Alfonso Blasco

Universidad de Burgos | UBU · Department of Physics

The exact analytical solution in closed form of a modified SIR system is presented. This is, to the best of our knowledge, the first closed-form solution for a three-dimensional deterministic compartmental model of epidemics. In this dynamical system the populations S(t) and R(t) of susceptible and recovered individuals are found to be generalized...

Any epidemiological compartmental model with constant population is shown to be a Hamiltonian dynamical system in which the total population plays the role of the Hamiltonian function. Moreover, some particular cases within this large class of models are shown to be bi-Hamiltonian. New interacting compartmental models among different populations, w...

Any epidemiological compartmental model with constant population is shown to be a Hamiltonian dynamical system in which the total population plays the role of the Hamiltonian function. Moreover, some particular cases within this large class of models are shown to be bi-Hamiltonian. New interacting compartmental models among different populations, w...

The generalization of (super)integrable Euclidean classical Hamiltonian systems to the two-dimensional sphere and the hyperbolic space by preserving their (super)integrability properties is reviewed. The constant Gaussian curvature of the underlying spaces is introduced as an explicit deformation parameter, thus allowing the construction of new int...

The generalization of (super)integrable Euclidean classical Hamiltonian systems to the two-dimensional sphere and the hyperbolic space by preserving their (super)integrability properties is reviewed. The constant Gaussian curvature of the underlying spaces is introduced as an explicit deformation parameter, thus allowing the construction of new int...

The generalization of (super)integrable Euclidean classical Hamiltonian systems to the two-dimensional sphere and the hyperbolic space by preserving their (super)integrability properties is reviewed. The constant Gaussian curvature of the underlying spaces is introduced as an explicit deformation parameter, thus allowing the construction of new int...

A method to construct integrable deformations of Hamiltonian systems of ODEs endowed with Lie-Poisson symmetries is proposed by considering Poisson-Lie groups as deformations of Lie-Poisson (co)algebras. Moreover, the underlying Lie-Poisson symmetry of the initial system of ODEs is used to construct integrable coupled systems, whose integrable defo...

We construct a constant curvature analogue on the two-dimensional sphere ${\mathbf S}^2$ and the hyperbolic space ${\mathbf H}^2$ of the integrable H\'enon--Heiles Hamiltonian $\mathcal{H}$ given by $$ \mathcal{H}=\dfrac{1}{2}(p_{1}^{2}+p_{2}^{2})+ \Omega \left( q_{1}^{2}+ 4 q_{2}^{2}\right) +\alpha \left( q_{1}^{2}q_{2}+2 q_{2}^{3}\right) , $$ wh...

A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional real Lie algebra of Hamiltonian vector fields with respect to a Poisson structure. We provide new algebraic/geometric techniques to easily determine the pro...

We study Lie-Hamilton systems on the plane, i.e. systems of first-order differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of planar Hamiltonian vector fields with respect to a Poisson structure. We start with the local classification of finite-dimensional rea...

The constant curvature analogue on the two-dimensional sphere and the hyperbolic space of the integrable Hénon-Heiles Hamiltonian H given by where Ω and α are real constants, is revisited. The resulting integrable curved Hamiltonian, Hκ, depends on a parameter κ which is just the curvature of the underlying space and allows one to recover H under t...

A new integrable generalization to the 2D sphere $S^2$ and to the hyperbolic space $H^2$ of the 2D Euclidean anisotropic oscillator Hamiltonian with Rosochatius (centrifugal) terms is presented, and its curved integral of the motion is shown to be quadratic in the momenta. In order to construct such a new integrable Hamiltonian $H_\kappa$, we will...

A class of integrable 3D Lotka-Volterra (LV) equations is shown to be a particular instance of Poisson-Lie dynamics on a family of solvable 3D Lie groups. As a consequence, the classification of all possible Poisson-Lie structures on these groups is shown to provide a systematic approach to obtain multiparametric integrable deformations of this LV...

