Zero Singularity Research Articles

The Steady State Bifurcation for General Network-Organized Reaction-Diffusion Systems and Its Application in a Metapopulation Epidemic Model

Because of the high dimensionality of reaction-diffusion systems on networks, exploration for the bifurcation properties of such systems has been far less thorough than their corresponding parabolic PDE systems. With the overcoming of this long-standing difficulty, we consider the steady state bifurcation for the general network-organized reaction-diffusion system and derive the algorithm of calculating the perturbation incorporated restricted system associated with the simple zero singularity, through seeking out the parameter dependent center manifolds belonging to the unique zero eigenvalue. Rigorous theoretical analysis for the third-order truncated restricted system shows that the complex underlying networks can trigger some bifurcation types differing from the usual supercritical and subcritical pitchfork bifurcations and so enrich the dynamical behaviors of reaction-diffusion process in networks. To illustrate the application for our obtained theoretical results, we conduct a strict steady state bifurcation analysis for a metapopulation epidemic system defined on four typical connected undirected underlying networks, and find that there exist some new multistability phenomena and hysteresis effects in the irregular small-word and scale-free lattice network.

Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition

A diffusive predator-prey model with nonlocal prey competition and the homogeneous Neumann boundary conditions is considered, to explore the effects of nonlocal reaction term. Firstly, conditions of the occurrence of Hopf, Turing, Turing-Turing and double zero bifurcations, are established. Then, several concise formulas of computing normal form at a double zero singularity for partial functional differential equations, are provided. Next, via analyzing normal form derived by utilizing these formulas, we find that diffusive predator-prey system admits interesting spatiotemporal dynamics near the double zero singularity, like tristable phenomenon that a stable spatially inhomogeneous periodic solution with the shape of \begin{document}$ \cos\omega_0 t\cos\frac{x}{l}- $\end{document} like which is unstable in model without nonlocal competition and also greatly different from these with the shape of \begin{document}$ \cos\omega_0 t+\cos\frac{x}{l}- $\end{document} like resulting from Turing-Hopf bifurcation, coexists with a pair of spatially inhomogeneous steady states with the shape of \begin{document}$ \cos\frac{x}{l}- $\end{document} like. At last, numerical simulations are shown to support theory analysis. These investigations indicate that nonlocal reaction term could stabilize spatially inhomogeneous periodic solutions with the shape of \begin{document}$ \cos\omega_0 t\cos\frac{kx}{l}- $\end{document} like for reaction-diffusion systems subject to the homogeneous Neumann boundary conditions.

SECOND-ORDER NORMAL FORMS FOR N-DIMENSIONAL SYSTEMS WITH A NILPOTENT POINT

Normal forms theory is one of the most powerful tools for the study of nonlinear differential equations, in particular, for stability and bifurcation analysis. Many works paid attention to normal forms associated with nilpotent Jacobian where the critical eigenvalues have algebraic multiplicity k ($ k>1 $) and geometric multiplicity one, and in particular, the case $ k>2 $ is more complicated for determining unfolding. Despite a lot of theoretical results on nilpotent normal forms have been obtained, computation developing can not satisfy practical applications. To our knowledge, no results have been reported on the computation of explicit formulas of the nilpotent normal forms for $ k>3 $ with perturbation parameters. The main difficulty is how to determine the complementary spaces of the Lie transformation. In this paper, we achieve the following results. (1) A simple dimension formula for the complementary space of the Lie transform; (2) a simple direct method to determine a basis of the complementary spaces; (3) a simple direct method to determine the projection of any vector to the complementary spaces. Using this method, the second-order normal forms for any n-dimensional nilpotent systems can be given easily. As an illustrative application, the normal forms for the vector field with triple-zero or four-fold zero singularity and functional differential equation with a triple-zero singularity are presented, and explicit formulas for the normal form coefficients with three or four unfolding parameters are obtained.

Bogdanov-Takens bifurcation in a neutral BAM neural networks model with delays.

In this study, the authors first discuss the existence of Bogdanov-Takens and triple zero singularity of a five neurons neutral bidirectional associative memory neural networks model with two delays. Then, by utilising the centre manifold reduction and choosing suitable bifurcation parameters, the second-order and the third-order normal forms of the Bogdanov-Takens bifurcation for the system are obtained. Finally, the obtained normal form and numerical simulations show some interesting phenomena such as the existence of a stable fixed point, a pair of stable non-trivial equilibria, a stable limit cycles, heteroclinic orbits, homoclinic orbits, coexistence of two stable non-trivial equilibria and a stable limit cycles in the neighbourhood of the Bogdanov-Takens bifurcation critical point.

