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In a previous work, we proved an index theorem for families of asymptotically hyperbolic discrete dynamical systems and obtained applications to bifurcation theory. A weaker and far more common assumption than asymptotic hyperbolicity is…
In this paper, we consider the data-driven discovery of stable dynamical models with a single equilibrium. The proposed approach uses a basis-function parameterization of the differential equations and the associated Lyapunov function. This…
Using properties of asymptotically almost periodic solutions we prove existence theorem for piece-wise continuous almost periodic solutions of differential equations with delay and impulses. We apply these results to study almost periodic…
As a first approach to the study of systems coupling finite and infinite dimensional natures, this article addresses the stability of a system of ordinary differential equations coupled with a classic heat equation using a Lyapunov…
Generic dynamical systems have `typical' Lyapunov exponents, measuring the sensitivity to small perturbations of almost all trajectories. A generic system has also trajectories with exceptional values of the exponents, corresponding to…
We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the the order of singularities of the coefficients and provide examples to illustrate the…
We propose to study a new type of Backward stochastic differential equations driven by a family of It\^o's processes. We prove existence and uniqueness of the solution, and investigate stability and comparison theorem.
A fundamental result that characterizes elliptic-hyperbolic equations of Tricomi type, the uniqueness of classical solutions to the open Dirichlet problem, is extended to a large class of elliptic-hyperbolic equations of Keldysh type. The…
We show that a linear Young differential equation generates a topological two-parameter flow, thus the notions of Lyapunov exponents and Lyapunov spectrum are well-defined. The spectrum can be computed using the discretized flow and is…
In the theory of neutral differential equations with pulse influence (neutral impulsive differential equations), there are many unsolved problems related to certain results in the theory of integral and integro-differential equations. In…
The threshold, or saturation phenomenon of spatially coupled systems is revisited in the light of Lyapunov's theory of dynamical systems. It is shown that an application of Lyapunov's direct method can be used to quantitatively describe the…
The aim of this paper is to establish the existence and uniqueness of the solution to a system of nonlinear fully coupled forward-backward doubly stochastic differential equations with Poisson jumps. Our system is Markovian in the sense…
We discuss several numerical methods for calculating Lyapunov exponents (a quantitative measure of chaos) in systems of ordinary differential equations. We pay particular attention to constrained systems, and we introduce a variety of…
We propose a novel framework for the Lyapunov analysis of an important class of hybrid systems, inspired by the theory of symbolic dynamics and earlier results on the restricted class of switched systems. This new framework allows us to…
Motivated by recent interest on Kirchhoff-type equations, in this short note we utilize a classical, yet very powerful, tool of nonlinear functional analysis in order to investigate the existence of positive eigenvalues of systems of…
We study proximal random dynamical systems of homeomorphisms of the circle without a common fixed point. We prove the existence of two random points that govern the behavior of the forward and backward orbits of the system. Assuming the…
The well established mixed monotone iterative technique that is used to study the existence and uniqueness of fractional order system is studied explicitly for impulsive system with Hilfer fractional order in this paper. The procedure of…
We develop a new theory for treating boundary problems for linear ordinary differential equations whose fundamental system may have a singularity at one of the two endpoints of the given interval. Our treatment follows an algebraic…
This paper presents some new propositions related to the fractional order $h$-difference operators, for the case of general quadratic forms and for the polynomial type, which allow proving the stability of fractional order $h$-difference…
We present a Lyapunov type approach to the problem of existence and uniqueness of general law-dependent stochastic differential equations. In the existing literature most results concerning existence and uniqueness are obtained under…