Related papers: Integrable Mappings from a Unified Perspective
Some aspects of the predictability problem in dynamical systems are reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy, Shannon entropy and algorithmic complexity is discussed. In particular, we emphasize how a…
Arguments inspired by algorithmic information theory predict an inverse relation between the probability and complexity of output patterns in a wide range of input-output maps. This phenomenon is known as \emph{simplicity bias}. By viewing…
This thesis deals with the geometric and integrable aspects associated with random matrix models. Its purpose is to provide various applications of random matrix theory, from algebraic geometry to partial differential equations of…
These notes offer a unified introduction to spectral methods for the study of complex systems. They are intended as an operative manual rather than a theorem-proof textbook: the emphasis is on tools, identities, and perspectives that can be…
Basic notions regarding classical integrable systems are reviewed. An algebraic description of the classical integrable models together with the zero curvature condition description is presented. The classical r-matrix approach for discrete…
Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the…
Many interesting physical systems have mathematical descriptions as finite-dimensional or infinite-dimensional Hamiltonian systems. Poincare who started the modern theory of dynamical systems and symplectic geometry developed a particular…
We consider integrable systems that are connected with orthogonal separation of variables in complex Riemannian spaces of constant curvature. An isomorphism with the hyperbolic Gaudin magnet, previously pointed out by one of us, extends to…
A new geometric procedure to construct symplectic methods for constrained mechanical systems is developed in this paper. The definition of a map coming from the notion of retraction maps allows to adapt the continuous problem to the…
We propose a unified theoretical framework for quantifying spatio-temporal interactions in a stochastic dynamical system based on information geometry. In the proposed framework, the degree of interactions is quantified by the divergence…
The positive and not completely positive maps of density matrices, which are contractive maps, are discussed as elements of a semigroup. A new kind of positive map (the purification map), which is nonlinear map, is introduced. The density…
In this work we investigate dynamical systems designed to approach the solution sets of inclusion problems involving the sum of two maximally monotone operators. Our aim is to design methods which guarantee strong convergence of…
The aim of this work is to establish the existence of invariant manifolds in complex systems. Considering trajectory curves integral of multiple time scales dynamical systems of dimension two and three (predator-prey models, neuronal…
Regarding the adjacency matrices of n-vertex graphs and related graph Laplacian, we introduce two families of discrete matrix models constructed both with the help of the Erdos-Renyi ensemble of random graphs. Corresponding matrix sums…
We discuss geometrical aspects of different dualities in the integrable systems of the Hitchin type and its various generalizations. It is shown that T duality known in the string theory context is related to the separation of variables…
The probability representation of quantum and classical statistical mechanics is discussed. Symplectic tomography, center-of-mass tomography, and spin tomography are studied. The connection of tomographic probabilities with dynamic…
This paper combines image metamorphosis with deep features. To this end, images are considered as maps into a high-dimensional feature space and a structure-sensitive, anisotropic flow regularization is incorporated in the metamorphosis…
We derive, and discuss the properties of, a symplectic map for the dynamics of bodies on nearly parabolic orbits. The orbits are perturbed by a planet on a circular, coplanar orbit interior to the pericenter of the parabolic orbit. The map…
This paper has a double goal, the first one is to make a slight survey of some theoretical results about the existence of positively invariant curves that allow to describe important properties of the set of bounded orbits and its boundary…
Divisibility of dynamical maps is visualized by trajectories in the parameter space and analyzed within the framework of collision models. We introduce ultimate completely positive (CP) divisible processes, which lose CP divisibility under…