Woodpecker-Like Chaotic Attractor from Cross-Coupled Exponential Nonlinearity: Dynamics, Circuit Realization and Random Number Generation

Abstract

This study presents a three-dimensional chaotic model whose entire nonlinearity consists of two exponential terms and which contains no product of state variables. The two exponentials are crosscoupled, each entering the equation of the other variable, while the third equation remains linear. Because no term contains a product of state variables, the flow is uniformly dissipative with a contraction rate fixed by the parameters alone. The model possesses three equilibria: one saddle with a one-dimensional unstable manifold and two saddle-foci with two-dimensional unstable manifolds, a configuration that supports Shilnikov-type dynamics. For four representative parameter regimes the largest Lyapunov exponents are 1.881, 1.541, 1.332 and 1.762, with Kaplan-Yorke dimensions of 2.041, 2.033, 2.028 and 2.038. The projection onto the (13 , ) plane resembles the profile of a woodpecker, which suggested the name used for the model; the same shape is obtained in all four regimes. The analysis covers the equilibria and their stability through the Routh-Hurwitz criterion and the Gershgorin circle theorem, bifurcation diagrams, two-parameter maps of the largest Lyapunov exponent and of entropy, finite-time outcome maps, and a step-size convergence study establishing that the exponential terms render the system stiff. An electronic realization is presented as a MultiSIM circuit simulation whose phase-portrait topology is compared with the numerical portraits. A pseudorandom number generator derived from the model is evaluated against the NIST SP 800-22 suite applied in its intended form, using one hundred independent sequences per regime together with the proportion and uniformity criteria, and its measured key space is 215 2 . Weak phase synchronization under low-amplitude periodic forcing is also examined using Poincaré sections and phase diffusion analysis.