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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. [ABSTRACT FROM AUTHOR]

In this paper we focus on a nonlocal reaction-diffusion population model. Such a model can be used to describe a single species which is diffusing, aggregating, reproducing and competing for space and resources, with the free boundary representing the expanding front. The main objective is to understand the influence of the nonlocal term in the form of an integral convolution on the dynamics of the species. Precisely, when the species successfully spreads into infinity as t→∞, it is proved that the species stabilizes at a positive equilibrium state under rather mild conditions. Furthermore, we obtain a upper bound for the spreading of the expanding front. [ABSTRACT FROM AUTHOR]

The paper is concerned with a predator-prey diffusive system subject to homogeneous Neumann boundary conditions, where the growth rate $(\frac{\alpha}{1+\beta v})$ of the predator population is nonlinear. We study the existence of equilibrium solutions and the long-term behavior of the solutions. The main tools used here include the super-sub solution method, the bifurcation theory and linearization method. [ABSTRACT FROM AUTHOR]

In this paper, a predator-prey model with nonmonotonic functional response is concerned. Using spectrum analysis and bifurcation theory, the bifurcating solution and its stability of the model are investigated. We discuss the bifurcation solution which emanates from the semi-trivial solution by taking the death rate as a bifurcation parameter. Furthermore, by fixed point’s index theory, the result of existence or nonexistence of positive steady states of the model is also obtained. [ABSTRACT FROM AUTHOR]

In this paper a method to rigorously compute several non trivial solutions of the Gray-Scott reaction-diffusion system defined on a 2-dimensional bounded domain is presented. It is proved existence, within rigorous bounds, of non uniform patterns significantly far from being a perturbation of the homogenous states. As a result, a non local diagram of families that bifurcate from the homogenous states is depicted, also showing coexistence of multiple solutions at the same parameter values. Combining analytical estimates and rigorous computations, the solutions are sought as fixed points of a operator in a suitable Banach space. To address the curse of dimensionality, a variation of the existing technique is presented, necessary to enable successful computations in reasonable time. [ABSTRACT FROM AUTHOR]

We consider a modification of the model proposed by Abrams and Strogatz to describe the death of a language when it competes with a stronger one within the same community of speakers. The modification opened the possibility of coexistence of both languages under some conditions, but so far it has not been possible to write down the expression of the equilibrium points. In this paper, we nontrivially use bifurcation theory to calculate under which conditions such coexistence arises; namely, we calculate the specific ranges of the parameters that describe the modified model to have this situation, paying special attention to the cases that yield a stable cohabitation of two monolingual populations along with a bilingual one. [ABSTRACT FROM AUTHOR]

Enumeration problems for the central configurations of the Newtonian $n$ body problem are hard for $n>3$ in $\mathbb{R}^{2}$ and $n>4$ in $\mathbb{R}^{3}$. These are problems in finding the numbers of classes of central configurations for all the masses in a parameter space of positive dimensions. Many results are obtained generically. That is, rigorous proofs of the counting problems only exists for parameters not at the bifurcation points. For the bifurcation points, only numerical evidences are provided due to the complexity of the problems. In this paper, we propose an algorithm that rigorously proves results on counting central configurations for all masses in one dimensional parameter spaces. Especially, we provide an approach to find all bifurcation points and count real roots at those points, known only implicitly. A spatial restricted $(4+1)$-body problem and a planar $(1+3)$-body problem are successfully applied by our method. All results except for the equal masses for the restricted $(4+1)$-body problem are new and the results for the planar $(1+3)$-body problem are new at the bifurcation points. [ABSTRACT FROM AUTHOR]

The biomathematical model is an important tool in exploring the dynamic behavior of different species. In this paper, we deal with a predator-prey model with Beddington-DeAngelis functional response and non-selective harvesting. We discuss the existence and stability of the local bifurcation solution, which emanates from the semi-trivial solution. Moreover, by using the boundedness of positive solutions and the global bifurcation theory, we find that the local bifurcation branch can be extended to the global bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, we use rigorous numerics to compute several global smooth branches of steady states for a system of three reaction-diffusion PDEs introduced by Iida et al. [J. Math. Biol. 53(4):617–641, 2006 ] to study the effect of cross-diffusion in competitive interactions. An explicit and mathematically rigorous construction of a global bifurcation diagram is done, except in small neighborhoods of the bifurcations. The proposed method, even though influenced by the work of van den Berg et al. [Math. Comput. 79(271):1565–1584, 2010 ], introduces new analytic estimates, a new gluing-free approach for the construction of global smooth branches and provides a detailed analysis of the choice of the parameters to be made in order to maximize the chances of performing successfully the computational proofs. [ABSTRACT FROM AUTHOR]

