The Completeness of Exponential Stability and Patient Recovery Dynamics in Mathematical Rehabilitation

Vollständiger Abstract

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This paper investigates the completeness of exponential stability analysis within a nonlinear mathematical rehabilitation framework, emphasizing its implications for patient recovery dynamics. Extending earlier studies that relied primarily on first-order linearization, we develop a fully articulated Taylor series expansion around the recovery equilibrium ( r* ), incorporating higher-order derivatives to capture subtle nonlinear effects. By presenting both operator-level and index-notation expansions, the study enhances analytical transparency and supports computational implementation. The results demonstrate that nonlinear contributions can meaningfully alter transient behavior, basin geometry, and robustness of recovery, underscoring the necessity of complete Taylor representations in rehabilitation modelling. This framework provides a rigorous foundation for future patient-specific model refinement, data-driven parameter estimation, and the design of optimized therapeutic strategies.