Limit cycles in Kukles systems of arbitrary degree with invariant ellipse
This paper is concerned with planar polynomial differential systems of Kukles type, that is, systems of the form \[ \dot{x} = -y, \quad \dot{y} = Q(x,y), \] where \(Q(x,y)\) is a real polynomial, which is assumed to be not constant and not divisible by \(y\). The degree of the system is defined as the degree of the polynomial \(Q(x,y)\) and will be denoted by \(n\). This work deals with the existence, uniqueness and location of limit cycles in a Kukles system with an invariant ellipse. The existence of invariant algebraic curves and its distribution on the real plane implies properties on the phase portrait of the system and, in particular, on the number of limit cycles. A result on the relation between invariant algebraic curves and existence limit cycles is, for instance, that a quadratic system with an invariant algebraic curve of degree \(2\) has at most one limit cycle, see \textit{C. Christopher} [Proc. Roy. Soc. Edinburgh Sect. A 112, 113--134 (1989; Zbl 0677.34034)]. In particular, C. Christopher shows that if a quadratic system has an invariant ellipse and a limit cycle, this limit cycle is the ellipse, it is unique and it is hyperbolic. Other related results appear in [Nonlinear Anal. 59, 673--693 (2004; Zbl 1076.34029)], see also the references therein, where an example of a cubic Kukles system with an invariant hyperbola and two small-amplitude limit cycles is given. The first main result of this work is a normal form for the Kukles systems of degree \(n\) with an invariant ellipse. Any system with these characteristics is shown to be affine-equivalent to the system: \[ \dot{x} = -y, \quad \dot{y} = -ac + bx + q_{n-2}(x,y) f(x,y), \tag{1} \] where \(f(x,y) = a-2acx+bx^2+y^2\), \(a\), \(b\) and \(c\) are real numbers such that \(b>0\), \(a<0\) and \(q_{n-2}(x,y)\) is a real polynomial of degree \(n-2\) such that \(q_{n-2}(0,0) = c\). We observe that the origin \((0,0)\) is a singular point of system (1), the algebraic curve \(f(x,y)=0\) is the invariant ellipse of system (1) and that it surrounds the origin of coordinates. Once the normal form (1) for the target systems is given, an analysis of several properties of them is provided. It is shown, for instance, that the ellipse \(f(x,y)=0\) contains no singular points. Moreover, the values of \(a\), \(b\), \(c\) and the coefficients of \(q_{n-2}(x,y)\) for which the origin is a singular point of focus type are described. Distinct particular forms of the polynomial \(q_{n-2}(x,y)\) are studied in relation with the existence of limit cycles and, among them, it is shown the existence of Kukles systems of arbitrarily high degree with an invariant ellipse as unique limit cycle. The last part of the paper contains the study of the number of limit cycles, of a system of the form (1), which bifurcate from the period annulus of a Hamiltonian system. More concretely, the systems taken into account are of the form: \[ \dot{x} = y = \frac{\partial H}{\partial y}, \quad \dot{y} = -b x + Q(x,y,\varepsilon) = - \frac{\partial H}{\partial x} + Q(x,y,\varepsilon), \tag{2} \] where \(\varepsilon>0\) is the perturbation parameter, the Hamiltonian function is \(H(x,y) = (bx^2+y^2)/2\) and \(Q(x,y,\varepsilon) = \varepsilon g_1(x,y) + \varepsilon^2 g_2(x,y)\), with \[ \begin{aligned} g_1(x,y) & = c(bx^2+y^2)+\widetilde{q}_{n-2}(x,y)(a+bx^2+y^2), \\ g_2(x,y) & = 2acx \left[c+\widetilde{q}_{n-2}(x,y)\right] \end{aligned} \] and \(\widetilde{q}_{n-2}(x,y)\) is a real polynomial without constant term. The main result of this part is the following: Let \(\widetilde{q}_{n-2}(x,y) = \sum_{i+j=1}^{n-2} q_{ij} x^{i} y^{j}\) and assume \(\sum_{i+j=0}^{[(n-3)/2]} q_{2i,2j+1}^2 \neq 0 ,\) then: (i) For \(k \geq 3\), if either \(n=2k\) or \(n=2k-1\), system (2) has at most \(k-1\) limit cycles, one of them given by an invariant ellipse. (ii) If \(n=5\) or \(n=6\), \(3bq_{03}+q_{21} \neq 0\) and \(a/2 \neq 2bq_{01} /(3bq_{03}+q_{21}) < 0\), system (2) has exactly two hyperbolic limit cycles. Both limit cycles collapse in the algebraic one when \(a/2 = 2bq_{01} /(3bq_{03}+q_{21}) .\) Numerical examples of statement (ii) and five limit cycles for \(n=11\) are also provided.
scientific article; zbMATH DE number 5161903