Dynamical Roles of Jacobian Feedback Loops and Qualitative Modeling —  2014 (Englisch)

New Frontiers of Multidisciplinary Research in STEAM-H (Science, Technology, Engineering, Agriculture, Mathematics, and Health)

Abstract

The chapter presents a mathematical methodology applicable to qualitative modeling of systems using the feedback loops encoded in the Jacobian matrix and described by the products of the Jacobian entries under cyclic permutations of the indices. The relation between these feedback loops and the Jacobian spectrum defines their dynamical properties. We determine the conditions of nondegeneracy and appearance of multiple equilibria in terms of feedback loops as well as the conditions of loop stability to induce the stability analysis of the systems. In particular we emphasize the applications to qualitative modeling in biological and biochemical sciences, economics. We present a complete loop analysis of the celebrated Lorenz and Rossler systems predicting their global dynamics.The methodology is proved efficient to assert the possibility of multistationarity, periodicity, self-sustained oscillations, and chaos using strictly the qualitative relations and assumptions of the systems, to achieve primarily qualitative understanding rather than quantitative numerical prediction. We also show the Jacobian loops technique is easy to implement and could quickly demarcate both parameter and phase spaces into exciting regions (limit cycle, multiple equilibria, chaotic behavior), non-exciting ones (single stable fixed points), hard-instance regions (ergodic behavior). Therefore as such the technique could be useful in surveying dynamical responses of models simulating physico-chemical, biological, biochemical, economical systems and game theory.

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