New Type Of Bifurcation Research Articles

Bifurcation of solutions through a contact manifold in bidisperse models

This research focuses on a hyperbolic system that describes bidisperse suspensions, consisting of two types of small particles dispersed in a viscous fluid. The dependence of solutions on the relative position of contact manifolds in the phase space is examined. The wave curve method serves as the basis for the first and second analyses. The former involves the classification of elementary waves that emerge from the origin of the phase space. Analytical solutions to prototypical Riemann problems connecting the origin with any point in the state space are provided. The latter focuses on semi-analytical solutions for Riemann problems connecting any state in the phase space with the maximum packing concentration line, as observed in standard batch sedimentation tests. When the initial condition crosses the first contact manifold, a bifurcation occurs. As the initial condition approaches the second manifold, another structure appears to undergo bifurcation, although it does not represent an actual bifurcation according to the triple shock rule. The study reveals important insights into the behavior of solutions in relation to these contact manifolds. This research sheds light on the existence of emerging quasi-umbilic points within the system, which can potentially lead to new types of bifurcations as crucial elements of the elliptic/hyperbolic boundary in the system of partial differential equations. The implications of these findings and their significance are discussed.

Bursting oscillation of a pendulum with irrational nonlinearity

The bursting oscillation has attracted lots of attention due to its complex dynamical behaviors. However, most models are based on polynomial nonlinearity, and the irrational nonlinearity has not been considered by researchers. Particularly, it is a typical nonlinear model that has been widely utilized in engineering fields, so it is urgent to investigate the bursting behaviors of the system. To realize this task, this paper explores a novel bursting oscillation of a pendulum with irrational nonlinearity. Three typical cases of excitation patterns are employed to investigate the complicated bursting behaviors. The mechanism of slow bursting response and the stability of equilibrium points are analyzed via the multiple time scale dynamics method. The three-parameter, two-parameter, and one-parameter bifurcations are proposed. Four new types of bifurcations, namely, forked-, spindle-, saddle- and camelback-shaped bifurcations are observed. The bursting trajectory and jumping characteristics are examined by employing the transformed phase. Finally, the multi-valued phenomenon is performed via the basin of attraction.

On the Pseudo-Bifurcation of Non-Convexity in the Feasible Region of AC Optimal Power Flow

The subject of optimal power flow (OPF) convexity has attracted great interest in the research community. In this brief, it will be shown that a new type of bifurcation can occur in OPF model to alter the convexity property; in particular, a pseudo-bifurcation of OPF feasible region non-convexity can occur when some upper or lower bounds of the inequality constraints are tightened or relaxed. Hence, caution must be taken to state whether a feasible region of an AC OPF model is convex/non-convex without specifying the set of inequality constraints. Numerical studies on several IEEE test systems are conducted to illustrate the impact of this type of pseudo-bifurcation on the feasible region. The applicability of a class of convex relaxation methods is examined via the convex/non-convex feasible region. A guideline derived from our numerical studies is provided to give conditions under which the convex relaxation method is applicable.

Bifurcations of a plane parallel flow with Kolmogorov forcing

We study the primary bifurcations of a plane parallel flow in a channel with Kolmogorov forcing. We find a new type of bifurcation with both the oscillation frequency and the amplitude of the growing mode being zero at the threshold. We call this a stationary drift bifurcation. The laminar steady flow can display different types of bifurcation depending on the forcing wave number of the base flow. This is in contrast to the case of doubly periodic boundary conditions for which the primary bifurcation is stationary.

Allee’s Effect Bifurcation in Generalized Logistic Maps

This paper concerns the study of the Allee effect on the dynamical behavior of a new class of generalized logistic maps. The fundamentals of the dynamics of this 4-parameter family of one-dimensional maps are presented. A complete classification of the nature and stability of its fixed points is provided. The main results relate to the Allee effect bifurcation: a new type of bifurcation introduced for this class of unimodal maps. A necessary and sufficient condition so that the Allee fixed point is a snap-back repeller is established. In addition, in the parameters space is defined an Allee’s effect region, which determines the existence of an essential extinction for the generalized logistic maps. Local and global bifurcations of generalized logistic maps are investigated.