Cosmic Initial Singularities in a Single Repeating Universe as Opposed to Their Behavior in a Multiverse

When the initial radius of the universe is set in four dimensions and if there is only ONE repeating universe, then the initial radii of the universe is R → 0 or gets very close to zero if we use the Einstein Equations modified by Stoica. The Einstein Equations are reset by Stoical in a formalism which removes in four dimensions, the big bang singularity pathology. So then the reason for Planck length no longer holds. This manuscript assumes a repeating single universe. We present entanglement entropy in the early universe with a shrinking scale factor, due to Muller and Lousto, and show that there are consequences due to initial entangled for a time dependent horizon radius in cosmology, with (flat space conditions) for conformal time. Even if the 3-dimensional spatial length goes to zero. Our new manuscript presentation sets as a starting point a cosmology with a non-zero Λ vacuum energy. The non-zero Λ vacuum energy, initial configuration of the universe permits us to keep in an information theory stand point (information theory), computational bits for our configuration of cosmological expansion. This assemblage of computational bits occurs in cosmological evolution even if in an initial four-dimensional cosmology, we have the initial radii of the universe R → 0. We also find that in the case of a multiverse, such considerations will not hold and that cosmic singularities have a more different characteristic in the multiverse setting than in the single universe repeated over and over again, i.e. using an argument borrowed and modified from Kauffman, the multiverse will not mandate “perfect” singularities. The existence of a multiverse may allow for non zero singularities in lieu with the Kauffman argument cited at the end of the document, plus the lower pre big bang temperatures which may allow for the survival of gravitons just before the onset of the cosmological expansion phase, if a multiverse exists embedding our present universe.

Zero singularities of codimension two in a delayed predator-prey diffusion system

In this paper, a delayed predator-prey diffusion system with Neumann boundary condition is addressed. By complex computation and rigorous analysis, it is obtained that the unique positive equilibrium of the system under consideration is a Bogdanov-Takens (B-T) singularity under certain conditions, whereas this is not true for the corresponding system without diffusion. By using the normal form theory and the center manifold reduction theorem for partial functional differential equations (PFDEs), the normal form of the B-T singularity is provided. Finally, numerical simulations are carried out to illustrate the theoretical results presented.

Characterizing the Codimension of Zero Singularities for Time-Delay Systems

The analysis of time-delay systems mainly relies on detecting and understanding the spectral values bifurcations when crossing the imaginary axis. This paper deals with the zero singularity, essentially when the zero spectral value is multiple. The simplest case in such a configuration is characterized by an algebraic multiplicity two and a geometric multiplicity one, known as the Bogdanov-Takens singularity. Moreover, in some cases the codimension of the zero spectral value exceeds the number of the coupled scalar-differential equations. Nevertheless, to the best of the author's knowledge, the bounds of such a multiplicity have not been deeply investigated in the literature. It is worth mentioning that the knowledge of such an information is crucial for nonlinear analysis purposes since the dimension of the projected state on the center manifold is none other than the sum of the dimensions of the generalized eigenspaces associated with spectral values with zero real parts. Motivated by a control-oriented problems, this paper provides an answer to this question for time-delay systems, taking into account the parameters' algebraic constraints that may occur in applications. We emphasize the link between such a problem and the incidence matrices associated with the Birkhoff interpolation problem. In this context, symbolic algorithms for LU-factorization for functional confluent Vandermonde as well as some classes of bivariate functional Birkhoff matrices are also proposed.

Qualitative analysis of a predator–prey system with double Allee effect in prey

Growing biological evidence from various ecosystems (Berec et al., 2007) suggests that the Allee effect generated by two or more mechanisms can act simultaneously on a single population. Surprisingly, prey–predator system incorporating multiple Allee effect is relatively poorly studied in literature. In this paper, we consider a ratio dependent prey–predator system with a double Allee effect in prey population growth. We have taken into account the case of the strong and the weak Allee effect separately and study the stability and complete bifurcation analysis. It is shown that the model exhibits the bi-stability and there exists separatrix curve(s) in the phase plane implying that dynamics of the system is very sensitive to the variation of the initial conditions. Generic normal forms at double zero singularity are derived by a rigorous mathematical analysis to show the different types of bifurcation of the proposed model system including Bogdanov–Takens bifurcation, saddle–node bifurcation curve, a Hopf bifurcation curve and a homoclinic bifurcation curve. The complete analysis of possible topological structures including elliptic, parabolic, or hyperbolic orbits, and any combination of them in a neighborhood of the complicated singular point (0,0) in the interior of the first quadrant is explored by using a blow up transformation. These structures have important implications for the global behavior of the model. Numerical simulation results are given to support our theoretical results. Finally, the paper concludes with a discussion of the ecological implications of our analytic and numerical findings.

