Chaotic bursting and multi-layer dynamics in a discrete fractional meminductor-coupled Hopfield neural network with mixed-saturation heterogeneous activations
N. Avinash · Nonlinear science. · 2026
Modelling brain-like memory and rhythmic processing requires dynamical systems that simultaneously capture long-term memory, synaptic plasticity, and frequency-multiplexed information transfer. This study introduces a five-dimensional discrete fractional Hopfield neural network coupled with a flux-controlled meminductor whose meminductance is periodically modulated by a sine of the internal flux state, and whose internal dynamics are governed by a harmonic-scale restoring nonlinearity. Heterogeneous mixed-saturation activations are employed: a hyperbolic tangent for the first and third neurons (fast exponential-tail saturation), an arctangent scaled activation 2 π arctan ( u 2 ) for the second neuron (algebraic-tail saturation, slower approach to bounds), and a softsign activation u 4 / ( 1 + | u 4 | ) for the fourth neuron (heaviest algebraic-tail saturation). This deliberately non-periodic but distinct-shape combination mirrors the coexistence of neuron populations with different transfer sharpness in cortical microcircuits; no intrinsic sinusoidal frequency is assigned to the activations themselves, the rhythmic content arising from the sine-modulated meminductance and the network coupling. The Caputo fractional difference operator endows the network with non-local memory effects and makes hereditary dependencies of biological synapses tractable in a discrete setting. Local stability of the unique equilibrium is established through the Cermak–Győri–Nechvátal criterion and verified numerically. Bifurcation analysis with respect to the fractional order ν and the meminductance parameters ζ 1 , ζ 2 uncovers windows of period-doubling, intermittent bursting, hyperchaos, and reverse cascades. Coexisting attractors, basin-of-attraction mosaics, and a composite approximate-entropy landscape are used to quantify the system’s multi-stability and dynamical complexity. A central new finding is the emergence of frequency-locked multi-layer attractors whose layer count is controlled jointly by the ratio ζ 1 / ζ 2 and by the fractional order, producing symmetric, asymmetric, and limit-cycle-cored multi-layer regimes. The biological consequences for mixed-saturation coupling, attentional gating, and neuromorphic encoding are discussed, alongside applications to chaos-based secure communication.