Pseudo-Pitchfork Bifurcation of Feasible Regions in Power Systems

Local bifurcations occur in power systems, causing changes in power system dynamic behaviors. These local bifurcations include the saddle-node bifurcation, Hopf bifurcation, and structure-induced bifurcation. This paper presents a new type of bifurcation that can occur in the optimal power flow (OPF) problem. This new type of bifurcation, termed pseudo-pitchfork bifurcation, involves bifurcations of the feasible components of the OPF problem and the disappearance of local optimal power flow solutions. The main features of this special type of bifurcation are demonstrated on several power systems with different loading condition parameters and different constraint parameters. Then the computation consideration and physical meaning of the pseudo-pitchfork bifurcation are roughly discussed. It is also demonstrated that a pseudo-pitchfork bifurcation occurring between feasible components can help us interpret the loss or birth of optimal power flow solutions and can lead to powerful numerical methods for computing high-quality optimal power flow solutions.

Impact adding bifurcation in an autonomous hybrid dynamical model of church bell

Abstract In this paper we present the bifurcation analysis of the yoke-bell-clapper system which corresponds to the biggest bell “Serce Lodzi” mounted in the Cathedral Basilica of St Stanislaus Kostka, Lodz, Poland. The mathematical model of the system considered in this work has been derived and verified based on measurements of dynamics of the real bell. We perform numerical analysis both by direct numerical integration and path-following method using toolbox ABESPOL (Chong, 2016). By introducing the active yoke the position of the bell-clapper system with respect to the yoke axis of rotation can be easily changed and it can be used to probe the system dynamics. We found a wide variety of periodic and non-periodic solutions, and examined the ranges of coexistence of solutions and transitions between them via different types of bifurcations. Finally, a new type of bifurcation induced by a grazing event - an “impact adding bifurcation” has been proposed. When it occurs, the number of impacts between the bell and the clapper is increasing while the period of the system’s motion stays the same.

Effects of hydrodynamic dispersion on the stability of buoyancy-driven porous media convection in the presence of first order chemical reaction

Carbon dioxide (text {CO}_2) sequestration is one of several long-term solutions suggested to decrease the amount of greenhouse gases in the atmosphere. Among different methods of carbon dioxide sequestration, the dissolution of text {CO}_2 in deep saline aquifers is considered one of the most effective. A significant number of studies are currently being carried out to provide a good understanding of the physical mechanisms involved in this type of storage. The present work focuses on the hydrodynamic part of the problem: setting a model for carbon dioxide-loaded flows in an idealised two-dimensional geometry. It considers the impact of hydrodynamic dispersion in porous media on the development of convective instabilities. Particular attention is paid to the mathematical form of the dispersion tensor widely used in porous media studies, and a new type of bifurcation is investigated. We show that the analysis of bifurcations from the no-flow steady-state solution is a continuous but non-smooth problem, which is a key feature of the analysis. Although the problem is non-smooth, it is also shown that the basic behaviours of linear stability analysis are observed in its solution.

Degenerate “Star” Bifurcations in a Twinkling Oscillator

We have studied a nonlinear spring-mass chain loaded by a quasistatic pull. The spring forces are assumed to be cubic with intervals of negative stiffness. Depending on the parameters, the system has multiple equilibria. The normal form and the bifurcation behaviors for the single- and two-degree-of-freedom systems are studied in detail. A new type of bifurcation, which we refer to as a star bifurcation, has been observed for the symmetric two-degree-of-freedom system. This bifurcation is of codimension-four for the undamped case and codimension-three or two for the damped case, depending on the form of the damping.

New bifurcations of basin boundaries involving Wada and a smooth Wada basin boundary

This paper demonstrates and analyses double heteroclinic tangency in a three-well potential model, which can produce three new types of bifurcations of basin boundaries including from smooth to Wada basin boundaries, from fractal to Wada basin boundaries in which no changes of accessible periodic orbits happen, and from Wada to Wada basin boundaries. In a model of mechanical oscillator, it shows that a Wada basin boundary can be smooth.

