m,n)system. We reveal some interesting bifurcation phenomena as follows. (1) ForD(2,1)system, the fractional solitary waves can be bifurcated from the trigonometric periodic waves and the elliptic periodic waves, and the kink waves can be bifurcated from the solitary waves and the singular waves. (2) ForD(1,2)system, the compactons can be bifurcated from the solitary waves, and the peakons can be bifurcated from the solitary waves and the singular cusp waves. (3) ForD(2,2)system, the solitary waves can be bifurcated from the smooth periodic waves and the singular periodic waves.

PT-symmetric coupler withχ(2)nonlinearity

We introduce the notion of a ${\cal PT}$-symmetric dimer with a $\chi^{(2)}$ nonlinearity. Similarly to the Kerr case, we argue that such a nonlinearity should be accessible in a pair of optical waveguides with quadratic nonlinearity and gain and loss, respectively. An interesting feature of the problem is that because of the two harmonics, there exist in general two distinct gain/loss parameters, different values of which are considered herein. We find a number of traits that appear to be absent in the more standard cubic case. For instance, bifurcations of nonlinear modes from the linear solutions occur in two different ways depending on whether the first or the second harmonic amplitude is vanishing in the underlying linear eigenvector. Moreover, a host of interesting bifurcation phenomena appear to occur including saddle-center and pitchfork bifurcations which our parametric variations elucidate. The existence and stability analysis of the stationary solutions is corroborated by numerical time-evolution simulations exploring the evolution of the different configurations, when unstable.

The Sliding Bifurcations in Planar Piecewise Smooth Differential Systems

In this paper we are mainly interested in the bifurcation phenomena for a class of planar piecewise smooth differential systems, where a new phenomenon, i.e. sliding heteroclinic bifurcation, is found. Furthermore we will show that the involved systems can present many interesting bifurcation phenomena, such as the (sliding) heteroclinic bifurcation, sliding (homoclinic) cycle bifurcation and semistable limit cycle bifurcation and so on. The system can have two hyperbolic limit cycles, which are bifurcated in one way from a semistable limit cycle, and in another way from a heteroclinic cycle and a sliding cycle. In the proof of our main results, we will use the geometric singular perturbation theory to analyze the dynamics near the sliding region.

Some Explicit Expressions and Interesting Bifurcation Phenomena for Nonlinear Waves in Generalized Zakharov Equations

Using bifurcation method of dynamical systems, we investigate the nonlinear waves for the generalized Zakharov equationsutt-cs2uxx=β(|E|2)xx, iEt+αExx-δ1uE+δ2|E|2E+δ3|E|4E=0,whereα,β,δ1,δ2,δ3, andcsare real parameters,E=E(x,t)is a complex function, andu=u(x,t)is a real function. We obtain the following results. (i) Three types of explicit expressions of nonlinear waves are obtained, that is, the fractional expressions, the trigonometric expressions, and the exp-function expressions. (ii) Under different parameter conditions, these expressions represent symmetric and antisymmetric solitary waves, kink and antikink waves, symmetric periodic and periodic-blow-up waves, and 1-blow-up and 2-blow-up waves. We point out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively. (iii) Five kinds of interesting bifurcation phenomena are revealed. The first kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up and 2-blow-up waves. The second kind is that the 2-blow-up waves can be bifurcated from the periodic-blow-up waves. The third kind is that the symmetric solitary waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the low-kink waves can be bifurcated from four types of nonlinear waves, the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves. The fifth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves. We also show that the exp-function expressions include some results given by pioneers.

Bifurcation Phenomena of Nonlinear Waves in a Generalized Zakharov-Kuznetsov Equation

We study the bifurcation phenomena of nonlinear waves described by a generalized Zakharov-Kuznetsov equationut+au2+bu4ux+γuxxx+δuxyy=0. We reveal four kinds of interesting bifurcation phenomena. The first kind is that the low-kink waves can be bifurcated from the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves. The second kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up waves, the symmetric solitary waves, and the 2-blow-up waves. The third kind is that the periodic-blow-up waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves.

Bifurcation of a Swelling Gel with a Mechanical Load and Geometric Constraint

We present an analysis of the bifurcation phenomenon of a gel in contact with a solvent. When a Mooney—Rivlin form-free energy function is introduced, an asymmetric swelling may appear for a gel swelling under uniaxial constraint or subjected to equal dead loads, which results in an interesting pitchfork bifurcation phenomenon. We present an analytical investigation of this problem based on the classical theory of continuum mechanics. The bifurcation points are obtained for different values of the chemical potential of the solvent molecules. The results demonstrate that the free swelling of the gel under uniaxial constraint will not result in the bifurcation unless further mechanical loads are applied.