In this paper we deal with the existence of positive solutions for the following nonlocal type of problems $$\everymath{\displaystyle} \left\{ \begin{array}{l@{\quad}l} -\Delta u = \frac{\sigma}{( \int_{\varOmega} g(u)\, dx )^p} f(u) & \mbox{in}\ \varOmega, \\[3mm] u>0 & \mbox{in}\ \varOmega, \\[1mm] u=0 & \mbox{on}\ \partial\varOmega, \end{array} \right. $$ where Ω is a bounded smooth domain in ℝ N ( N≥1), f, g are continuous positive functions, σ>0 and p∈ℝ. We give sufficient conditions on the functions f and g in order to have existence of positive solutions. [ABSTRACT FROM AUTHOR]

This paper is devoted to study the bifurcation phenomenon for scalar conservation laws with flux functions involving discontinuous coefficients. In order to deal with it, the special Cauchy initial data are taken and the interactions of stationary wave discontinuities with shock waves and rarefaction waves are considered in detail. The global solutions of this special Cauchy problem are constructed completely when the bifurcation phenomena appear in their solutions. [ABSTRACT FROM AUTHOR]

The phenomenon of resonance will be dealt with from the viewpoint of dynamical systems depending on parameters and their bifurcations. Resonance phenomena are associated to open subsets in the parameter space, while their complement corresponds to quasi-periodicity and chaos. The latter phenomena occur for parameter values in fractal sets of positive measure. We describe a universal phenomenon that plays an important role in modelling. This paper gives a summary of the background theory, veined by examples. [ABSTRACT FROM AUTHOR]

This paper is concerned with a two-species predator-prey reaction-diffusion system with Beddington-DeAngelis functional response and subject to homogeneous Neumann boundary conditions. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation in detail, the asymptotic stability of the positive constant steady-state solution and the existence of local Hopf bifurcations are investigated. Also, it is shown that the appearance of the diffusion and homogeneous Neumann boundary conditions can lead to the appearance of codimension two Bagdanov-Takens bifurcation. Moreover, by applying the normal form theory and the center manifold reduction for partial differential equations (PDEs), the explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions is given. Finally, numerical simulations supporting the theoretical analysis are also included. [ABSTRACT FROM AUTHOR]

This paper studied the stability and Hopf bifurcation of a type of protein synthesis system with time delay and negative feedback. Firstly, it is proved theoretically that the time delay, nonlinearity in the protein production and the cooperativity in the negative feedback are key factors to generate circadian oscillation; Taking time delay as a parameter, we obtained the critical value of the time delay that Hopf bifurcation generates. Secondly, based on the center manifold and normal form theorem, we derived the formulas for determining the stability of bifurcating periodic solutions and the supercritical or subcritical Hopf bifurcation. Finally, the matlab program is used to simulate the results. [ABSTRACT FROM AUTHOR]

By using Lie symmetry analysis and the method of dynamical systems for the extended mKdV equation, the all exact solutions based on the Lie group method are given. Especially, the bifurcations and traveling wave solutions are obtained. To guarantee the existence of the above solutions, all parameter conditions are determined. Furthermore, the exact analytic solutions are considered by using the power series method. Such solutions for the equation are important in both applications and the theory of nonlinear science. [ABSTRACT FROM AUTHOR]

We study a model of three interacting species in a food chain composed by a prey, an specific predator and a generalist predator. The capture of the prey by the specific predator is modelled as a modified Holling-type II non-differentiable functional response. The other predatory interactions are both modelled as Holling-type I. Moreover, our model follows a Leslie-Gower approach, in which the function that models the growth of each predator is of logistic type, and the corresponding carrying capacities depend on the sizes of their associated available preys. The resulting model has the form of a set of nonlinear ordinary differential equations which includes a non-differentiable term. By means of topological equivalences and suitable changes of parameters, we find that there exists an Allee threshold for the survival of the prey population in the food chain, given, effectively, as a critical level for the generalist predator. The dynamics of the model is studied with analytical and computational tools for bifurcation theory. We present two-parameter bifurcation diagrams that contain both local phenomena (Hopf, saddle-node transcritical, cusp, Bogdanov-Takens bifurcations) and global events (homoclinic and heteroclinic connections). In particular, we find that two types of heteroclinic cycles can be formed, both of them containing connections to the origin. One of these cycles is planar involving the absence of the specific predator. In turn, the other heteroclinic cycle is formed by connections in the full three-dimensional phase space. [ABSTRACT FROM AUTHOR]

