Summary
Some results of Poincaré and Dulac concerning non-isolated periodic orbits and singular cycles in the plane are here extended to certain classes of autonomous analytic ordinary differential equations of higher dimension. The equations in these classes are then shown to have only isolated periodic orbits provided that all their critical points satisfy a simple condition. A further condition at infinity can ensure that the equation has only finitely many periodic orbits.
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Smith, R.A. Certain differential equations have only isolated periodic orbits. Annali di Matematica pura ed applicata 137, 217–244 (1984). https://doi.org/10.1007/BF01789396
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DOI: https://doi.org/10.1007/BF01789396