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An asymptotic stability proof and a port-Hamiltonian physics-informed neural network approach to chaotic synchronization

Behnam Babaeian, Marius E. Yamakou

TL;DR

The paper addresses chaotic synchronization in a biophysically enriched five-dimensional Hindmarsh–Rose neuron model with electromagnetic induction and memristive autapse. It combines rigorous dynamical-systems analysis, deriving a linearized error system and a quadratic Lyapunov function to prove global asymptotic synchronization, along with a Helmholtz-based synchronization energy H and its rate dH/dt to quantify energetic costs. It also introduces a port–Hamiltonian physics-informed neural network (pH–PINN) that learns the energy landscape and interconnection/dissipation structure from data, enforcing both conservation and dissipation through specialized losses; the learned H and energy rate align closely with analytical benchmarks. Numerical results confirm CS and demonstrate consistent Lyapunov–Hamiltonian diagnostics across parameter sweeps, supporting an energy-aware framework for understanding and controlling nonlinear neuronal synchronization. The work bridges rigorous stability proofs with energy-consistent data-driven modeling, with potential impact on energy-efficient control of complex neural networks and other dissipative, chaotic systems.

Abstract

We study chaotic synchronization in a five-dimensional Hindmarsh-Rose neuron model augmented with electromagnetic induction and a switchable memristive autapse. For two diffusively coupled neurons, we derive the linearized error dynamics and prove global asymptotic stability of the synchronization manifold using a quadratic Lyapunov function. Verifiable sufficient conditions follow from Sylvester's criterion, and convergence is established using Barbalat's lemma. A Helmholtz decomposition splits the error field into conservative and dissipative parts, yielding a closed-form synchronization energy and its dissipation law, which quantify the energetic cost of synchrony. Numerical simulations confirm complete synchronization, monotone decay of the synchronization energy, and close agreement between Lyapunov and Hamiltonian diagnostics across parameter sweeps. Building on these results, we introduce a port-Hamiltonian physics-informed neural network that embeds the conservative/dissipative structure through tailored losses and structural priors. The learned Hamiltonian and energy-rate match analytical benchmarks, providing an energy-aware, data-driven template for modeling and control of nonlinear neuronal synchronization.

An asymptotic stability proof and a port-Hamiltonian physics-informed neural network approach to chaotic synchronization

TL;DR

The paper addresses chaotic synchronization in a biophysically enriched five-dimensional Hindmarsh–Rose neuron model with electromagnetic induction and memristive autapse. It combines rigorous dynamical-systems analysis, deriving a linearized error system and a quadratic Lyapunov function to prove global asymptotic synchronization, along with a Helmholtz-based synchronization energy H and its rate dH/dt to quantify energetic costs. It also introduces a port–Hamiltonian physics-informed neural network (pH–PINN) that learns the energy landscape and interconnection/dissipation structure from data, enforcing both conservation and dissipation through specialized losses; the learned H and energy rate align closely with analytical benchmarks. Numerical results confirm CS and demonstrate consistent Lyapunov–Hamiltonian diagnostics across parameter sweeps, supporting an energy-aware framework for understanding and controlling nonlinear neuronal synchronization. The work bridges rigorous stability proofs with energy-consistent data-driven modeling, with potential impact on energy-efficient control of complex neural networks and other dissipative, chaotic systems.

Abstract

We study chaotic synchronization in a five-dimensional Hindmarsh-Rose neuron model augmented with electromagnetic induction and a switchable memristive autapse. For two diffusively coupled neurons, we derive the linearized error dynamics and prove global asymptotic stability of the synchronization manifold using a quadratic Lyapunov function. Verifiable sufficient conditions follow from Sylvester's criterion, and convergence is established using Barbalat's lemma. A Helmholtz decomposition splits the error field into conservative and dissipative parts, yielding a closed-form synchronization energy and its dissipation law, which quantify the energetic cost of synchrony. Numerical simulations confirm complete synchronization, monotone decay of the synchronization energy, and close agreement between Lyapunov and Hamiltonian diagnostics across parameter sweeps. Building on these results, we introduce a port-Hamiltonian physics-informed neural network that embeds the conservative/dissipative structure through tailored losses and structural priors. The learned Hamiltonian and energy-rate match analytical benchmarks, providing an energy-aware, data-driven template for modeling and control of nonlinear neuronal synchronization.

Paper Structure

This paper contains 13 sections, 4 theorems, 33 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical model and dynamics
  3. Stable chaotic synchronization energy: A dynamical systems approach
  4. Asymptotic stability proof of the synchronization manifold
  5. Synchronization energy
  6. Numerical simulations and discussion
  7. A port-Hamiltonian physics-informed neural network (pH-PINN) approach
  8. Physics-informed neural networks (PINNs)
  9. Hamiltonian neural networks (HNNs)
  10. Port-Hamiltonian neural networks (pHNN)
  11. Port--Hamiltonian physics-informed neural network (pH-PINN)
  12. Training and Results
  13. Summary and conclusions

Key Result

Theorem 3.2

The coupled chaotic systems in Eq. coupled_eq achieve CS if the five ordered principal minors in Eq. sub_determinants are all sufficiently positive.

Figures (12)

  • Figure 1: Chaotic time series of the $x$ variable (a) and corresponding phase portrait in the $x-y$ plane (b) with $k = 0.87$, $m = 0.5$ and $\rho = 0.7$.
  • Figure 2: Bifurcation diagrams and the two largest Lyapunov exponents versus $k$ at $\rho = 0.7$ for different values of $m$: (a)$m = 1$, (b)$m = 0.5$, (c)$m = 0.25$.
  • Figure 3: Bifurcation diagrams and the two largest Lyapunov exponents versus $\rho$ at $k = 0.87$ for different values of $m$: (a)$m = 1$, (b)$m = 0.5$, (c)$m = 0.25$.
  • Figure 4: Two-dimensional dynamical maps for different parameter pairs: (a)$(k,m)$ at fixed $\rho = 0.7$; (b)$(\rho,m)$ at fixed $k = 0.87$; (c1–c3)$(\rho,k)$ at fixed $m = 1$, $0.5$, and $0.25$, respectively.
  • Figure 5: (a)-(e): Time evolution of the five synchronization-error components $(e_x,e_y,e_z,e_u,e_\phi)$, respectively, showing convergence to zero.
  • ...and 7 more figures

Theorems & Definitions (6)

  • Definition 3.1
  • Theorem 3.2
  • proof
  • Lemma 3.3: Sylvester's Criterion
  • Lemma 3.4
  • Lemma 3.5