Zero singularity of codimension two or three in a four-neuron BAM neural network model with multiple delays

In this paper, a four-neuron BAM neural network model with multiple delays is considered. The existence conditions under which that the origin of the system is Bogdanov–Takens (B–T) or triple zero singularity are given. By choosing the connected weights as bifurcation parameters and using the center manifold reduction and the normal form theory and the formula developed by Xu and Huang (J Differ Equ 244:582–598 2008) and Qiao et al. (Chinese Ann Math Ser A 31:59–70 2010), the versal unfoldings and the normal forms for this singularity were given to analyze the behaviors of the system. This paper is a further study of paper Cao and Xiao (IEEE Trans Neural Netw 18:416–430 2007).

Cosmic Initial Singularities in a Single Repeating Universe as Opposed to Their Behavior in a Multiverse

When initial radius of the universe in four dimensions and there is only ONE repeating universe then Rinitial→0 or gets very close to zero if Stoica actually derived Einstein equations in a formalism which remove in four dimensions the big bang singularity pathology. So then the reason for Planck length no longer holds. This assumes a repeating single universe. We present entanglement entropy in the early universe with a shrinking scale factor, due to Muller and Lousto , and show that there are consequences due to initial entanged for a time dependent horizonradius rH in cosmology, with (flat space conditions) rH=η for conformal time . Even if the 3 dimensional spatial length goes to zero. This construction preserves a minimum non zero ∧ vacuum energy, and in doing so keep the bits, for computational bits cosmological evolution even if in four dimensions we have Rinitial→0. We also find that in the case of a multiverse, that such considerations will not hold and that cosmic singularities have a different characteristic in the multiverse setting than in the single universe repeated over and over again. i.e. using an argument borrowed and modified from Kauffman, the multiverse will not mandate ‘perfect’ singularities. The existence of a multiverse may allow for non zero singularities in lieu with the Kauffman argument cited at the end of the document, plus the lower pre big bang temperatures which may allow for the survivial of gravitons just before the onset of the cosmological expansion phase, if a multiverse exists embedding our present universe.

Triple-zero singularity of a Kaldor–Kalecki model of business cycles with delay

In this manuscript, we study triple zero singularity of a Kaldor–Kalecki model of business cycles with delay in both the gross product and the capital stock. By using the frameworks of Campbell–Yuan [1] and Faria–Magalhães [2, 3], the normal form on the center manifold is derived for this singularity and hence the corresponding bifurcation diagrams such as Hopf, BT, zero-Hopf, and homoclinic bifurcations are obtained. An example is given to verify some theoretical results.

Bogdanov-Takens and Triple Zero Bifurcations of a Delayed Modified Leslie-Gower Predator Prey System

A delayed modified Leslie-Gower predator prey system with nonlinear harvesting is considered. The existence conditions that an equilibrium is Bogdanov-Takens (BT) or triple zero singularity of the system are given. By using the center manifold reduction, the normal form theory, and the formulae developed by Xu and Huang, 2008 and Qiao et al., 2010, the normal forms and the versal unfoldings for this singularity are presented. The Hopf bifurcation of the system at another interior equilibrium is analyzed by taking delay (small or large) as bifurcation parameter.

Codimension two bifurcation in a simple delayed neuron model

This paper reports a simple delayed neuron model. Bogdanov-Takens bifurcation is studied by using the center manifold reduction and the normal form method for retarded functional differential equation. We get the versal unfolding of the norm forms at double zero singularity and show that the model can exhibit pitchfork, Hopf, homoclinic bifurcation and saddle node bifurcation of periodic orbits. Some numerical simulations are given to support the analytic results.

