Tightly coupled AI, ESG reporting and green supply-chain programs can create multiple stable organizational regimes; speeding up managerial adjustment can destabilize high-performing configurations and even trigger oscillations and chaos, so governance must manage both complementarity strength and the pacing of responses.

Main Finding

A bounded three-dimensional discrete nonlinear model of corporate AI-investment maturity (A), ESG disclosure quality (E), and green supply chain collaboration (G) shows that structural complementarity and organizational responsiveness are distinct. The same structural parameters can support multiple interior equilibria (low, unstable intermediate, high complementarity), but increasing the speed of organizational adjustment (responsiveness µ) can destabilize desirable equilibria and generate quasiperiodicity, intermittent chaotic windows, and re-entrant period-3 behavior. Deterministic chaos (positive Lyapunov exponent) and multistability (coexisting attractors for different initial conditions) imply that faster adjustment is not always beneficial; effective governance must manage both reinforcement strengths and the pacing/damping of responses.

Key Points

• Model structure:
  • Three endogenous, bounded states A, E, G ∈ (0, 1) represent sustained maturity/quality levels (not one-period monetary flows).
  • Cross-domain reinforcement uses Hill (sigmoidal) functions to capture activation thresholds and saturation.
  • Net incentives ΠA, ΠE, ΠG combine baseline incentives r, cross-domain gains β, and self-limiting costs c.
  • Updates use a logit-logistic adaptive rule: ℓ(x) = ln[x/(1−x)] adjusted by µ·Π then mapped back via σ. This keeps dynamics smooth and confined to (0,1) without hard projection.
• Fixed points and separable roles:
  • Interior fixed points satisfy ΠA = ΠE = ΠG = 0; their locations depend on structural parameters (r, β, c, Hill parameters) but not on responsiveness µ.
  • Under baseline calibration, three interior fixed points exist: stable low, unstable intermediate, stable high (i.e., multistability).
• Stability and bifurcations:
  • Local linear stability analyzed via Schur–Jury conditions (discrete analogue of Routh–Hurwitz).
  • Loss of stability occurs through Neimark–Sacker bifurcations at distinct µ thresholds: the low-regime loses stability via a subcritical Neimark–Sacker; the high-regime loses stability via a supercritical Neimark–Sacker.
  • Beyond the high-regime bifurcation, increasing responsiveness produces quasiperiodic motion on invariant tori, torus breakdown producing chaotic windows, intermittent chaos, and re-entrant stable period-3 attractors.
• Nonlinear dynamics diagnostics:
  • Positive leading Lyapunov exponent confirms deterministic chaos in parts of parameter space.
  • Branch-aware multi-start diagnostics and basin computations reveal coexistence of attractors (quasiperiodic/chaotic dynamics can coexist with stable period-3 or fixed-point attractors depending on initial conditions).
• Core insight: structural complementarity (β, Hill shape) determines feasible equilibria; organizational responsiveness (µ) controls dynamic stability and can induce complex, potentially undesirable dynamics even when equilibria remain economically bounded.

Data & Methods

• Nature of the study: theoretical / computational dynamical-systems analysis (no empirical microdata).
• Model equations:
  • Hill-type reinforcement HX(x) = bX + (1−bX) x^{nX} / (θX^{nX} + x^{nX}), with bX∈(0,1), θX∈(0,1), nX>1.
  • Net incentives: ΠA = rA + βG HG(G) − cA A (and analogously for ΠE, ΠG).
  • Bounded adaptive update: X_{t+1} = σ[ℓ(X_t) + µX ΠX(X_t, ·)] for X∈{A,E,G}, where ℓ is logit and σ is logistic.
• Analytical tools:
  • Invariance proof showing (0,1)^3 is forward-invariant under updates.
  • Fixed-point algebra: location independent of µ.
  • Local stability: Schur–Jury criteria applied to Jacobian at fixed points.
  • Bifurcation classification: Neimark–Sacker normal form analysis to determine super- vs subcritical transitions.
• Numerical methods and diagnostics:
  • Baseline calibration and parameter sweeps (primarily varying µ responsiveness coefficients) to trace stability boundaries.
  • Time-series exploration, Poincaré-style inspections for quasiperiodicity.
  • Lyapunov exponent computations to detect chaos.
  • Multi-start simulations to map basins and coexisting attractors; identification of period-3 windows and intermittent chaotic subwindows.
• Modeling choices justified:
  • Discrete-time updates fit decision-interval adjustments (budgets, reporting, contracts).
  • Logit-logistic update avoids boundary artefacts and models saturation/friction near capacity limits.

Implications for AI Economics

• Distinguish structure from dynamics: empirically observed complementarities between AI, ESG disclosure, and green supply-chain activities (β, Hill-shape) determine possible equilibria, but management/practice choices about how rapidly to adjust (µ) determine whether those equilibria are stable and attainable in practice.
• Policy and governance design:
  • Rapid ramp-up of AI investment, disclosure systems, or supply-chain programs can destabilize an otherwise valuable sustainability architecture; regulators and boards should consider pacing and damping mechanisms (phased rollouts, regulatory transition windows, standards harmonization) to avoid oscillations or chaotic behavior in organizational outcomes.
  • Multistability implies path dependence and lock-in: early conditions, shocks, or coordination failures can leave firms stuck in low-complementarity regimes even when a high regime is structurally feasible. Targeted interventions (subsidies, coordination platforms, standards) should consider basin boundaries, not just long-run equilibria.
• Empirical testing suggestions:
  • Operationalize responsiveness µ via observable frequencies and magnitudes of budget/reporting/collaboration adjustments (e.g., revision size per quarter, time-to-implement governance changes); test whether faster-adjusting firms show greater volatility in ESG and supply-chain outcomes.
  • Look for nonlinearity and thresholds (Hill-type activation) in cross-domain returns: quantify diminishing marginal returns and potential half-activation points by estimating interactions between AI capability metrics, disclosure quality measures, and collaboration indices.
  • Search for early warning signals of instability (increasing variance, autocorrelation, emergence of cycles) in longitudinal firm- or supply-chain-level panels around reforms or rapid digitalization events.
• Modeling & empirical extensions:
  • Incorporate stochastic shocks (regulatory changes, supply disruptions) to evaluate noise-induced escapes between basins.
  • Endogenize heterogeneity and networks (multiple firms, suppliers) to study coordination failures and systemic stability.
  • Calibrate the model to firm-level or supply-chain datasets (AI spending cycles, ESG audit quality, supplier-collaboration indices) to estimate structural β and responsiveness µ and evaluate policy levers.

In short: the paper formalizes a plausible AI–ESG–green collaboration feedback loop and shows that organizational responsiveness—how quickly firms push updates across these domains—matters critically for stability. Faster adjustment is not uniformly better; governance and managerial choices must explicitly account for dynamic effects (pacing, damping, coordination) to realize sustainable, stable complementarities.