Globally Asymptotically Stable Equilibrium Points in Kukles Systems
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Globally Asymptotically Stable Equilibrium Points in Kukles Systems Fabio Scalco Dias1
· Luis Fernando Mello1
Received: 29 May 2020 / Accepted: 16 October 2020 © Springer Nature Switzerland AG 2020
Abstract The problem of determining the basin of attraction of equilibrium points is of great importance for applications of stability theory. In this article, we address the global asymptotic stability problem of an equilibrium point of an ordinary differential equation on the plane. More precisely, we study equilibrium points of Kukles systems from the global asymptotic stability point of view. First of all, we classify the Kukles systems satisfying the assumptions: the origin is the unique equilibrium point which is locally asymptotically stable, and the divergence is negative except possibly at the origin. Then, for each of such Kukles system, we prove that the origin is globally asymptotically stable. Poincaré compactification is used to study the systems on the complements of compact sets. Keywords Global asymptotic stability · Kukles system · Poincaré compactification Mathematics Subject Classification 34D23 · 34D05 · 34A26
1 Introduction and Statement of the Main Result Consider an ordinary differential equation on the plane x =
B
dx = P(x, y), dt
y =
dy = Q(x, y), dt
Fabio Scalco Dias [email protected] Luis Fernando Mello [email protected]
1
Instituto de Matemática e Computação, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, CEP 37.500–903, Itajubá, MG, Brazil 0123456789().: V,-vol
(1)
94
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F.S. Dias, L.F. Mello
where t is the independent variable and the vector field F : R2 −→ R2 , F(x, y) = (P(x, y), Q(x, y)) is of class C 1 . As usual, we identify the ordinary differential equation or system (1) with the vector field F. Assume that this differential equation has a locally asymptotically stable equilibrium point p0 ∈ R2 which we take to be the origin. So, it is well defined the basin of attraction of p0 , B( p0 ), as the set of initial conditions whose solutions tend to p0 . An important problem in the Qualitative Theory of Differential Equations is the following: to give checkable conditions for the global asymptotic stability of p0 = (0, 0), that is, B(0, 0) = R2 . When this is the case, we say that (1) is globally asymptotically stable (GAS). Related to the above problem, we can cite the challenging Markus–Yamabe conjecture [15] (solved at the beginning of 1990s independently by Feßler [6], Glutsyuk [10] and Gutiérrez [11]): if for all (x, y) ∈ R2 the eigenvalues of D F(x, y) have negative real parts, then the origin is globally asymptotically stable. The condition on the eigenvalues of D F(x, y), equivalent to tr D F(x, y) < 0 and det D F(x, y) > 0, ∀(x, y) ∈ R2 , can be very difficult to verify and, of course, is not necessary for GAS. In fact, for F(x, y) = (−x + x y, −y), the eigenvalues of D F(x, y) are −1 and −1 + y, for all (x, y) ∈ R2 , but F is GAS, since L(x, y) = ln(1 + x 2 ) + y 2 is a strict Lyapunov function [1]. However, in general, there is
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