Saddle-nodes and period-doublings of Smale horseshoes: a case study near resonant homoclinic bellows

In unfoldings of resonant homoclinic bellows interesting bifurcation phenomena occur: two suspensed Smale horseshoes can collide and disappear in saddle-node bifurcations (all periodic orbits disappear through saddle-node bifurcations, there are no other bifurcations of periodic orbits), or a suspended horseshoe can go through saddle-node and period-doubling bifurcations of the periodic orbits in it to create an additional doubled horseshoe.

Characteristics of electron ponderomotive acceleration by a laser pulse in vacuum

The ponderomotive acceleration scenario (PAS) for slow-electron acceleration by an intense laser pulse has been studied in detail using three-dimensional test-particle simulations. The relativistic Newton–Lorentz equation has been solved numerically for a linearly polarized laser. The results show that the PAS can accelerate slow electrons to an energy close to 1 MeV for a laser intensity a0 = 3 (around 1019 W cm−2 for a Ti : sapphire laser) and 24 MeV for a0 = 20. These data are consistent with the experimental observation. The net gain comes from an asymmetry in the fields experienced by the electron during its acceleration and deceleration stages. It is found that electrons are scattered isotropically in the transverse direction for moderate laser intensity. The output bunches exhibit relatively wide angular and energy dispersions. We have also discussed an interesting bifurcation phenomenon in the energy–angle correlation spectra. According to our simulations, the maximal electron-energy gain in the PAS regime is approximately proportional to the laser intensity and the laser beam width and inversely proportional to the laser pulse duration. Physical interpretations based on the ponderomotive potential model as well as Lorentz–Newton force analyses are presented.

MATHEMATICAL ANALYSIS AND CHALLENGES ARISING FROM MODELS OF TUMOR GROWTH

In the last four decades, various cancer models have been developed in which the evolution of the densities of cells (abnormal, normal, or dead) and the concentrations of biochemical species are described in terms of differential equations. In this paper, we deal with tumor models in which the tumor occupies a well-defined region in space; the boundary of this region is held together by the forces of cell-to-cell adhesion. We shall refer to such tumors as "solid" tumors, although they may sometimes consist of fluid-like tissue, such as in the case of brain tumors (e.g. gliomas) and breast tumors. The most common class of solid tumors is carcinoma: a cancer originating from epithelial cells, that is, from the closely packed cells which align the internal cavities of the body. Models of solid tumors must take spatial effects into account, and are therefore described in terms of partial differential equations (PDEs). They also need to take into account the fact that the tumor region is changing in time; in fact, the tumor region, say Ω(t), and its boundary Γ(t), are unknown in advance. Thus one needs to determine both the unknown "free boundary" Γ(t) together with the solution of the PDEs in Ω(t). These types of problems are called free boundary problems. The models described in this paper are free boundary problems, and our primary interest is the spatial/geometric features of the free boundary. Some of the basic questions we shall address are: What is the shape of the free boundary? How does the free boundary behave as t → ∞? Does the tumor volume increase or shrink as t → ∞? Under what conditions does the tumor eventually become dormant? Finally, we shall explore the dependence of the free boundary on some biological parameters, and this will give rise to interesting bifurcation phenomena. The structure of the paper is as follows. In Secs. 1 and 2 we consider models in which all the cells are of one type, they are all proliferating cells. The tissue is modeled either as a porous medium (in Sec. 1) or as a fluid medium (in Sec. 2). The models are extended in Secs. 3 and 4 to include three types of cells: proliferating, quiescent, and dead. Finally, in Sec. 5 we outline a general multiphase model that includes gene mutations.

Synchronization phenomena in four simple chaotic oscillators full-coupled by capacitors

Synchronization phenomena in simple chaotic oscillators full-coupled by capacitors with tetrahedral structure are investigated. A simple three-dimensional autonomous oscillator is considered as a chaotic subcircuit. We consider two types of systems by changing the coupling points. By carrying out numerical simulations and circuit experiments, entirely different synchronization modes can be observed. Further, we pay attention to the region of occurrence of each attractor of subcircuit, and also pay attention to the shapes of attractors generated by each chaotic oscillator in order to investigate synchronization states and bifurcation to synchronization of chaos. In the proposed systems, there are many interesting synchronization and bifurcation phenomena including different types of synchronization not hitherto reported. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 83(7): 79–90, 2000

Border-collision bifurcations in the buck converter

Interesting bifurcation phenomena are observed for the current feedback-controlled buck converter. We demonstrate that most of these bifurcations can be categorized as border-collision bifurcations. A method of predicting the local bifurcation structure through the construction of a normal form is applied. This method applies to many power electronic circuits as well as other piecewise smooth systems.