Magnetohydrodynamic evolution of magnetic skeletons

The heating of the solar corona is probably due to reconnection of the highly complex magnetic field that threads throughout its volume. We have run a numerical experiment of an elementary interaction between the magnetic field of two photospheric sources in an overlying field that represents a fundamental building block of the coronal heating process. The key to explaining where, how and how much energy is released during such an interaction is to calculate the resulting evolution of the magnetic skeleton. A skeleton is essentially the web of magnetic flux surfaces (called separatrix surfaces) that separate the coronal volume into topologically distinct parts. For the first time, the skeleton of the magnetic field in a three-dimensional numerical magnetohydrodynamic experiment is calculated and carefully analysed, as are the ways in which it bifurcates into different topologies. A change in topology normally changes the number of magnetic reconnection sites. In our experiment, the magnetic field evolves through a total of six distinct topologies. Initially, no magnetic flux joins the two sources. Then, a new type of bifurcation, called a global double-separator bifurcation , takes place. This bifurcation is probably one of the main ways in which new separators are created in the corona (separators are field lines at which three-dimensional reconnection takes place). This is the first of five bifurcations in which the skeleton becomes progressively more complex before simplifying. Surprisingly, for such a simple initial state, at the peak of complexity there are five separators and eight flux domains present.

Nonlinear Behavior of Reactor−Separator Networks: Influence of the Energy Balance Formulation

The dynamic behavior of a stand alone flash with a constant heating rate and a corresponding reactor−separator system with recycle is analyzed by means of different model formulations which are frequently applied. It is shown that the predictions of the different model formulations are rather close for the stand alone flash. In particular, in all cases, a stable steady state is predicted. In contrast to this, it is shown that the dynamics of the corresponding reactor−separator system can depend on the specific model formulation. For a rigorous model with a dynamic energy balance, a new type of bifurcation is found, where the eigenvalues have a pole with a change of sign. Similar effects are reported for systems with negligible and finite transportation delay of the recycle. For the latter, multiple periodic solutions are found for a model with a quasistatic energy balance, which will disappear because of the new type of bifurcation if a rigorous dynamic energy balance is applied.

Crossing Bifurcations and Unstable Dimension Variability

A crisis is a global bifurcation in which a chaotic attractor has a discontinuous change in size or suddenly disappears as a scalar parameter of the system is varied. In this Letter, we describe a global bifurcation in three dimensions which can result in a crisis. This bifurcation does not involve a tangency and cannot occur in maps of dimension smaller than 3. We present evidence of unstable dimension variability as a result of the crisis. We then derive a new scaling law describing the density of the new portion of the attractor formed in the crisis. We illustrate this new type of bifurcation with a specific example of a three-dimensional chaotic attractor undergoing a crisis.

Multistability of reentrant rhythms in an ionic model of a two-dimensional annulus of cardiac tissue

The dynamics of reentry in a model of a two-dimensional annulus of homogeneous cardiac tissue, with a Beeler-Reuter type formulation of the membrane ionic currents, is examined. The bifurcation structure of the sustained propagated solutions is described as a function of Rin and Rout, the inner and outer radii of the annulus. The transition from periodic to quasiperiodic reentry occurs at a critical Rin, which first diminishes and then saturates as Rout is increased. The reduction of the critical Rin is a consequence of the increase of the wave-front curvature. There is a range of Rin below the critical radius in which two distinct quasiperiodic solutions coexist. Each of these solutions disappears at its own specific value of Rin, and their annihilation is preceded by a new type of bifurcation leading to a regime of propagation with transient successive detachments of the wave front from the inner border of the annulus.

Gravity travelling waves for two superposed fluid layers, one being of infinite depth: a new type of bifurcation.

In this paper, we study the travelling gravity waves in a system of two layers of perfect fluids, the bottom one being infinitely deep, the upper one having a finite thickness h. We assume that the flow is potential and the dimensionless parameters are the ratio between densities rho = rho(2)/rho(1) and lambda = gh/c(2). We study special values of the parameters such that lambda(1 - rho) is near 1(-), where a bifurcation of a new type occurs. We formulate the problem as a spatial reversible dynamical system, where U = 0 corresponds to a uniform state (velocity c in a moving reference frame), and we consider the linearized operator around 0. We show that its spectrum contains the entire real axis (essential spectrum), with, in addition, a double eigenvalue in 0, a pair of simple imaginary eigenvalues +/-ilambda at a distance O(1) from 0, and for lambda(1 - rho) above 1, another pair of simple imaginary eigenvalues tending towards 0 as lambda(1 - rho) --> 1(+). When lambda(1 - rho) <or= 1, this pair disappears into the essential spectrum. The rest of the spectrum lies at a distance at least O(1) from the imaginary axis. We show in this paper that for lambda(1 - rho) close to 1(-) there is a family of periodic solutions similar to that in the Lyapunov-Devaney theorem (despite the resonance due to the point 0 in the spectrum). Moreover, showing that the full system can be seen as a perturbation of the Benjamin-Ono equation, coupled with a nonlinear oscillation, we also prove the existence of a family of homoclinic connections to these periodic orbits, provided that these ones are not too small.

