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Due to existence of periodic windows, chaotic systems undergo numerous bifurcations as system parameters vary, rendering it hard to employ an analytic continuation, which constitutes a major obstacle for its effective analysis or…
In this paper we apply techniques from nonstandard analysis to study expansive dynamical systems. Among other results, we provide a necessary and sufficient condition for an expansive homeomorphism on a compact metric space to admit…
This paper investigates the asymptotic behaviour of solutions to certain infinite systems of ordinary differential equations. In particular, we use results from ergodic theory and the asymptotic theory of $C_0$-semigroups to obtain a…
Systems of ordinary differential equations (or dynamical forms in Lagrangian mechanics), induced by embeddings of smooth fibered manifolds over one-dimensional basis, are considered in the class of variational equations. For a given…
Asymptotic formula is derived for the behavior of the fundamental solution of the second-order elliptic self-adjoint operator with a piecewise-smooth coefficient in front of the senior derivatives near the discontinuity surface of the…
A general formalism of the relation between geometric phases produced by circularly evolving interacting spin systems and their criticality behavior is presented. This opens up the way for the use of geometric phases as a tool to study…
We formulate symmetric versions of classical variational principles. Within the framework of non-smooth critical point theory, we detect Palais-Smale sequences with additional second order and symmetry information. We discuss applications…
Rough path theory provides one with the notion of signature, a graded family of tensors which characterise, up to a negligible equivalence class, and ordered stream of vector-valued data. In the last few years, use of the signature has…
The paper is devoted to a comprehensive second-order study of a remarkable class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to…
In a previous paper [1] it was discussed the viability of functional analysis using as a basis a couple of generic functions, and hence vectorial decomposition. Here we complete the paradigm exploiting one of the analysis methodologies…
A new method is proposed for integrating the equations of motion of an elastic filament. In the standard finite-difference and finite-element formulations the continuum equations of motion are discretized in space and time, but it is then…
We introduce new machine-learning techniques for analyzing chaotic dynamical systems. The primary objectives of the study include the development of a new and simple method for calculating the Lyapunov exponent using only two trajectory…
A geometric method to analyze nonlinear oscillations is discussed. We consider a nonlinear oscillation modeled by a second order ordinary differential equation without specifying the function form. By transforming the differential equation…
We describe recent nonlinear analytic approximation tools in the classical setting of Hardy spaces in the upper half plane and show how to transfer them to the higher dimensional real setting of harmonic functions in upper half spaces. It…
Although asymptotic analyses of undirected network models based on degree sequences have started to appear in recent literature, it remains an open problem to study statistical properties of directed network models. In this paper, we…
We analyze the asymptotics of the dimension of components of the Chow variety as degree increases. By analogy with the divisor case, the main goal is to relate the asymptotic behavior with the positivity of the corresponding cycle classes.…
Let $v$ be an odd real polynomial (i.e. a polynomial of the form $\sum_{j=1}^\ell a_jx^{2j-1}$). We utilize sets of iterated differences to establish new results about sets of the form $\mathcal…
We report a systematic study of the dynamics of chain formation in electrorheological (ER) fluids using Brownian Dynamics simulations. The parameters of the system such as applied electric field, polarizability, dipole moment, friction…
These notes develop aspects of perturbation theory of matrices related to so-called diagonalisation schemes. Primary focus is on constructive tools to derive asymptotic expansions for small/large parameters of eigenvalues and…
This article presents a general description of dynamical systems using the language of enriched functors and enriched natural transformations. This framework is essential to establish the equivalence of three descriptions of dynamics -- a…