arXiv subjects

Ábel Garab

Publications and source records attributed to Ábel Garab.

5 recordsLinked to original sources

arXiv2026

Oscillation results for first order neutral delay differential equations with several positive and negative coefficients

We provide sufficient criteria for the oscillation of all solutions of neutral delay differential equations of the form \[ \left[x(t) - \sum_{i=1}^{N_r}R_i(t)x(t - r_i(t)) \right]' + \sum_{i=1}^{N_p}P_i(t)x(t - τ_i(t)) - \sum_{i=1}^{N_q}Q_i(t)x(t - δ_i(t))=0, \] with both positive and negative terms and time-variable delays. Our results improve and generalize several existing criteria available in the literature that address restricted cases, such as constant delays or the absence of negative coefficients. Under additional assumptions on slowly varying parameters, we derive sharper oscillation conditions. We demonstrate the applicability of our findings through illustrative examples.

math.DS↗

arXiv2025

Morse decomposition for semi-dynamical systems with an application to systems of state-dependent delay differential equations

Understanding the structure of the global attractor is crucial in the field of dynamical systems, where Morse decompositions provide a powerful tool by partitioning the attractor into finitely many invariant Morse sets and gradient-like connecting orbits. Building on Mallet-Paret's pioneering use of discrete Lyapunov functions for constructing Morse decompositions in delay differential equations, similar approaches have been extended to various delay systems, also including state-dependent delays. In this paper, we develop a unified framework assuming the existence and some properties of a discrete Lyapunov function for a semi-dynamical system on an arbitrary metric space, and construct a Morse decomposition of the global attractor in this general setting. We demonstrate that our findings generalize previous results; moreover, we apply our theorem to a cyclic system of differential equations with threshold-type state-dependent delay.

math.DS↗

arXiv2025

Morse decomposition of scalar differential equations with state-dependent delay

We consider state-dependent delay differential equations of the form $$\dot{x}(t) = f(x(t), x(t - r(x_t))),$$ where $f$ is continuously differentiable and fulfills a negative feedback condition in the delayed term. Under suitable conditions on $r$ and $f$, we construct a Morse decomposition of the global attractor, giving some insight into the global dynamics. The Morse sets in the decomposition are closely related to the level sets of an integer valued Lyapunov function that counts the number of sign changes along solutions on intervals of length of the delay. This generalizes former results for constant delay. We also give two major types of state-dependent delays for which our results apply.

math.DS↗

arXiv2025

Discrete Lyapunov functional for cyclic systems of differential equations with time-variable or state-dependent delay

We consider nonautonomous cyclic systems of delay differential equations with variable delay. Under suitable feedback assumptions, we define an (integer valued) Lyapunov functional related to the number of sign changes of the coordinate functions of solutions. We prove that this functional possesses properties analogous to those established by Mallet-Paret and Sell for the constant delay case and by Krisztin and Arino for the scalar case. We also apply the results to equations with state-dependent delays.

math.DS↗

arXiv2012

Local stability implies global stability for the 2-dimensional Ricker map

Consider the difference equation $x_{k+1}=x_k e^{α-x_{n-d}}$ where $α$ is a positive parameter and d is a non-negative integer. The case d = 0 was introduced by W.E. Ricker in 1954. For the delayed version d >= 1 of the equation S. Levin and R. May conjectured in 1976 that local stability of the nontrivial equilibrium implies its global stability. Based on rigorous, computer aided calculations and analytical tools, we prove the conjecture for d = 1.

math.DS↗