We discuss the dynamics of a fractional order discrete neuron model with electromagnetic flux coupling. The discussed neuron model is a simple one-dimensional map which is modified by considering flux coupling. We consider a discrete fractional order memristor to mimic the effects of electromagnetic flux on the neuron model. The bifurcation dynamics of the fractional order neuron map show an inverse period-doubling route to chaos as a function of control parameters, namely the fractional order of the map and the flux coupling coefficient. The bifurcation dynamics of the systems are derived both in the time and frequency domains. We present a two-parameter phase diagram using the Lyapunov exponent to categorize the various dynamics present in the system. In addition to the Lyapunov exponent, we use the entropy of the model to distinguish the various dynamics of the systems. To investigate the network behavior of the fractional order neuron map, a lattice array of $N\times N$ nodes is constructed and external periodic stimuli are applied to the network. The formation of spiral waves in the network and the impact of various parameters, like the fractional order, flux coupling coefficient and the coupling strength on the wave propagation are also considered in our analysis.

The subject of this research is a coupled system of nonlinear viscoelastic wave equations with distributed delay components, infinite memory and Balakrishnan-Taylor damping. Assume the kernels $g_i:{\mathbf{R}}_{+}\to {\mathbf{R}}_{+}$ holds true the below

$$g_i^{\prime }(t)\le -{\zeta }_i(t)G_i(g_i(t)),\mathrm{\forall }t\in {\mathbf{R}}_{+},\mathrm{f}\mathrm{o}\mathrm{r}i=1,2,$$

in which ${\zeta }_i$ and $G_i$ are functions. We demonstrate the stability of the system under this highly generic assumptions on the behaviour of $g_i$ at infinity and by dropping the boundedness assumptions in the historical data.

This study examines the pace at which solutions to a Bresse system in combination with the Cattaneo law of heat conduction and the dispersed delay term degradation. We establish our major finding utilizing the energy approach in the Fourier space.

This work focused on the analysis of a nonlinear wave equation of Hartree-type that includes a distributed delay term, where the delay effects are governed with fractional conditions. Such a formulation allows the model to incorporate long-range memory effects and anomalous dissipation phenomena, which are characteristic of complex media. The model captures complex memory and nonlocal interaction effects that arise in various physical systems, such as quantum mechanics and nonlinear optics. In particular, the fractional delay mechanism provides a more accurate description of hereditary effects than classical integer-order delay models. We worked under a framework that allows for initial data with negative energy and imposed suitable assumptions on the kernel functions and nonlinear terms. Using energy methods and a concavity argument, we rigorously proved that the solution to the system cannot exist globally in time and must blow up in finite time. Compared with the classical Hartree wave equation without delay or fractional effects, our results show that the combined presence of distributed delay and fractional damping significantly enhances the instability mechanism.

A nonlinear viscoelastic Kirchhoff-type equation with a logarithmic nonlinearity, Balakrishnan-Taylor damping, dispersion and distributed delay terms is studied. We establish the global existence of the solutions of the problem and by the energy method we prove an explicit and general decay rate result under suitable hypothesis.

This paper is an extension of our earlier research^[1]. In this article, we discussed the fractionally damped plate equation with infinite memory. We proved that solutions are global under different initial and boundary conditions. Subsequently, we investigated the exponential stability of the solutions through Lyapunov functionals in order to establish stability conditions sufficient for stability analysis.