Dynamical Systems on Leibniz Algebroids

In this paper we study the differential systems on Leibniz algebroids. We introduce a class of almost metriplectic manifolds as a special case of Leibniz manifolds. Also, the notion of almost metriplectic algebroid is introduced. These types of algebroids are used in the presentation of associated differential systems. We give some interesting examples of differential systems on algebroids and the orbits of the solutions of corresponding systems are established.Comment: 14 pages, 6 figures, the paper will be presented at The 5-th Conference of Balkan Society of Geometers, August 29-September 2, 2005 Mangalia, Romani

Analysis of a 3D chaotic system

A 3D nonlinear chaotic system, called the T system, is analyzed in this paper. Horseshoe chaos is investigated via the heteroclinic Shilnikov method constructing a heteroclinic connections between the saddle equilibrium points of the system. Partially numerical computations are carried out to support the analytical results

Leibniz Dynamics with Time Delay

In this paper we show that several dynamical systems with time delay can be described as vector fields associated to smooth functions via a bracket of Leibniz structure. Some examples illustrate the theoretical considerations.Comment: 15 pages, 2 figures, it will be presented at The 5-th Conference of Balkan Society of Geometers, August 29- September 2, 2005, Mangalia, Romani

Synchronization and secure communication using some chaotic systems of fractional differential equations

Using Caputo fractional derivative of order $\alpha,$ $\alpha\in (0,1),$ we consider some chaotic systems of fractional differential equation. We will prove that they can be synchronized and anti-synchronized using suitable nonlinear control function. The synchronized or anti-synchronized error system of fractional differential equations is used in secure communication.Comment: 10 pages, 6 figure

Stochastic generalized fractional HP equations and applications

In this paper we established the condition for a curve to satisfy stochastic generalized fractional HP (Hamilton-Pontryagin) equations. These equations are described using Ito integral. We have also considered the case of stochastic generalized fractional Hamiltonian equations, for a hyperregular Lagrange function. From the stochastic generalized fractional Hamiltonian equations, Langevin generalized fractional equations were found and numerical simulations were done.Comment: 14 pages, 10 figures, the paper will be presented at The International Conference of Differential Geometry and Dynamical Systems DGDS-2009/October 8-11, 2009, University Politehnica of Bucharest, Romani

Dissipative Mechanical Systems with Delay

The idea of dissipative mechanical system with delay is proposed. The paper studies the phenomenon of dissipation with delay for Euler-Poincare systems on Lie algebras or equivalently, for Lie-Poisson systems on the duals of Lie algebras. The study was suggested by the work [2] and it is ended with a discussion regarding the stability and the Hopf bifurcations for the free rigid body with delay.Comment: 33 page

Rent seeking games with tax evasion

We consider the static and dynamic models of Cournot duopoly with tax evasion. In the dynamic model we introduce the time delay and we analyze the local stability of the stationary state. There is a critical value of the delay when the Hopf bifurcation occurs.Comment: 8 pages, 4 figures, the paper was presented at Pannonian Applied Mathematical Meetings, 31 may-3june, 200

A dynamic p53-mdm2 model with delay kernel

Specific activator and repressor transcription factors which bind to specific regulator DNA sequences, play an important role in gene activity control. Interactions between genes coding such transcripion factors should explain the different stable or sometimes oscillatory gene activities characteristic for different tissues. In this paper, the dynamic P53-Mdm2 interaction model with distributed delays and weak kernel, is investigated. Choosing the delay or the kernel's coefficient as a bifurcation parameter, we study the direction and stability of the bifurcating periodic solutions. Some numerical examples are finally given for justifying the theoretical results.Comment: 23 pages, 8 figures, the paper was presented at "Conference Francophone sur la Modelisation Mathematique en Biologie et en Medecine", Craiova, Roumanie,12-14 July, 200

Hopf bifurcation analysis of pathogen-immune interaction dynamics with delay kernel

The aim of this paper is to study the steady states of the mathematical models with delay kernels which describe pathogen-immune dynamics of many kinds of infectious diseases. In the study of mathematical models of infectious diseases it is important to predict whether the infection disappears or the pathogens persist. The delay kernel is described by the memory function that reflects the influence of the past density of pathogen in the blood. By using the coefficient of kernel k, as a bifurcation parameter, the models are found to undergo a sequence of Hopf bifurcation. The direction and the stability criteria of bifurcation periodic solutions are obtained by applying the normal form theory and the center manifold theorems. Some numerical simulation examples for justifying the theoretical results are also given.Comment: 18 pages, 12 figures, the paper was presented at "Conference Francophone sur la Modelisation Mathematique en Biologie et en Medecine", Craiova, Roumanie,12-14 July, 200

The analysis of stochastic stability of stochastic models that describe tumor-immune systems

In this paper we investigate some stochastic models for tumor-immune systems. To describe these models, we used a Wiener process, as the noise has a stabilization effect. Their dynamics are studied in terms of stochastic stability in the equilibrium points, by constructing the Lyapunov exponent, depending on the parameters that describe the model. Stochastic stability was also proved by constructing a Lyapunov function. We have studied and and analyzed a Kuznetsov-Taylor like stochastic model and a Bell stochastic model for tumor-immune systems. These stochastic models are studied from stability point of view and they were represented using the second Euler scheme and Maple 12 software.Comment: 23 pages, 30 figure