H. Merdan

Stability and bifurcation analysis of two-neuron network with discrete and distributed delays

In this paper, we give a detailed Hopf bifurcation analysis of a recurrent neural network system involving both discrete and distributed delays. Choosing the sum of the discrete delay terms as a bifurcation parameter the existence of Hopf bifurcation is demonstrated. In particular, the formulae determining the direction of the bifurcations and the stability of the bifurcating periodic solutions are studied by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.

Hopf bifurcation analysis of a general non-linear differential equation with delay

This work represents Hopf bifurcation analysis of a general non-linear differential equation involving time delay. A special form of this equation is the Hutchinson-Wright equation which is a mile stone in the mathematical modeling of population dynamics and mathematical biology. Taking the delay parameter as a bifurcation parameter, Hopf bifurcation analysis is studied by following the theory in the book by Hazzard et al. By analyzing the associated characteristic polynomial, we determine necessary conditions for the linear stability and Hopf bifurcation. In addition to this analysis, the direction of bifurcation, the stability and the period of a periodic solution to this equation are evaluated at a bifurcation value by using the Poincare normal form and the center manifold theorem. Finally, the theoretical results are supported by numerical simulations.

Hopf bifurcation analysis of a system of coupled delayed-differential equations

In this paper, we have considered a system of delay differential equations. The system without delayed arises in the Lengyel-Epstein model. Its dynamics are studied in terms of local analysis and Hopf bifurcation analysis. Linear stability is investigated and existence of Hopf bifurcation is demonstrated via analyzing the associated characteristic equation. For the Hopf bifurcation analysis, the delay parameter is chosen as a bifurcation parameter. The stability of the bifurcating periodic solutions is determined by using the center manifold theorem and the normal form theory introduced by Hassard et al. (1981) [7]. Furthermore, the direction of the bifurcation, the stability and the period of periodic solutions are given. Finally, the theoretical results are supported by some numerical simulations.

Stability analysis of a general discrete-time population model involving delay and Allee effects

Abstract In this paper, we investigate stability conditions of equilibrium points of a general delay difference population model with and without Allee effects which occur at low population density. The analysis demonstrates that Allee effects have both stabilizing and destabilizing effects on population dynamics including time delay.

Renormalization group methods for nonlinear parabolic equations

Renormalization group (RG) methods are described for determining the key exponents related to the decay of solutions to nonlinear parabolic differential equations. Higher order (in the small coefficient of the nonlinearity) methods are developed. Exact solutions and theorems in some special cases confirm the RG results.

A mathematical model for asset pricing

Asset price dynamics is studied by using a system of ordinary differential equations which is derived by utilizing a new excess demand function introduced by Caginalp (2005) [4] for a market involving more information on demand and supply for a stock rather than their values at a particular price. Derivation is based on the finiteness of assets (rather than assuming unbounded arbitrage) in addition to investment strategies that are based on not only price momentum (trend) but also valuation considerations. For this new model and the older models which were extracted using the classical excess demand function by Caginalp and Balenovich (1994,1999) [2,3], time evolutions of asset price are compared through numerical simulations.

STABILITY ANALYSIS OF A LOTKA–VOLTERRA TYPE PREDATOR–PREY SYSTEM INVOLVING ALLEE EFFECTS

We present a stability analysis of steady-state solutions of a continuous-time predator– prey population dynamics model subject to Allee effects on the prey population which occur at low population density. Numerical simulations show that the system subject to an Allee effect takes a much longer time to reach its stable steady-state solution. This result differs from that obtained for the discrete-time version of the same model. doi:10.1017/S1446181111000630

Delay Effects on the Dynamics of the Lengyel–Epstein Reaction-Diffusion Model

We investigate bifurcations of the Lengyel–Epstein reaction-diffusion model involving time delay under the Neumann boundary conditions. We first give stability and Hopf bifurcation analysis of the ordinary differential equation (ODE) models, including delay associated with this model. Later, we extend this analysis to the partial differential equation (PDE) model. We determine conditions on parameters of both models to have Hopf bifurcations. Bifurcation analysis for both models show that Hopf bifurcations occur by regarding the delay parameter as a bifurcation parameter. Using the normal form theory and the center manifold reduction for partial functional differential equations, we also determine the direction of the Hopf bifurcations and the stability of bifurcating periodic solutions for the PDE model. Finally, we perform some numerical simulations to support analytical results obtained for the ODE models.

Renormalization Methods And Interface Problems

The application of renormalization techniques to interface problems is considered after a brief review of the methodology. We study the standard sharp interface problem in the quasi-static limit (time derivative set to zero in the heat equation) for large time. The characteristic length, R(t), behaves as t where β has values in the continuous spectrum [1/3, 1/2] when the dynamical undercooling is nonzero, and β ∈ [1/3,∞) when the undercooling is set at zero. The value of β = 1 obtained by Jasnow and Vinals is extracted from this spectrum as a consequence of boundary conditions that impose a plane wave. In almost all of these cases, the capillarity length is an irrelevant variable for large time, in sharp contrast to its role in linear stability for short time.

Global Stability Analysis of a General Scalar Difference Equation

We consider a general first order scalar difference equation with and without Allee effect. The model without Allee effect represents asexual reproduction of a species while the model including Allee effect represents sexual reproduction. We analyze global stabilities of both models analytically and compare the results obtained. Numerical simulations are included to support the analytical results. We conclude that Allee effect decreases global stability of a nonnegative fixed point of the model. This result is different from the local stability behavior of the same fixed point of the model.