Dynamics

Diffusion on curved surfaces deviates from the flat case due to geometrical corrections in the evolution of its moments, such as the geodesic mean square displacement. Moreover, anomalous diffusion is widely used to model transport in disordered, confined, or crowded environments and can be described by a temporal subordination scheme, leading to a time-fractional diffusion equation. In this work, we analyze the dynamics of time subordinated anomalous diffusion on curved surfaces. By using a generalized Taylor expansion with fractional derivatives in the Caputo sense, we express the moments as a temporal power series and show that the anomalous exponent couples with curvature terms, leading to a competition between geometric and anomalous effects. This coupling indicates a mechanism through which curvature modulates anomalous transport.

In this study, we explore the emergence of generalized synchronization (GS) in arrays of Hindmarsh–Rose (HR) neurons that are coupled through memristive synapses. We design coupling functions utilizing active memristors to facilitate GS in a bidirectionally coupled two-neuron memristive neural network (MNN). Our analysis employs a nearest neighbor (NN) approach. Our findings indicate that there is a threshold coupling strength for the active memristive synapses required to achieve GS. Additionally, we investigate how memristor parameters affect the temporal characteristics of synchronized neuronal firing patterns. Specifically, we discover that the interburst interval (IBI) is directly proportional to the coupling strength of the memristive synapses, while the interspike interval (ISI) is inversely proportional to this strength.

We provide an analytical framework for describing the propagation of light in waveguide arrays, considering both infinite and semi-infinite cases. The interaction up to second neighbors is taken into account, which provides a more realistic setup. We show that these solutions follow a distinctive structural pattern. This pattern reflects a transition from conventional Bessel functions to the lesser-known one-parameter generalized Bessel functions, offering new insights into the propagation dynamics in these systems.