The Method of Holomorphic Foliations in Planar Periodic Systems: The Case of Riccati Equations

Abstract

We develop a new method of investigation of periodic solutions of planar holomorphic polynomial differential equations with trigonometric coefficients. This method relies on analysis of geometry of leaves of certain holomorphic foliations in CP 1 _CP 1 . We concentrate our attention on the case of Riccati equations. We determine the dynamics of the non-autonomous equations, using some local invariants of holomorphic vector fields defining the foliations. In particular, new examples of Riccati equations without periodic solutions are presented.

Periodic solutions of quaternionic-valued ordinary differential equations

“…Very recently, Zoladek [43,44] has used the method of holomorphic foliations to construct examples of equations of the form…”

Positive oriented periodic solutions of the first-order complex ODE with polynomial nonlinear part

“…In last Section 6 we give a few applications of our main theorem to the complex Riccati equation and make a comparison with other results in this direction (Lloyd [14], Hassan [12], Campos and Ortega [10], Miklaszewski [20], andŻoładek [22]). We also study the positive oriented periodic solutions on the base of the Nielsen fixed-point theory, see [1,2,7,8,11] and [3][4][5].…”

“…Let us consider (6.14) for n = 2 (Riccati equation) (6.21) where the coefficient 22) belongs to the algebra C 0 + (T) for given period T > 0. Assume that real R,r 0 > 0.…”

Planar nonautonomous polynomial equations: The Riccati equation

“…The complex ones were considered in [7,8]. The problem of nonexistence of periodic solutions was investigated in [9][10][11][12][13]. There are some papers (e.g.…”

“…(1) which have no periodic solutions do not fulfill this condition (cf. [2,[9][10][11][12][13]). Moreover, the critical line condition implies in the case of coefficients or period being big enough the existence of two periodic solutions (cf.…”