Papers by Anca Veronica Ion
Journal of Dynamics and Differential Equations, Feb 28, 2012
In computing the third order terms of the series of powers of the center manifold at an equilibri... more In computing the third order terms of the series of powers of the center manifold at an equilibrium point of a scalar delay differential equation, some problems occur at the term w21z 2 z. More precisely, in order to determine the values at 0, respectively -r of the function w21( . ), an algebraic system of equations must be solved. We show that the two equations are dependent, hence the system has an infinity of solutions. Then we show how we can overcome this lack of uniqueness and provide a formula for w21(0).
arXiv (Cornell University), Nov 7, 2011
For systems of delay differential equations the Hopf bifurcation was investigated by several auth... more For systems of delay differential equations the Hopf bifurcation was investigated by several authors. The problem we solve here is that of the possibility of emergence of a codimension two bifurcation, namely the Bautin bifurcation, for some such systems.
arXiv (Cornell University), Aug 8, 2012
In a previous work we investigated the existence of Hopf degenerate bifurcation points for a diff... more In a previous work we investigated the existence of Hopf degenerate bifurcation points for a differential delay equation modeling leukemia and we actually found Hopf points of codimension two for the considered problem. If around such a point we vary two parameters (the considered problem has five parameters), then a Bautin bifurcation should occur. In this work we chose a Hopf point of codimension two for the considered problem and perform numerical integration for parameters chosen in a neighborhood of the bifurcation point parameters. The results show that, indeed, we have a Bautin bifurcation in the chosen point. Acknowledgement.
arXiv (Cornell University), Jan 23, 2014
Some errors contained in the author's previous article "An example of Bautin-type bifurcation in ... more Some errors contained in the author's previous article "An example of Bautin-type bifurcation in a delay differential equation", JMAA, 329(2007), 777-789, are listed and corrected. I. To determine whether the bifurcation point presents a higher order degeneracy or is a proper Bautin bifurcation point, we computed the second Lyapunov coefficient for the reduced on the center manifold problem. For this we needed w 21 (0), w 21 (-r), where w 21 (•) ∈ C([-r, 0], R) is a coefficient of the series of powers of the function whose graph is the center manifold The two algebraic equations that yield w 21 (0) and w 21 (-r) proved to be dependent, and at that moment we have chosen arbitrarily w 21 (0) = 0 and we computed w 21 (-r) from one of the two equations. This is a mistake and we want to correct it here. By studying more carefully the problem of computing w 21 (0) and w 21 (-r), we found out that these can be uniquely determined by using a perturbation technique. This result was published in [2]. The formula obtained there for w 21 (0), adapted to problem (1), is
Analele Stiintifice Ale Universitatii Ovidius Constanta-seria Matematica, Nov 1, 2015
For the model of periodic chronic myelogenous leukemia considered by Pujo-Menjouet, Mackey et al.... more For the model of periodic chronic myelogenous leukemia considered by Pujo-Menjouet, Mackey et al., model consisting of two delay differential equations, the equation for the density of so-called "resting cells" was studied from numerical and qualitative point of view in several works. In this paper we focus on the equation for the density of proliferating cells and study it from a qualitative point of view.
arXiv (Cornell University), Jan 26, 2010
The paper is devoted to the study of stability of equilibria of a delay differential equation tha... more The paper is devoted to the study of stability of equilibria of a delay differential equation that models leukemia. The equation was previously studied in [5] and [6], where the emphasis is put on the numerical study of periodic solutions. Some stability results for the equilibria are also presented in these works, but they are incomplete and contain some errors. Our work aims to complete and to bring corrections to those results. Both Lyapunov first order approximation method and second Lyapunov method are used.
arXiv (Cornell University), Mar 12, 2014
For the model of periodic chronic myelogenous leukemia considered by Pujo-Menjouet, Mackey et al.... more For the model of periodic chronic myelogenous leukemia considered by Pujo-Menjouet, Mackey et al., model consisting of two delay differential equations, the equation for the density of so-called "resting cells" was studied from numerical and qualitative point of view in several works. In this paper we focus on the equation for the density of proliferating cells and study it from a qualitative point of view.
A Comparative Study of Non-Fickean Diusion in Binary Fluids
We consider a non-Fickean diusion model for binary mixtures. Here, the flux is not governed by Fi... more We consider a non-Fickean diusion model for binary mixtures. Here, the flux is not governed by Fick’s law, it is governed by an evolution equation, derived from the partial balance momenta under the hypothesis of “small” diusion velocities. We apply this model to a binary non-reactive mixture with zero average velocity at thermal equilibrium. In particular, Fick’s model is recovered as a first order perturbation of the non-Fickean model.
Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2015
For the model of periodic chronic myelogenous leukemia considered by Pujo-Menjouet, Mackey et al.... more For the model of periodic chronic myelogenous leukemia considered by Pujo-Menjouet, Mackey et al., model consisting of two delay differential equations, the equation for the density of so-called “resting cells” was studied from numerical and qualitative point of view in several works. In this paper we focus on the equation for the density of proliferating cells and study it from a qualitative point of view.
Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2015
A mathematical model for the water ow on a hill covered by variable distributed vegetation is pro... more A mathematical model for the water ow on a hill covered by variable distributed vegetation is proposed in this article. The model takes into account the variation of the geometrical properties of the terrain surface, but it assumes that the surface exhibits large curvature radius. After describing some theoretical properties for this model, we introduce a simplified model and a well-balanced numerical approximation scheme for it. Some mathematical properties with physical relevance are discussed and finally, some numerical results are presented.
Approximate inertial manifolds for systems of ordinary differential equations
New results in the stability study of non-autonomous evolution equations in Banach spaces
The work extends some previous results of the first author, results concerning the Lyapunov stabi... more The work extends some previous results of the first author, results concerning the Lyapunov stability of the zero solution of some nonautonomous nonlinear evolution equation.
In this paper we set up a numerical algorithm for computing the flow of a class of pseudo-plastic... more In this paper we set up a numerical algorithm for computing the flow of a class of pseudo-plastic fluids. The method uses the finite vol-ume technique for space discretization and a semi-implicit two steps backward differentiation formula for time integration. As primitive variables the algorithm uses the velocity field and the pressure field. In this scheme quadrilateral structured primal-dual meshes are used. The velocity and the pressure fields are discretized on the primal mesh and the dual mesh respectively. A certain advantage of the method is that the velocity and pressure can be computed without any artificial boundary conditions and initial data for the pressure. Based on the numerical algorithm we have written a numerical code. We have also performed a series of numerical simulations.
Journal of Mathematical Analysis and Applications, 2007
Some errors contained in the author's previous article "An example of Bautin-type bifurcation in ... more Some errors contained in the author's previous article "An example of Bautin-type bifurcation in a delay differential equation", JMAA, 329(2007), 777-789, are listed and corrected.
This paper continues the work contained in two previous papers, devoted to the study of the dynam... more This paper continues the work contained in two previous papers, devoted to the study of the dynamical system generated by a delay differential equation that models leukemia. Here our aim is to identify degenerate Hopf bifurcation points. By using an approximation of the center manifold, we compute the first Lyapunov coefficient for Hopf bifurcation points. We find by direct computation, in some zones of the parameter space (of biological significance), points where the first Lyapunov coefficient equals zero. For these we compute the second Lyapunov coefficient, that determines the type of the degenerate Hopf bifurcation.
We consider a delay differential equation that occurs in the study of chronic myelogenous leukemi... more We consider a delay differential equation that occurs in the study of chronic myelogenous leukemia. After shortly reminding some previous results concerning the stability of equilibrium solutions, we concentrate on the study of stability of periodic solutions emerged by Hopf bifurcation from a certain equilibrium point. We give the algorithm for approximating a center manifold at a typical point (in the parameter space) of Hopf bifurcation (and an unstable manifold in the vicinity of such a point, where such a manifold exists). Then we find the normal form of the equation restricted to the center manifold, by computing the first Lyapunov coefficient. The normal form allows us to establish the stability properties of the periodic solutions occurred by Hopf bifurcation.
When studying a general system of delay differential equation with a single constant delay, we en... more When studying a general system of delay differential equation with a single constant delay, we encounter a certain lack of uniqueness in determining the coefficient of one of the third order terms of the series defining the center manifold. We solve this problem by considering a perturbation of the considered problem, perturbation that allows us to remove the singularity. The result generalizes a similar result obtained for scalar differential equations (J. Dyn. Diff. Eqns., 24/2012).
When studying a general system of delay differential equation with a single constant delay, we en... more When studying a general system of delay differential equation with a single constant delay, we encounter a certain lack of uniqueness in determining the coefficient of one of the third order terms of the series defining the center manifold. We solve this problem by considering a perturbation of the considered problem, perturbation that allows us to remove the singularity. The result generalizes a similar result obtained for scalar differential equations (J. Dyn. Diff. Eqns., 24/2012).
The paper deals with the approximation of some nonlinear diffusion equations with source terms an... more The paper deals with the approximation of some nonlinear diffusion equations with source terms and nonhomogeneous Dirichlet boundary conditions and initial conditions. The approximation scheme consists in the discretization of space derivative operators while the time differentiation is kept continous. As result the solution of the partial differential equations is approximate by the solution of a system of ordinary differential equations. We provide the bounds for the solutions of the discrete model that are independent of the mesh size of triangulation. 1.
In the framework of the inflnite-dimensional dynamical systems theory, a family of approximate in... more In the framework of the inflnite-dimensional dynamical systems theory, a family of approximate inertial manifolds (a.i.m.s) for a problem modelling the Fick- ian difiusion of a substance into a Newtonian ∞uid is constructed. Estimates of the distance between these manifolds and the exact solution of the problem are given, proving that, at large times, the solution is kept in some very narrow neighbourhoods of the a.i.m.s. The concept of approximate inertial manifold (a.i.m.) arose in the frame- work of the theory of inertial manifolds. First deflned in (3), inertial manifolds are flnite-dimensional (at least) Lipschitz invariant manifolds, that attract ex- ponentially all trajectories of an evolution equation. For many evolution equations the existence of an inertial manifold is not yet proved, since the proof requires the existence of a certain large between two succesive eigenvalues of the linear part (the so-called spectral gap) and this requirement is not fulfllled (7). Even if ...
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Papers by Anca Veronica Ion