Bistability and Basin Classification in Competitive Adaptive Systems: A Structural Framework and Scalar Index for Regime-Shift Analysis

This paper introduces a structural class of two-dimensional competitive adaptive systems (CRTI-class systems) defined by competitive coupling, self-limitation, and a compact invariant domain. Using Poincaré index theory and the Stable Manifold Theorem, we show that systems in this class admitting two boundary attractors necessarily contain an interior saddle point whose stable manifold partitions state space into two qualitatively distinct basins corresponding to adaptive and compression-dominated outcomes.   We further define a scalar index T = R/\Phi, where R denotes adaptive response capacity and \Phi denotes structural compression, and prove that T acts as a local first-order basin classifier in a neighbourhood of the saddle, without constituting a geometric distance to the separatrix.   Operational estimators for R and \Phi are derived from linear response theory and spectral covariance analysis, enabling empirical application to multivariate time series. High-dimensional reduction, thermodynamic interpretation, and cross-domain universality are explicitly identified as open problems.   The framework provides a mathematically grounded substrate for regime-shift analysis while maintaining clear limits of validity.   bistability, basin of attraction, separatrix, competitive dynamical systems, regime shifts, early warning signals, complex adaptive systems, structural compression, adaptive capacity, Poincaré index, stable manifold, critical transitions, dynamical systems theory, CRTI