Introduction
This research explores a dynamic graph rewrite model of coupled oscillators, representing autocatalytic cycles of frequency and phase. This is inspired by the Kauffman-Guerin autocatalytic particle transformations in the Particle Apothecary Model defined in Kauffman, Guerin (2024) Did the Universe Construct Itself?, where spacetime graphs grow autocatalytically.
Background and Motivation
In the Kauffman-Guerin model, spacetime emerges from autocatalytic transformations, suggesting a self-constructing universe. Coupled oscillators are known to exhibit complex behaviors such as synchronization and resonance, which can be mapped to autocatalytic cycles in frequency and phase.
By leveraging spectral graph theory, this project aims to analyze resonances and modes within these dynamic graphs, offering insights into the emergent properties of spacetime growth. The research bridges concepts from:
- Nonlinear dynamics of coupled oscillators
- Spectral graph theory
- Autocatalytic network formation
- Emergent spacetime models
Research Objectives
- Model coupled oscillators as autocatalytic cycles with frequency and phase states.
- Apply spectral graph theory and Laplacian analysis to explore resonance and mode dynamics.
- Investigate the relationship between eigenvalue spectra and autocatalytic growth patterns.
- Relate findings to the Kauffman-Guerin DUCI model's autocatalytic particle transformations.
Mathematical Formulation
1. Dynamic Graph Model of Coupled Oscillators
The dynamic graph rewrite model is represented as a time-varying graph \(G(t) = (V, E(t))\) where each node \(i\) has the following state variables:
Edge Dynamics:
Coupling between nodes \(i\) and \(j\) can be represented as:
\[
\frac{d^2x_i}{dt^2} + \gamma_i\frac{dx_i}{dt} + \omega_i^2x_i = \sum_j K_{ij}(x_j - x_i)
\]
where:
- \(K_{ij}\) is the coupling strength between nodes \(i\) and \(j\)
- \(\gamma_i\) is the damping coefficient for node \(i\)
Energy of the System:
\[
E = \sum_{i=1}^{n} \left[ \frac{1}{2}m_i\left(\frac{dx_i}{dt}\right)^2 + \frac{1}{2}k_ix_i^2 \right] + \sum_{i,j} \frac{1}{2}K_{ij}(x_j - x_i)^2
\]
Spectral Graph Theory and Laplacian Analysis
Laplacian Matrix Formulation
The Laplacian matrix of a graph is defined as:
\[ L = D - A \]
where:
- \(D\) is the degree matrix
- \(A\) is the adjacency matrix
For weighted graphs, the weighted Laplacian is:
\[ L_{ij} =
\begin{cases}
\sum_{k \neq i} w_{ik} & \text{if } i = j \\
-w_{ij} & \text{if } i \neq j \text{ and } (i,j) \in E \\
0 & \text{otherwise}
\end{cases}
\]
Eigenvalue and Eigenvector Analysis
- Eigenvalues \(\lambda_k\) correspond to normal modes of oscillation
- Small eigenvalues indicate global, systemic modes
- Larger eigenvalues correspond to localized modes
- The eigenvector \(v_k\) corresponding to eigenvalue \(\lambda_k\) represents the spatial distribution of the mode
The eigenvalue equation is:
\[ L v_k = \lambda_k v_k \]
Resonance Detection and Mode Analysis
Natural Frequencies
Resonance occurs when node frequencies match or harmonically relate to Laplacian eigenfrequencies:
\[ \omega_i \approx \sqrt{\lambda_k} \]
Eigenvalue Spectrum Analysis
- Eigenvalue Spacing: Anomalous spacing indicates resonance or mode coupling
- Participation Ratio: Measures localization of eigenmodes
\[ P_k = \frac{(\sum_i |v_{ki}|^2)^2}{\sum_i |v_{ki}|^4} \]
where \(v_{ki}\) is the \(i\)-th component of the \(k\)-th eigenvector
- Eigenvector Analysis: Eigenvectors correspond to spatial modes of oscillation, revealing phase coherence and amplitude distribution among nodes
Dynamic Graph Rewrite and Harmonic Analysis
Time-varying Laplacian
The time-dependent Laplacian \(L(t) = D(t) - A(t)\) captures dynamic coupling changes in the network.
Fourier Analysis
Applying Fourier transforms to node oscillations extracts dominant frequencies and phases:
\[ \hat{x}_i(\omega) = \int_{-\infty}^{\infty} x_i(t) e^{-i\omega t} dt \]
Autocatalytic Cycles
Autocatalytic cycles emerge when the oscillation of one node catalyzes changes in connected nodes, which in turn affect the original node. These cycles can be detected through:
- Cycle detection in the graph
- Phase synchronization patterns
- Frequency entrainment behaviors
Relation to Kauffman-Guerin DUCI Model
The Kauffman-Guerin DUCI (Did the Universe Construct Itself?) model proposes that spacetime emerges from autocatalytic particle transformations. Our research extends this concept by:
- Mapping oscillator nodes to autocatalytic motifs in as expressed in DUCI paper
- Interpreting frequency/phase synchronization as analogous to least action pathway selection
- Using dynamic graph rewrites to model the autocatalytic growth of spacetime
- Relating eigenvalue spectra to emergent spacetime properties
Methodology and Implementation
- Construct dynamic graphs where nodes are oscillators with states of frequency, amplitude, and phase.
- Define autocatalytic cycles influencing node state updates via dynamic graph rewrites.
- Utilize spectral graph theory, particularly Laplacian eigenvalues and eigenvectors, to analyze resonances and modes.
- Simulate the system using agentscript.org, visualizing resonance patterns and spectral evolution.
Simulation Framework
\[
\begin{align}
x_i(t+\Delta t) &= x_i(t) + v_i(t)\Delta t \\
v_i(t+\Delta t) &= v_i(t) + \left[-\gamma_i v_i(t) - \omega_i^2 x_i(t) + \sum_j K_{ij}(x_j(t) - x_i(t))\right]\Delta t
\end{align}
\]
The graph structure itself evolves according to node state-dependent rules:
\[
K_{ij}(t+\Delta t) = f(K_{ij}(t), \omega_i, \omega_j, \phi_i, \phi_j, A_i, A_j)
\]
Outcomes and Significance
This research is focused on::
- The relationship between network topology and emergent oscillator dynamics
- How autocatalytic cycles in coupled oscillators lead to self-organizing behavior
- Potential mechanisms for spacetime emergence in the DUCI framework
- Novel approaches to analyzing complex systems through spectral graph theory
The findings contributes to fundamental understanding in complex systems theory, network science, and theoretical physics, particularly relating to emergent phenomena and self-organization.