Stratified Topological Behavioural Metastrcture

This work introduces a stratified geometric framework for the analysis of dynamical systems based on locally stable behaviours linked to catastrophe theory, Whitney unfolding, and stratified spaces. The central construction is a compactified space, the Stratified Pinched Double Torus (SPDT), obtained through one-point compactification of the Whitney unfolding domain of the fold catastrophe while preserving equilibrium strata and incidence relations. From this global structure, we derive a reduced representation, the Stratified Topological Behavioural Metastructure (STBM), obtained via transversal projections onto stratified double-disc configurations. This representation encodes system dynamics in terms of four structurally stable regimes associated with distinct phases of behaviour: emergence, adaptation, structural growth, and consolidation/re-emergence. The framework integrates local stability theory with global geometric organisation, allowing equilibrium curves, bifurcations, and transition zones to be represented as stratified structures across compactified domains. A finite structural modality emerges, characterised by four dominant scaling regimes that govern stable unfolding behaviour. Transversal fibre structures provide a geometric interpretation of transitions between regimes, while zero-dimensional strata act as organising centres for reconfiguration and cyclic re-emergence. The resulting construction supports a unified representation of stability, transition, and resilience within a single stratified topology. The proposed framework establishes a bridge between catastrophe theory, stratified spaces, and behavioural modelling, offering a coherent geometric language for complex system dynamics across scales.