Advanced Differential Equations and Dynamical Systems

Dynamical systems described by differential equations can undergo sudden qualitative changes in behavior — called bifurcations — when parameters cross critical thresholds, and understanding exactly when and how these transitions occur is a central problem in mathematics. Research in this area concentrates on planar systems, where trajectories evolve in two dimensions, with particular attention to limit cycles: isolated closed orbits that nearby trajectories either spiral toward or away from. A long-standing open question, part of Hilbert's sixteenth problem, asks how many limit cycles a polynomial vector field of a given degree can possess, and modern work extends this to piecewise linear and discontinuous systems, where abrupt switches in the governing equations introduce new geometric complexity. Active directions include characterizing bifurcations near nilpotent singularities — points where the linearization provides almost no useful information — and developing tools such as Abelian integrals and Darboux integrability to detect and count limit cycles with precision.

Works

54,521

Total citations

446,346

Keywords

BifurcationsPlanar SystemsPiecewise LinearLimit CyclesPolynomial Vector FieldsHopf Bifurcation