The conditions of existence of robust homoclinic cycles for G-equivariant vector fields in R 4 with G a finite group are investigated. Depending on the action of G, such cycles are either of type A, B or C. We first introduce a notion of minimal admissible group. The existence of robust homoclinic cycles for vector fields which are equivariant by such a group is generic. Then we show that for type A cycles, the number n of equilibria is either even or equal to three. In the case of type B cycles, n can only be equal to two, three or six. Finally, for those of type C, n is either four or eight. Moreover, we provide expressions for the generators of the minimal admissible groups for each of the above mentioned cycles and we show the explicit form of vector fields generating three type A homoclinic cycles with six, eight and 24 equilibria, respectively.

The problem of a classification of robust homoclinic cycles in low-dimensional spaces has been frequently asked in recent years. In this paper, we resume the results in R 3 and R 4 and we solve the problem in R 5 in the case of orientation-preserving group actions.

No abstract

This paper explores the properties of a family of bivariate copulas based on a new approach using the counter-monotonic shock method. The resulting copula covers the full range of negative dependence induced by one parameter. Expressions for the copula and density are derived and many theoretical properties are examined thoroughly, including explicit expressions for prominent measures of dependence, namely Spearman’s rho, Kendall’s tau and Blomqvist’s beta. The convexity properties of this copula are presented, together with explicit expressions of the mixed moments. Estimation of the dependence parameter using the method of moments is considered, then a simulation study is carried out to evaluate the performance of the suggested estimator. Finally, an application of the proposed copula is illustrated by means of a real data set on air quality in New York City.

No abstract