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Unsteady, Non-Linear, and Chaotic Systems
Published in L.M.B.C. Campos, Non-Linear Differential Equations and Dynamical Systems, 2019
The aim of the qualitative theory of differential equations is to obtain some properties of the solution without explicitly solving the differential equation. The main advantage of this approach is that the qualitative theory applies to much wider classes of differential equations than those that can be solved. The examples of the qualitative theory of differential equations (section 9.1) include: (i) existence theorems stating that a certain class of functions provides at least one solution; (ii) unicity theorems stating initial or boundary conditions that ensure that the solution is unique; and (iii) regularity theorems indicating properties of the solution, such as differentiability or continuity with regard to parameters. The bifurcations are an example of non-regularity and/or non-unicity, since beyond the bifurcation point several solutions may exist and be stable or unstable. The stability or instability of solutions can be established (section 9.2) by the qualitative theory for fairly general classes of differential equations. The singularities of the differential equation or of its solutions may limit the methods of solution that can be applied. Thus the qualitative methods are very useful for preliminary investigation of the properties of fairly wide classes of differential equations; the inherent limitation is the lack of quantitative results that are essential in many scientific and technical applications.
Transcritical bifurcation at infinity in planar piecewise polynomial differential systems with two zones
Published in Dynamical Systems, 2022
Denis de Carvalho Braga, Jaume Llibre, Luis Fernando Mello
In the classical Qualitative Theory of Differential Equations, it is usual to study the global behaviour of the phase portrait of a given planar polynomial differential system by means of the Poincaré compactification [11]. When we apply this construction to a polynomial vector field G on , we obtain a new vector field on through the central projections and its extension to the Poincaré sphere is everywhere analytic and analytically equivalent to G in each hemisphere.