Abstract
Relative stability can be defined as the stability of equilibriumrays. Hence standard method of studying local stability by means of the spectrum of the Jacobian matrix at an equilibriumpoint cannot be used. This problem can often be circumvented by deriving from the original one a new system, having as state variables certainratios between the original variables, so that the equilibrium set will become a point and the standard method will apply. However, the Jacobian matrix at that point will be formed on the basis of a linear combination of the rows of the Jacobian matrix of the original system at a point of its equilibrium set and will normally be much more difficult to analyze. Here a method for studying local relative stability on the basis of the latter matrix is provided.
Riassunto
La stabilità relativa può esser definita come la stabilità diraggi di equilibrio. Quindi il metodo consueto consistente nello studiare la stabilità locale per mezzo dello spettro della matrice giacobiana in unpunto di equilibrio non può esser usato. Tuttavia questo problema spesso può esser superato qualora dal sistema originale possa esser derivato un altro sistema, avente come variabili di sato certi rapporti tra le variabili originarie, cosicché l'insieme di equilibrio del nuovo sistema si riduca ad un punto. In tal caso la matrice giacobiana in tale punto sarà formata sulla base di combinazioni lineari delle righe della matrice giacobiana del sistema originario in un punto del suo insieme di equilibrio, cosicché lo studio qualitativo della prima matrice sarà di regola molto più difficile di quello della seconda. Qui si propone un metodo per studiare la stabilità relativa locale sulla base della seconda matrice.
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I wish to thank M. Bianchi and E. Venini for their comments on previous drafts of this paper.
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Boggio, L. On local relative stability. With special reference to economic applications. Rivista di Matematica per le Scienze Economiche e Sociali 16, 3–15 (1993). https://doi.org/10.1007/BF02086759
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DOI: https://doi.org/10.1007/BF02086759