Linear and Affine Dynamics in the Plane

Bjarne S. Jensen ()
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Bjarne S. Jensen: University of Southern Denmark (SDU)

Chapter Chapter 29 in The Elements and Dynamic Systems of Economic Growth and Trade Models, 2025, pp 1059-1113 from Springer

Abstract: Abstract This chapter applies the results of the general theory of homogeneous dynamics, Chap. 28 , to the special case, where m = 1 $$m=1$$ and the governing functions are linear. The linear system is of course well known, but our solution technique, based on calculus instead of linear algebra, is novel and helpful in obtaining relevant classification of solutions and the phase portraits. Rather than the traditional dichotomy, stable/unstable, we illustrate the use of the kinematic stability concepts for the ratio and coordinate solutions in global and generic versions. Bifurcation and structural stability are related to the regions in the parameter space, as are the kinematic stability properties. The solutions are obtained without coordinate transformations, and the main types of solutions for linear dynamics are classified directly with the parameters ( a , b , c , d $$a,b,c,d$$ ), without the calculation of the traditional eigenvalues. The parametric extension to affine dynamics supplements the linear dynamics.

Keywords: Linear system of homogeneous dynamics; Director function; Classification of Calculus solutions; Initial value problems; General solution; Hyperconic phase portraits; Kinematic stability concepts; Bifurcation and structural stability (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-52493-6_29

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DOI: 10.1007/978-3-031-52493-6_29

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