The integrable perturbations of the two-dimensional integrable Hénon-Heiles Hamiltonians of KdV type are revisited by making use of their underlying sl(2,R)⊕h3 Poisson symmetry (co)algebra. As an application, a straightforward N-dimensional integrable generalization of all these systems is constructed by using N-dimensional symplectic realizations...

The loop coproduct approach to integrable systems introduced in [1, 2] is reformulated in terms of maps between Poisson manifolds. This reformulation is a generalization of that of the coalgebra symmnetry method given in [3] by using Poisson maps.

All real three dimensional Poisson-Lie groups are explicitly constructed and fully classified under group automorphisms by making use of their one-to-one correspondence with the complete classification of real three-dimensional Lie bialgebras given in [X. Gomez, J. Math. Phys. vol. 41, p. 4939 (2000)]. Many of these 3D Poisson-Lie groups are non-co...

All possible Poisson-Lie (PL) structures on the 3D real Lie group generated by a dilation and two commuting translations are obtained. Its classification is fully performed by relating these PL groups with the corresponding Lie bialgebra structures on the corresponding "book" Lie algebra. By construction, all these Poisson structures are quadratic...

The Hamiltonian structure of a class of three-dimensional (3D) Lotka-Volterra (LV) equations is revisited from a novel point of view by showing that the quadratic Poisson structure underlying its integrability structure is just a real three-dimensional Poisson-Lie group. As a consequence, the Poisson coalgebra map that is given by the group multipl...

The three integrable two-dimensional Henon-Heiles systems and their integrable perturbations are revisited. A family of new integrable perturbations is found, and N-dimensional completely integrable generalizations of all these systems are constructed by making use of sl(2,R)+h(3) as their underlying Poisson symmetry algebra. In general, the proced...

A constructive method to obtain integrable Hamiltonians with N degrees of freedom is presented. This approach is based on the h 6 Poisson coalgebra and allows us to construct two new families of nonlinear integrable perturbations of the N‐dimensional oscillator. The first one is a family of integrable perturbations depending on N parameters and...

Two new families of completely integrable perturbations of the N-dimensional isotropic harmonic oscillator Hamiltonian are presented. Such perturbations depend on arbitrary functions and N free parameters and their integrals of motion are explicitly constructed by making use of an underlying h_6-coalgebra symmetry. Several known integrable Hamilton...

A wide class of Hamiltonian systems with N degrees of freedom and endowed with, at least, (N-2) functionally independent integrals of motion in involution is constructed by making use of the two-photon Lie-Poisson coalgebra. The set of (N-2) constants of the motion is shown to be a universal one for all these Hamiltonians, irrespectively of the dep...

The coalgebra approach to the construction of classical integrable systems from Poisson coalgebras is reviewed, and the essential role played by symplectic realizations in this framework is emphasized. Many examples of Hamiltonians with either undeformed or q-deformed coalgebra symmetry are given, and their Liouville superintegrability is discussed...

The construction of N-dimensional (ND) integrable systems from coalgebras is reviewed. In the case of Poisson coalgebras, a necessary condition for the integrability of the ND coalgebra Hamiltonian coming from a given coalgebra is obtained in terms of the dimension of the symplectic realization and the number of nonlinear Casimir functions. From th...

The construction of integrable systems from symplectic realizations of Poisson coalgebras with Casimir elements is revisited. Several examples of Hamiltonians with either undeformed or 'quantum' coalgebra symmetry are given, and their Li- ouville integrability is discussed. The essential role of symplectic realizations in this context is emphasized...

Deformed algebras are used to construct effective Hamiltonians for several nonlinear quantum problems. In all the cases here presented, the deforma-tion parameter can be fitted in such a way that the dynamical properties of the physical models are faithfully reproduced, despite the strong alge-braic solvability of the corresponding effective Hamilt...

Esta Tesis presenta nuevos sistemas hamiltonianos clásicos completamente integrables N dimensionales, construidos mediante la técnica de simetría de coálgebra. Primeramente se ha obtenido la condición necesaria de integrabilidad para una representación simpléctica de cualquier coálgebra de Poisson. Esta condición ha sido utilizada sistemáticamente...