Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain.
Departamento de Matemática, Instituto de Matemática, Estatística e Computação Científica (IMECC), Universidade Estadual de Campinas (UNICAMP), Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz, 13083-859 Campinas, SP, Brazil.
Chaos. 2023 Jul 1;33(7). doi: 10.1063/5.0138309.
The generalized Chazy differential equation corresponds to the following two-parameter family of differential equations x⃛+|x|qx¨+k|x|qxx˙2=0, which has its regularity varying with q, a positive integer. Indeed, for q=1, it is discontinuous on the straight line x=0, whereas for q a positive even integer it is polynomial, and for q>1 a positive odd integer it is continuous but not differentiable on the straight line x=0. In 1999, the existence of periodic solutions in the generalized Chazy differential equation was numerically observed for q=2 and k=3. In this paper, we prove analytically the existence of such periodic solutions. Our strategy allows to establish sufficient conditions ensuring that the generalized Chazy differential equation, for k=q+1 and any positive integer q, has actually an invariant topological cylinder foliated by periodic solutions in the (x,x˙,x¨)-space. In order to set forth the bases of our approach, we start by considering q=1,2,3, which are representatives of the different classes of regularity. For an arbitrary positive integer q, an algorithm is provided for checking the sufficient conditions for the existence of such an invariant cylinder, which we conjecture that always exists. The algorithm was successfully applied up to q=100.
广义 Chazy 微分方程对应于以下两个参数家族的微分方程：$x'+\vert x\vert qx''+k\vert x\vert qxx''=0$，其中$q$为正整数，正则性随$q$变化。事实上，当$q=1$时，它在直线$x=0$上是不连续的，而当$q$为正偶数时，它是多项式的，当$q\gt1$为正奇数时，它在直线$x=0$上是连续但不可微的。1999 年，人们在广义 Chazy 微分方程中数值观察到了$q=2$和$k=3$时的周期解的存在性。在本文中，我们从分析上证明了这种周期解的存在性。我们的策略允许建立充分条件，以确保广义 Chazy 微分方程，对于$k=q+1$和任何正整数$q$，实际上在$(x,x',x'')$空间中具有由周期解组成的不变拓扑圆柱。为了阐述我们方法的基础，我们首先考虑$q=1,2,3$，它们代表不同正则性的类别。对于任意正整数$q$，我们提供了一种算法，用于检查存在这样一个不变圆柱的充分条件，我们推测这些条件总是存在的。该算法成功地应用于$q=100$。
