A Lyapunov stability proof and a port-Hamiltonian physics-informed neural network for chaotic synchronization in memristive neurons

We study chaotic synchronization in a 5D Hindmarsh--Rose neuron model augmented with electromagnetic induction and a switchable memristive autapse. For two diffusively coupled identical neurons, we derive the transverse error dynamical system and analyze local synchronization via the linearized error system around the synchronization manifold. A quadratic Lyapunov function yields explicit sufficient conditions for (i) asymptotic stability when the memristive switching remains dissipative and (ii) practical stability with an explicit ultimate bound under non-dissipative switching. We complement this with a Hamiltonian-based viewpoint: a Helmholtz decomposition of the linearized error vector field provides a closed-form synchronization Hamiltonian and its rate identity. Numerical simulations corroborate convergence or ultimate boundedness of the synchronization errors and an overall decay of the synchronization Hamiltonian and its instantaneous rate toward zero after transients, and show consistent trends between Lyapunov- and Hamiltonian-based diagnostics across parameters. Finally, we propose the first port-Hamiltonian physics-informed neural network (pH-PINN) that learns this synchronization Hamiltonian and its rate from data while preserving conservative/dissipative structure, achieving close agreement with the analytical expressions.