Related papers: Complex ODEs, singularity theory and dynamics
Enlarging on Parts I and II we write more equations in the desired format of the extended abstract theory of composites. We focus on a multitude of full dynamic equations, including equations where the medium is moving or otherwise changing…
We study the classification of singularities of holomorphic foliations and non-integrable one-forms under the hypothesis of transversality with real hypersurfaces.
We show how singularities shape the evolution of rational discrete dynamical systems. The stabilisation of the form of the iterates suggests a description providing among other things generalised Hirota form, exact evaluation of the…
This is a collection of articles, written as sections, on arithmetic properties of differential equations, holomorphic foliations, Gauss-Manin connections and Hodge loci. Each section is independent from the others and it has its own…
This is an almost self-contained monograph (containing some new results) on left-orderable groups which mostly rely on dynamical and probabilistic aspects, but also on geometric, combinatorial, analytic, and topological ones. This new…
I review a number of cognate issues that, taken together, pertain to the creation of a non-reductionistic theory of multiscale coordination and present one candidate theory based on the principle of dynamical similarity.
This papers studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs). It presents a new framework using discrete ODEs as a central tool for computation and provides several implicit characterizations…
In this short note we discuss ordinary differential equations which linearize upon one (or more) differentiations. Although the subject is fairly elementary, equations of this type arise naturally in the context of integrable systems.
Complex systems are characterized by specific time-dependent interactions among their many constituents. As a consequence they often manifest rich, non-trivial and unexpected behavior. Examples arise both in the physical and non-physical…
This monograph, written for educational purposes, serves as an introduction to the concept of integrability as it applies to systems of differential equations (both ordinary and partial) as well as to vector-valued fields. The general cases…
This article summarises a Web-book on "Complexity" that was developed to introduce undergraduate students to interesting complex systems in the biological, physical and social sciences, and the common tools, principles and concepts used for…
Enlarging on Parts I, II, and III we write more equations in the desired format of the extended abstract theory of composites. We focus on a multitude of equations involving higher order derivatives. The motivation is that results and…
Given a fibration over the circle, we relate the eigenspace decomposition of the algebraic monodromy, the homological finiteness properties of the fiber, and the formality properties of the total space. In the process, we prove a more…
We review properties of closed meromorphic $1$-forms and of the foliations defined by them. We present and explain classical results from foliation theory, like index theorems, the existence of separatrices, and resolution of singularities…
A class of neural networks that gained particular interest in the last years are neural ordinary differential equations (neural ODEs). We study input-output relations of neural ODEs using dynamical systems theory and prove several results…
These notes derive a number of technical results on nonlinear contraction theory, a comparatively recent tool for system stability analysis. In particular, they provide new results on the preservation of contraction through system…
We study the complex Dulac map for a holomorphic foliation of the complex plane, near a non-degenerate singularity (both eigenvalues of the linearization are nonzero) with two separatrices. Following the well-known results of Y. Il'yashenko…
We study some classes of semi-linear differential equations including both well-posed and ill-posed cases that can generate cocycles (or cocycle correspondences with generating cocycles). Under exponential dichotomy condition with other…
A correspondence between the orbits of a system of 2 complex, homogeneous, polynomial ordinary differential equations with real coefficients and those of a polygonal billiard is displayed. This correspondence is general, in the sense that…
This thesis is designed to be a self-contained exposition of the neurobiological and mathematical aspects of sensory perception, memory, and learning with a bias towards olfaction. The final chapters introduce a new approach to modeling…