Non-isolated Singular Point Research Articles

On bifurcation of non-isolated singular points arising in a problem of two-phase fluid motion in a pipe

The dynamics of gas suspension in a pipe is studied. It is described by a system of three autonomous differential equations, whose equilibrium (singular) points are not isolated, but form several continuous curves. One of these curves belongs to the transonic plane. The dynamics in a neighborhood of this plane is interesting from the physical viewpoint. In this work, the dynamics in a neighborhood of the curves of singular points is investigated. The main tools of this work are the center manifold theorem and the reduction principle. A peculiarity of the system is that the curves of singular points intersect each other. Moreover, at the intersection point a bifurcation occurs. Results of the work are the description of the dynamics in a neighborhood of singular points, the proof that a curve of singular points belongs to a center manifold and the description of the bifurcation, which occurs at the intersection point of the curves of singular points.

Periodic Solutions Bifurcating From a Curve of Singularity of the Jerk System

We investigate a periodic solution which bifurcates from a curve of the singularity of the jerk system in .More precisely, we give the explicit states for the existence of a periodic solution of the jerk system with a nonisolated singular point, where for each singular point has a simple pair of purely imaginary and one zero eigenvalues. We recall for this point of singularity as a zero-Hopf (z-H) singular point. The coefficients in the jerk system are described for which the z-H singularity occur at each point of that curve of singularity. We show that for each point at that curve of singularity there is only one family of parameters which exhibits such type of singular points. The method of averaging in the second order is utilized to determine one periodic solution which bifurcates from any point of that curve of singularity. As far as, we realize that this investigation is the study on bifurcations from a curve of nonisolated z-H singularity to provide a periodic solution via the method of averaging. Under a generic small perturbation at the parameters, we prove that a periodic solution will be bifurcated at any point that located on a curve of a singularity of the jerk system.

An analytic study of the Vaidyanathan chaotic dynamics in Lorentzian metric which has no similar dynamics in Riemannian metric

In this paper we investigate behaviour at infinity of a physical 3-dimensional chaotic system via Poincare compactification method. This system has been introduced by Vaidyanathan et al. (2017a). We plot the phase portrait of the system for parameters a and b which appear in the nonlinear part of the system. We will see a set of non-isolated singular points at infinity is a hyperbolic set by considering a Lorentzian metric g on R2 and it is not a hyperbolic set in the sense of Riemannian metrics. We compute a first integral for the resulted system and we prove there is at most a generalised rational first integral when one of its parameters is equal to zero.

A note on local integrability of differential systems

For an n-dimensional local analytic differential system x˙=Ax+f(x) with f(x)=O(|x|2), the Poincaré nonintegrability theorem states that if the eigenvalues of A are not resonant, the system does not have an analytic or a formal first integral in a neighborhood of the origin. This result was extended in 2003 to the case when A admits one zero eigenvalue and the other are non-resonant: for n=2 the system has an analytic first integral at the origin if and only if the origin is a non-isolated singular point; for n>2 the system has a formal first integral at the origin if and only if the origin is not an isolated singular point. However, the question of whether the system has an analytic first integral at the origin provided that the origin is not an isolated singular point remains open.

Bifurcation Approach to Analysis of Travelling Waves in Some Taxis–Cross-Diffusion Models

An overview of recently obtained authors’ results on traveling wave solutions of some classes of PDEs is presented. The main aim is to describe all possible travelling wave solutions of the equations. The analysis was conducted using the methods of qualitative and bifurcation analysis in order to study the phase-parameter space of the corresponding wave systems of ODEs. In the first part we analyze the wave dynamic modes of populations described by the “growth - taxis - diffusion polynomial models. It is shown that “suitable nonlinear taxis can affect the wave front sets and generate non-monotone waves, such as trains and pulses, which represent the exact solutions of the model system. Parametric critical points whose neighborhood displays the full spectrum of possible model wave regimes are identified; the wave mode systematization is given in the form of bifurcation diagrams. In the second part we study a modified version of the FitzHugh-Nagumo equations, which model the spatial propagation of neuron firing. We assume that this propagation is (at least, partially) caused by the cross-diffusion connection between the potential and recovery variables. We show that the cross-diffusion version of the model, besides giving rise to the typical fast travelling wave solution exhibited in the original “diffusion FitzHugh-Nagumo equations, additionally gives rise to a slow traveling wave solution. We analyze all possible traveling wave solutions of the model and show that there exists a threshold of the cross-diffusion coefficient (for a given speed of propagation), which bounds the area where “normal impulse propagation is possible. In the third part we describe all possible wave solutions for a class of PDEs with cross-diffusion, which fall in a general class of the classical Keller-Segel models describing chemotaxis. Conditions for existence of front-impulse, impulse-front, and front-front traveling wave solutions are formulated. In particular, we show that a non-isolated singular point in the ODE wave system implies existence of free-boundary fronts.

