Adjusting Dynamics of Hopfield Neural Network via Time-variant Stimulus

TL;DR

The paper investigates modulating Hopfield neural network dynamics with time-variant stimuli to generate rich multi-scroll chaotic attractors, validated by FPGA experiments and extended to a chaos-based image encryption scheme. It introduces the Adjusted Hopfield Neural Network (AHNN) and analyzes boundedness, equilibrium stability, Lyapunov spectra, and bifurcations under WMS, SVS, and CS, finding that appropriate stimuli dramatically broaden chaotic regimes while CS can suppress chaos. The AHNN is implemented on an FPGA using RK-4 with a piecewise Taylor approximation for $\tanh$, and its chaotic sequences enable a Cat-map-based image encryption method with a key space around $10^{127}$ and strong statistical security metrics. Overall, the work provides a practical route to dynamic control of nonlinear networks and demonstrates a secure, chaos-based multimedia communication approach driven by time-variant stimuli.

Abstract

As a paradigmatic model for nonlinear dynamics studies, the Hopfield Neural Network (HNN) demonstrates a high susceptibility to external disturbances owing to its intricate structure. This paper delves into the challenge of modulating HNN dynamics through time-variant stimuli. The effects of adjustments using two distinct types of time-variant stimuli, namely the Weight Matrix Stimulus (WMS) and the State Variable Stimulus (SVS), along with a Constant Stimulus (CS) are reported. The findings reveal that deploying four WMSs enables the HNN to generate either a four-scroll or a coexisting two-scroll attractor. When combined with one SVS, four WMSs can lead to the formation of an eight-scroll or four-scroll attractor, while the integration of four WMSs and multiple SVSs can induce grid-multi-scroll attractors. Moreover, the introduction of a CS and an SVS can significantly disrupt the dynamic behavior of the HNN. Consequently, suitable adjustment methods are crucial for enhancing the network's dynamics, whereas inappropriate applications can lead to the loss of its chaotic characteristics. To empirically validate these enhancement effects, the study employs an FPGA hardware platform. Subsequently, an image encryption scheme is designed to demonstrate the practical application benefits of the dynamically adjusted HNN in secure multimedia communication. This exploration into the dynamic modulation of HNN via time-variant stimuli offers insightful contributions to the advancement of secure communication technologies.

Figures (21)

• Figure 1: Phase diagram of network \ref{['eq:OHNN']}: (a) two-scroll attractors; (b) coexisting one-scroll attractors.
• Figure 2: The structure of AHNN with time-variant stimuli.
• Figure 3: Phase diagram of AHNN with four WMSs: (a) four-scroll chaotic attractor; (b) two-scroll coexisting attractors.
• Figure 4: The time series of AHNN with $P(t) = \pm 1$: (a) $x_1-x'_1$; (b) $x_2-x'_2$; (c) $x_3-x'_3$.
• Figure 5: Phase diagram of AHNN with four WMSs and one SVS: (a) eight-scroll attractor; (b) four-scroll attractor; (c) four-scroll attractor with different $A_1$; (d) two-scroll attractor with different $A_1$.

Theorems & Definitions (2)

• proof
• proof