Abstract model of rhythm synchronization driven by asymmetric dissipation under VNAE.
Rather than pursuing biological realism, the objective is to expose a geometric mechanism of global convergence driven by asymmetric dissipation.
We reinterpret synchronization as a manifestation of asymmetric dissipation.
This example does not aim to compete with classical biological oscillator models (e.g. Kuramoto, Winfree). Instead, it provides a structural explanation for convergence in heterogeneous systems.
We consider a network of ( N ) abstract oscillatory units, each represented by a scalar phase variable ( \phi_i(t) ).
The dynamics are given by:
dφ/dt = − L φ − Θ φ
where:
- φ ∈ ℝⁿ is the vector of phase states
- L is the graph Laplacian encoding network coupling
- Θ = diag(θ₁, …, θₙ) is a diagonal matrix of asymmetric dissipation parameters
- The Laplacian term promotes coordination between neighboring units.
- The asymmetric dissipation term introduces heterogeneous contraction rates.
- No sinusoidal coupling, frequency tuning, or phase-locking assumptions are required.
In this framework, we can see it is clear and understandable that synchronization emerges as a consequence of global dissipation geometry, not parameter fine-tuning.
Within the VNAE framework:
- Asymmetry is not treated as a perturbation or defect.
- Heterogeneity induces an effective positive curvature in the system’s state space.
- Global convergence follows structurally, even without symmetric equilibria.
In this sense, synchronization appears as an operational manifestation of positive curvature, rather than a classical phase-locking phenomenon.
The same structural principle applies to:
- abstract neural synchronization
- circadian rhythm coordination
- coupled chemical oscillators
- distributed dynamical systems beyond biology
The simulation is implemented in R, Python, Julia, and Matlab, using:
- explicit Euler integration
- simple ring network topology
- heterogeneous dissipation parameters
The code demonstrates how asymmetric agents converge to a synchronized regime without symmetry assumptions.
MIT License