15 research outputs found
Bifurcation analysis of the Topp model
In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao
Existence and Multiplicity of Solutions of Functional Differential Equations
The first part of the memory goes through those discoveries related to Green’s functions.
In order to do that, first we recall some general results concerning involutions which will help
us understand their remarkable analytic and algebraic properties. Chapter 1 will deal about
this subject while Chapter 2 will give a brief overview on differential equations with involutions
to set the reader in the appropriate research framework.
In Chapter 3 we start working on the theory of Green’s functions for functional differential
equations with involutions in the most simple cases: order one problems with constant coefficients
and reflection. Here we solve the problem with different boundary conditions, studying
the specific characteristics which appear when considering periodic, anti-periodic, initial or arbitrary
linear boundary conditions. We also apply some very well known techniques (lower
and upper solutions method or Krasnosel’skiĭ’s Fixed Point Theorem, for instance) in order to
further derive results.
Computing explicitly the Green’s function for a problem with nonconstant coefficients is
not simple, not even in the case of ordinary differential equations. We face these obstacles in
Chapter 4, where we reduce a new, more general problem containing nonconstant coefficients
and arbitrary differentiable involutions, to the one studied in Chapter 3.
To end this part of the work, we have Chapter 5, in which we deepen in the algebraic nature
of reflections and extrapolate these properties to other algebras. In this way, we do not
only generalize the results of Chapter 3 to the case of -th order problems and general twopoint
boundary conditions, but also solve functional differential problems in which the Hilbert
transform or other adequate operators are involved.
The last chapters of this part are about applying the results we have proved so far to some
related problems. First, in Chapter 6, setting again the spotlight on some interesting relation
between an equation with reflection and an equation with a -Laplacian, we obtain some results
concerning the periodicity of solutions of that first problem with reflection. Chapter 7
moves to a more practical setting. It is of the greatest interest to have adequate computer
programs in order to derive the Green’s functions obtained in Chapter 5 for, in general, the
computations involved are very convoluted. Being so, we present in this chapter such an algorithm,
implemented in Mathematica. The reader can find in the appendix the exact code of
the program.
In the second part of the Thesis we use the fixed point index to solve four different kinds
of problems increasing in complexity: a problem with reflection, a problem with deviated arguments
(applied to a thermostat model), a problem with nonlinear Neumann boundary conditions
and a problem with functional nonlinearities in both the equation and the boundary
conditions.
As we will see, the particularities of each problem make it impossible to take a common
approach to all of the problems studied. Still, there will be important similarities in the different
cases which will lead to comparable results
Rendiconti dell'Istituto di Matematica dell'Università di Trieste. An International Journal of Mathematics. Vol. 46 (2014)
Rendiconti dell’Istituto di Matematica dell’Università di Trieste was founded in 1969 by Arno Predonzan, with the aim of publishing original research articles in all fields of mathematics and has been the first Italian mathematical journal to be published also on-line. The access to the electronic version of the journal is free. All published articles are available on-line. The journal can be obtained by subscription, or by reciprocity with other similar journals. Currently more than 100 exchange agreements with mathematics departments and institutes around the world have been entered in
Rendiconti dell'Istituto di Matematica dell'Università di Trieste. An International Journal of Mathematics. Vol. 44 (2012)
Rendiconti dell’Istituto di Matematica dell’Università di Trieste was founded in 1969 by Arno Predonzan, with the aim of publishing original research articles in all fields of mathematics and has been the first Italian mathematical journal to be published also on-line. The access to the electronic version of the journal is free. All published articles are available on-line. The journal can be obtained by subscription, or by reciprocity with other similar journals. Currently more than 100 exchange agreements with mathematics departments and institutes around the world have been entered in
Mathematical control theory and Finance
Control theory provides a large set of theoretical and computational tools with applications in a wide range of fields, running from ”pure” branches of mathematics, like geometry, to more applied areas where the objective is to find solutions to ”real life” problems, as is the case in robotics, control of industrial processes or finance. The ”high tech” character of modern business has increased the need for advanced methods. These rely heavily on mathematical techniques and seem indispensable for competitiveness of modern enterprises. It became essential for the financial analyst to possess a high level of mathematical skills. Conversely, the complex challenges posed by the problems and models relevant to finance have, for a long time, been an important source of new research topics for mathematicians. The use of techniques from stochastic optimal control constitutes a well established and important branch of mathematical finance. Up to now, other branches of control theory have found comparatively less application in financial problems. To some extent, deterministic and stochastic control theories developed as different branches of mathematics. However, there are many points of contact between them and in recent years the exchange of ideas between these fields has intensified. Some concepts from stochastic calculus (e.g., rough paths) have drawn the attention of the deterministic control theory community. Also, some ideas and tools usual in deterministic control (e.g., geometric, algebraic or functional-analytic methods) can be successfully applied to stochastic control. We strongly believe in the possibility of a fruitful collaboration between specialists of deterministic and stochastic control theory and specialists in finance, both from academic and business backgrounds. It is this kind of collaboration that the organizers of the Workshop on Mathematical Control Theory and Finance wished to foster. This volume collects a set of original papers based on plenary lectures and selected contributed talks presented at the Workshop. They cover a wide range of current research topics on the mathematics of control systems and applications to finance. They should appeal to all those who are interested in research at the junction of these three important fields as well as those who seek special topics within this scope.info:eu-repo/semantics/publishedVersio