Linearization: Difference between revisions

Revision as of 13:20, 1 June 2023

Revision as of 02:39, 9 June 2023

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===Stability analysis===

In [[stability theory|stability]] analysis of [[Autonomous system (mathematics)|autonomous systems]], one can use the [[eigenvalue]]s of the [[Jacobian matrix and determinant|Jacobian matrix]] evaluated at a [[hyperbolic equilibrium point]] to determine the nature of that equilibrium. This is the content of the [[linearization theorem]]. For time-varying systems, the linearization requires additional justification.<ref>{{cite journal |first=G. A. |last=Leonov |first2=N. V. |last2=Kuznetsov |title=Time-Varying Linearization and the Perron effects |journal=[[International Journal of Bifurcation and Chaos]] |volume=17 |issue=4 |year=2007 |pages=1079–1107 |doi=10.1142/S0218127407017732 |bibcode=2007IJBC...17.1079L }}</ref> In the context of [[network dynamics]] the system's fixed-point behavior is captured by its Jacobian matrix, as derived from the [[dynamic Jacobian ensemble]].<ref>{{Cite journal|last=C. Meena, C. Hens, S. Acharyya, S. Haber, S. Boccaletti and B. Barzel|date=2023|title=Emergent stability in complex network dynamics|journal=Nature Physics|volume= |issue= |pages= |doi=10.1038/s41567-023-02020-8}}</ref> This family of Jacobians links the <math>i,j</math> terms of the Jacobian to the degrees of nodes <math>i</math> and <math>j</math> in the network.

===Microeconomics===