The classical theorem of Hopf Bifurcation for the study of periodic solutions of ordinary differential equations (ODEs) at an equilibrium point has become the standard mathematical tool for reducing higher dimensional dynamical systems to lower dimensional systems, while at the same time preserving the emanating dynamics. The attraction for this tool is due to the fact that the classical Hopf t...

In this paper, the bifurcation analysis for Burgers mapping has been studied. The existence and stability of the fixed points of this map was derived. The conditions of existence for pitchfork bifurcation, flip bifurcation and Neimark-Sacker bifurcation are derived by using center manifold theorem and bifurcation theory. The control of the map around stable NeimarkSacker bifurcation has been ac...

For three specific singularly perturbed two-component reaction diffusion systems in a bounded 2-D domain admitting localized multi-spot patterns, we provide a detailed analysis of the parameter values for the onset of temporal oscillations of the spot amplitudes. The two key bifurcation parameters in each of the RD systems are the reaction-time parameter τ and the inhibitor diffusivity D. In th...

We study a mathematical model describing the dynamics of a pluripotent stem cell population involved in the blood production process in the bone marrow. This model is a differential equation with a time delay. The delay describes the cell cycle duration and is uniformly distributed on an interval. We obtain stability conditions independent of the delay. We also show that the distributed delay c...

A model of two coupled demand-supply systems, depending on 4 parameters is considered. We found that the dynamical system associated with this model may have at most two symmetric and at most two nonsymmetric equilibria as the parameters vary. The topological type of equilibria is established and the locus in the parameter space of the values corresponding to nonhyperbolic equilibria is determi...

A mathematical model is proposed to study the role of distributed delay on plankton ecosystem in the presence of a toxic producing phytoplankton. The model includes three state variables, namely, nutrient concentration, phytoplankton biomass, and zooplankton biomass. The release of toxic substance by phytoplankton species reduces the growth of zooplankton and this plays an important role in pla...

The existence of oscillatory solutions for a certain class of scalar first order delay-differential equations is proved. An application to a delay logistic equation arising in certain models for population variation of a single specie in a constant environment with limited resources for growth is considered. It is known (cf. [1, 2]) that all solutions of the delay logistic equations (1) N'(t) =...

We consider G-equivariant semilinear parabolic equations where G is a finite-dimensional possibly non-compact symmetry group. We treat periodic forcing of relative equilibria and resonant periodic forcing of relative periodic orbits as well as Hopf bifurcation from relative equilibria to relative periodic orbits using LyapunovSchmidt reduction. Our main interest are drift phenomena caused by re...

Many experiments reveal that Daphnia and its microparasite populations vary strongly in density and typically go through pronounced cycles. To better understand such dynamics, we formulate a simple two dimensional autonomous ordinary differential equation model for Daphnia magnamicroparasite infection with dose-dependent infection. This model has a basic parasite production number R0 = 0, yet i...

This paper provides an innovative practical realization of a memristor based chaotic circuit. The first part discusses the mathematical analysis of the proposed system, including calculation of an eigenvalues, bifurcation diagram and largest Lyapunov exponents. Another parts deal with circuitry realization and the influence of parasitic properties of active elements. The circuit simulations obt...