Singularities of a geodesic flow on surfaces with a cuspidal edge

This paper is a study of singularities of geodesic flows on surfaces with nonisolated singular points that form a smooth curve (like a cuspidal edge). The main results of the paper are normal forms of the corresponding direction field on the tangent bundle of the plane of local coordinates and the projection of its trajectories to the surface.

Families of traveling impulses and fronts in some models with cross-diffusion

An analysis of traveling wave solutions of partial differential equation (PDE) systems with cross-diffusion is presented. The systems under study fall in a general class of the classical Keller–Segel models to describe chemotaxis. The analysis is conducted using the theory of the phase plane analysis of the corresponding wave systems without a priory restrictions on the boundary conditions of the initial PDE. Special attention is paid to families of traveling wave solutions. Conditions for existence of front–impulse, impulse–front, and front–front traveling wave solutions are formulated. In particular, the simplest mathematical model is presented that has an impulse–impulse solution; we also show that a non-isolated singular point in the ordinary differential equation (ODE) wave system implies existence of free-boundary fronts. The results can be used for construction and analysis of different mathematical models describing systems with chemotaxis.

On the Behavior of Integral Surfaces of a Pfaff Equation with a Nonclosed Singular Curve

λ = 0, 1. The general solutions of Eqs. (2)–(5) can be found in [1]. One can readily see that Eqs. (2)–(5) have at most two isolated singular points. Obviously, nonisolated singular points of these equations fill a hyperbola, parabola, or two intersecting straight lines. By making the change of variables y = y1 − z1, z = y1 + z1 in Eq. (2), we obtain ( a1x+ λx + a2y 1 − a2z 1 ) dx+ 2 (b1y1 + b2xy1) dy1 − 2 (b1z1 + b2xz1) dz1 = 0. (2′)

Implicit differential equations and vector fields with non-isolated singular points

Vector fields with singularities that are not isolated, but form a smooth submanifold of the phase space of codimension 2 are studied. Fields of this kind occur, for instance, in the analysis of implicit differential equations. Furthermore, under slight perturbations of the original problem the variety of singular points does not disappear or degenerate, but merely deforms. Results on the structure of invariant manifolds of such fields are obtained, along with smooth normal forms for certain cases.

On stability when a singular point is non-isolated

Assuming that m zero roots correspond to the non-isolated singular point of the equations of disturbed motion, the problem of stability when some of the remaining roots are pure imaginary, while others have a negative real part in inessentially singular cases, is considered. This situation is typical for non-holonomic systems. The stability and instability theorems obtained are used to study the stability of the rotations of a celtic stone on the boundary of the domain of stability.

Bifurcation in the neighbourhood of a non-isolated singular point

The Lyapunov-Schmidt method for bifurcation problems has, until recently, been applied only to operator equations whose singular points are isolated in the solution set of the equation. For bifurcation at a multiple eigenvalue involving several parameters, however, singular points are often non-isolated. In this paper, the case of intersecting curves of singular points is considered. Under natural hypotheses on these curves, and assuming suitable transversality conditions on the first order nonlinearity of the operator, it is shown that the solution set of the equation may be completely determined locally in terms of the solutions of associated finite dimensional polynomial equations.

Phase plane analysis of non-linear systems using weighted linearization

An analysis based on weighted linear approximations is developed for certain non-linear ordinary differential equations, and is applied to the description of singular points and trajectories in the phase plane. This technique is compared with the analysis of Poincaré based on Taylor series linearization where the latter is applicable. Numerical results are obtained for a quadratic friction problem, and compared with the first-order Krylov-Bogoliubov-Mitropolsky result. The new technique is also successfully applied to problems of Lienard, Keil and Malgarini, and to a special case of a non-isolated singular point, where Taylor series linearizations are inapplicable or yield qualitatively incorrect results.