Ivković Graph-Coupled Chaotic Dynamical Networks for Bounded High-Dimensional Nonlinear Parametric Embedding: A Mathematical, Digital-Simulation and Analogue-Hybrid Framework

Extended author version v1.1 of the paper presented at the 4th LINK IT & EdTech International Scientific Conference. The revision strengthens the mathematical treatment of well-posedness, volume contraction, absorbing boundedness, global existence, synchronization-manifold invariance and master-stability decomposition; clarifies the finite-time representation interpretation; adds Lyapunov convergence and divergence-sum consistency checks; positions the framework relative to physical reservoir and oscillator computing; and documents the analogue-hybrid implementation path. Abstract: This paper presents Ivković Graph-Coupled Chaotic Dynamical Networks (GCCDNs) as a mathematical, numerical and analogue-hybrid framework for bounded high-dimensional finite-time nonlinear parametric representation. The construction strengthens the author’s 2024 N-dimensional attractor proposal by replacing independent stacking with explicit graph-Laplacian diffusive coupling. The canonical instance is a network of N Lorenz generators coupled through the x-coordinate, yielding a 3N-dimensional autonomous nonlinear system. Under a finite undirected nonnegatively weighted graph and nonnegative coupling strength, the paper establishes local well-posedness, volume contraction, an explicit absorbing-set estimate and global existence; it also identifies the invariant complete-synchronization manifold and its master-stability variational decomposition. Numerical certification is formulated in terms of the full finite-time Lyapunov spectrum, hyperchaotic capacity, Kaplan–Yorke diagnostics, synchronization non-collapse, solver refinement and a divergence-sum consistency check. For computational use, the finite-time map is treated as a representation operator rather than a strict topological embedding unless injectivity and rank conditions are separately established. The analogue-hybrid path is documented through Lorenz-generator prototypes, a three-module nine-state architecture and a proposed analogue Laplacian coupling stage. The work does not claim universal chaos, universal AI superiority or completed hardware validation; its contribution is a falsifiable graph-controlled dynamical framework and a reproducible program for testing whether intermediate, bounded and non-collapsed regimes can provide useful nonlinear representations. Keywords: chaotic dynamical systems; graph-coupled networks; Lorenz system; nonlinear parametric representation; hyperchaos; Lyapunov spectrum; synchronization; physical reservoir computing; oscillator computing; analogue-hybrid computing; numerical certification.