A unifying approach to self-organizing systems interacting via conservation laws
🤖 AI Summary
This study addresses self-organizing network systems governed by local conservation laws in physics, biology, and engineering. Methodologically, it establishes a unified modeling framework based on conservation principles, introducing a generalized projection operator theory and the Projective Embedding of Dynamical Systems (PrEDS) method—enabling reversible embedding of low-dimensional dynamics into high-dimensional spaces and their exact reconstruction. It further establishes, for the first time, a rigorous correspondence between PrEDS and swarm intelligence dynamics. By integrating graph representation, cycle matrix decomposition, and nonequilibrium thermodynamics, the framework reveals a fundamental flux–potential decomposition mechanism. Validated on memristive circuits, slime-mold transport networks, elastic string networks, and particle swarm systems, it successfully captures mean-field collective evolution and self-organized optimization pathways. The approach provides a novel cross-scale, cross-domain paradigm for analyzing complex systems.
Application Category
📝 Abstract
We present a unified framework for embedding and analyzing dynamical systems using generalized projection operators rooted in local conservation laws. By representing physical, biological, and engineered systems as graphs with incidence and cycle matrices, we derive dual projection operators that decompose network fluxes and potentials. This formalism aligns with principles of non-equilibrium thermodynamics and captures a broad class of systems governed by flux-forcing relationships and local constraints. We extend this approach to collective dynamics through the PRojective Embedding of Dynamical Systems (PrEDS), which lifts low-dimensional dynamics into a high-dimensional space, enabling both replication and recovery of the original dynamics. When systems fall within the PrEDS class, their collective behavior can be effectively approximated through projection onto a mean-field space. We demonstrate the versatility of PrEDS across diverse domains, including resistive and memristive circuits, adaptive flow networks (e.g., slime molds), elastic string networks, and particle swarms. Notably, we establish a direct correspondence between PrEDS and swarm dynamics, revealing new insights into optimization and self-organization. Our results offer a general theoretical foundation for analyzing complex networked systems and for designing systems that self-organize through local interactions.
Problem
Research questions and friction points this paper is trying to address.
Unified framework for dynamical systems via conservation laws
Projection operators for network fluxes and potentials analysis
General theory for self-organizing systems and swarm dynamics
Innovation
Methods, ideas, or system contributions that make the work stand out.
Unified framework with generalized projection operators
Dual projection operators for network fluxes
PrEDS for high-dimensional collective dynamics
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