Zero singularities in a ring network with two delays

In this paper, we consider a neural network model consisting of three neurons with delayed self- and nearest-neighbor connections. We provide multiple bifurcations of the zero solution of the system near zero eigenvalue singularity. Taking the coupling coefficients as the bifurcation parameters, four kinds of zero singularities are demonstrated through center manifold reduction and normal form calculation.

CODIMENSION ONE BIFURCATION OF EQUIVARIANT NEURAL NETWORK MODEL WITH DELAY

In this paper, we consider double zero singularity of a symmetric BAM neural network model with a time delay. Based on the normal form approach and the center manifold theory, we obtain the normal form on the centre manifold with double zero singularity. Some numerical simulations support our analysis results.

Simple zero singularity analysis in a coupled FitzHugh–Nagumo neural system with delay

A FitzHugh–Nagumo (FHN) model with delayed coupling is considered to investigate the steady state bifurcation due to the coupling strength and delay. The center manifold reduction and normal form method are employed to study the bifurcation from zero singularity. We show that the coupling strength can induce both the pitchfork and transcritical bifurcations and the coupled delay has the most significant impact on these bifurcations. An example is applied to display the above results. Our investigation will contribute to the understanding of the influence of the time delay in signal transmission on the dynamics of coupled neurons.

Zero singularities of codimension two and three in delay differential equations

We give conditions under which a general class of delay differential equations has a point of Bogdanov–Takens or a triple zero bifurcation. We show how a centre manifold projection of the delay equations reduces the dynamics to two- or three-dimensional systems of ordinary differential equations. We put these equations in normal form and determine how the coefficients of the normal forms depend on the original parameters in the model. Finally we apply our results to two neural models and compare the predictions of the theory with numerical bifurcation analysis of the full equations. One model involves a transcritical bifurcation, hence we derive and analyse the appropriate unfoldings for this case.

Multiple stabilities analysis in a magnetic bearing system with time delays

Abstract A kind of magnetic bearing system with time delay is considered. Firstly, multiple stabilities of the model is investigated. According to the analysis results, the bifurcation diagram is drawn in the appropriate parameter plane. Then the center manifold reduction and normal form computation for simple zero singularity are performed and detailed bifurcation analysis are carried out. Finally, some numerical simulations are presented to illustrate the results found.

Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling

Normal form theory is one of the most power tools for the study of nonlinear differential equations, in particular, for stability and bifurcation analysis. Recently, many researchers have paid attention to further reduction of conventional normal forms (CNF) to so called the simplest normal form (SNF). However, the computation of normal forms has been restricted to systems which do not contain perturbation parameters (unfolding). The computation of the SNF is more involved than that of CNFs, and the computation of the SNF with unfolding is even more complicated than the SNF without unfolding. Although some author mentioned further reduction of the SNF, no results have been reported on the exact computation of the SNF of systems with perturbation parameters. This paper presents an efficient method for computing the SNF of differential equations with perturbation parameters. Unlike CNF theory which uses an independent nonlinear transformation at each order, this approach uses a consistent nonlinear transformation through all order computations. The particular advantage of the method is able to provide an efficient recursive formula which can be used to obtain the nth-order equations containing the nth-order terms only. This greatly saves computational time and computer memory. The recursive formulations have been implemented on computer systems using Maple. As an illustrative example, the SNF for single zero singularity is considered using the new approach.

New aspects in the unfolding of the nilpotent singularity of codimension three

This paper is concerned with three-dimensional vector fields and more specifically with the study of dynamics in unfoldings of the nilpotent singularity of codimension three. The ultimate goal is to understand the dynamics and bifurcations in the unfolding of the singularity. However, it is clear from the literature that the bifurcation diagram is very complicated and a complete study is far beyond the current possibilities, not only from a theoretical point of view but also from a numerical point of view, despite recent developments of computational methods for dynamical systems. Since all complicated dynamical behaviour is known to be of small amplitude, shrinking to the singularity for parameter values tending to the bifurcation parameter, the aim in this paper is especially to focus on a different aspect that might be interesting in the study of global bifurcation problems in the presence of such a nilpotent singularity of codimension three. The notion is introduced of 'traffic regulator' and the specific sets called the 'inset' and 'outset', which give a new framework for studying a transition map in a cylindrical neighbourhood of the singularity that contains all the non-trivial dynamics that can bifurcate from the singularity, focusing on the domains on which the transition map is defined. A list is also given of open problems which are believed to be helpful for future investigation of the bifurcations from the nilpotent triple zero singularity in 3.