Non–existence of truly solitary waves in water with small surface tension

This paper considers permanent capillary–gravity waves on the free surface of a two–dimensional incompressible inviscid fluid of finite depth. It is shown that there are no solitary-wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals. The proof is based upon an estimate of a constant that is related to the approximation of the solution, if it exists, near its singularity. The approximation satisfies a fourth–order nonlinear ordinary differential equation when the solution is extended to the complex plane. Then the non–existence of truly solitary waves is obtained by using a contradiction on this constant.

On a new type of bifurcation of limit cycles for a planar cubic system

On a new type of bifurcation of limit cycles for a planar cubic system

Theory of the periodic orbits of a chaotic quantum well

A theory is developed for the periodic orbits of an electron trapped in a rectangular potential well under the influence of an electric field normal to the barriers and a magnetic field. When the magnetic field is parallel to the electric field the dynamics of an electron in the well is integrable; however, when it is tilted by an angle $\ensuremath{\theta}$ the system undergoes a transition to chaos. Motivated by recent experimental and theoretical studies of magnetotunneling in quantum wells that emphasize the role of periodic orbits, we present here a unified theory of all the periodic orbits within the well that are of relevance to experiments. We define the appropriate scaled variables for the problem, which we divide into two qualitatively different cases, the single-barrier model (depending on two parameters) and the double-barrier model (depending on three parameters). We show that in both cases all relevant orbits are related to bifurcations of period-one traversing orbits. A full analytic theory is derived for the period and stability of these traversing orbits; and analytic and numerical results are obtained for the important period-two and period-three orbits. An unusual feature of the classical mechanics of the double-barrier is a discontinuity in the classical Poincar\'e map, which leads to a new type of bifurcation that we term a cusp bifurcation. We show that all the periodic orbits that traverse the well exist only in finite intervals of voltage and magnetic field, appearing and disappearing in bifurcations. These intervals are shown to correspond to the appearance of new resonance peaks in the experimental data, laying the foundation for a quantitative semiclassical treatment of the system.

Basin Fractalization Due to Focal Points in a Class of Triangular Maps

For a class of rational triangular maps of a plane, characterized by the presence of points in which a component assumes the form [Formula: see text], a new type of bifurcation is evidenced which creates loops in the boundaries of the basins of attraction. In order to explain such bifurcation mechanism, new concepts of focal point and line of focal values are defined, and their effects on the geometric behavior of the map and of its inverses are studied in detail. We prove that the creation of loops, which generally constitute the boundaries of lobes of the basins issuing from the focal points, is determined by contacts between basin boundaries and the line of focal values. A particular map is proposed for which the sequence of such contact bifurcations occurs, causing a fractalization of basin boundaries. Through the analytical and the numerical study of this example new structures of the basins of attraction are evidenced, characterized by fans of stable sets issuing from the focal points, assuming the shape of lobes and arcs, the latter created by the merging of lobes due to contacts between the basin boundaries and the critical curve LC.

Grazing bifurcations in impact oscillators.

Impact oscillators demonstrate interesting dynamical features. In particular, new types of bifurcations take place as such systems evolve from a nonimpacting to an impacting state (or vice versa), as a system parameter varies smoothly. These bifurcations are called grazing bifurcations. In this paper we analyze the different types of grazing bifurcations that can occur in a simple sinusoidally forced oscillator system in the presence of friction and a hard wall with which the impacts take place. The general picture we obtain exemplifies universal features that are predicted to occur in a wide variety of impact oscillator systems.