OSCILLATORY BEHAVIOR OF SECOND-ORDER DIFFERENTIAL EQUATIONS VIA GENERALIZED LOCAL DERIVATIVES

Abstract

This work investigates the oscillatory behavior of solutions to a class of second-order differential equations governed by the Generalized Local Derivative (N^α_F). Unlike traditional models limited to power-law kernels, our framework employs an arbitrary functional kernel F(t, α) to describe systems with varying memory and dissipation properties. By utilizing the Riccati transformation technique and integral averaging, we establish new oscillation criteria for both divergent and convergent cases of the kernel-weight integral. Our analysis demonstrates that the property of oscillation is a functional consequence of the interaction between the kernel structure, the potential term q(t), and the conductance a(t). We validate these theoretical findings through numerical simulations in the phase plane, revealing the critical influence of the kernel’s decay rate on the system’s ability to sustain cyclic dynamics.