In “Counting central configurations at the bifurcation points,” we proposed an algorithm to rigorously count central configurations in some cases that involve one parameter. Here, we improve our algorithm to consider three harder cases: the planar (3+1)-body problem with two equal masses; the planar 4-body problem with two pairs of equal masses which have an axis of symmetry containing one pair of them; the spatial 5-body problem with three equal masses at the vertices of an equilateral triangle and two equal masses on the line passing through the center of the triangle and being perpendicular to the plane containing it.While all three problems have been studied in two parameter cases, numerical observations suggest new results at some points on the bifurcation curves. Applying the improved version of our algorithm, we count at those bifurcation points. As a result, for the (3+1)-body problem, we identify three points on the bifurcation curve where there are 8 central configurations, which adds to the known results of 8,9,10 ones. For our 4-body case, at the bifurcation points, there are 3 concave central configurations, which adds to the known results of 2,4 ones. For our 5-body case, at the bifurcation point, there is 1 concave central configuration, which adds to the known results of 0,2 ones. [ABSTRACT FROM AUTHOR]

Our work is concerned with the Hopf bifurcation for a class of 3-dimensional quadratic symmetric vector field. By making use of singular values methods and computing carefully, we give the expressions of the first four focal values of the two singular point that system has. Both singular symmetric points can become fine foci of fourth order at the same time. Moreover, we obtain that from each one can bifurcate 4 small limit cycles under a certain coefficients' perturbed condition, consequently, at most eight limit cycles can appear by simultaneous Hopf bifurcation. [ABSTRACT FROM AUTHOR]

The mathematical model is proposed to simulate the dynamics of rabies transmissions in the raccoon population where juveniles stay with their mother and become adults until they establish their own habitats. The basic reproduction number of rabies transmission is formulated and is shown to be a threshold value of disease invasion. The bifurcation direction from the disease-free equilibrium is proved to be forward when the basic reproduction number passes through unity for spatial homogenous environment. The global stability of the disease-free steady state is also studied. [ABSTRACT FROM AUTHOR]

We study a weakly stratified Kolmogorov flow under the effect of a small linear drag. We perform a linear stability analysis of the basic state. We construct the finite dimensional dynamical system deriving from the truncated Fourier mode approximation. Using the Reynolds number as bifurcation parameter we build the corresponding diagram up to Re=100. We observe the coexistence of three coherent structures. [ABSTRACT FROM AUTHOR]

Bifurcation of limit cycles is discussed for three-dimensional Lotka-Volterra competitive systems. A recursion formula for computation of the singular point quantities is given for the corresponding Hopf bifurcation equation. Some new results are obtained for 6 classes 26-31 in Zeeman's classification, especially, an example with four limit cycles in class 29 is given for the first time. The algorithm applied here is effective for solving the above general cyclicity. [ABSTRACT FROM AUTHOR]

Given a family of nonlinear control systems, where the Jacobian of the driver vector field at one equilibrium has a simple zero eigenvalue, with no other eigenvalues on the imaginary axis, we split it into two parts, one of them being a generic family, where it is possible to control the stationary bifurcations: saddle-node, transcritical and pitchfork bifurcations, and the other one being a non-generic family, where it is possible to control the transcritical and pitchfork bifurcations. The polynomial control laws designed are given in terms of the original control system. The center manifold theory is used to simplify the analysis to dimension one. Finally, the results obtained are applied to two underactuated mechanical systems: the pendubot and the pendulum of Furuta. [ABSTRACT FROM AUTHOR]

It is well-known that on a versal deformation of the Takens–Bogdanov bifurcation is possible to find dynamical systems that undergo saddle-node, Hopf, and homoclinic bifurcations. In this document a nonlinear control system in the plane is considered, whose nominal vector field has a double-zero eigenvalue, and then the idea is to find under which conditions there exists a scalar control law such that be possible establish a priori, that the closed-loop system undergoes any of the three bifurcations: saddle-node, Hopf or homoclinic. We will say then that such system undergoes the controllable Takens–Bogdanov bifurcation. Applications of this result to the averaged forced van der Pol oscillator, a population dynamics, and adaptive control systems are discussed. [ABSTRACT FROM AUTHOR]