Advanced Differential Equations and Dynamical Systems

Dynamical systems theory investigates how the qualitative behavior of differential equations changes as parameters vary, with bifurcation theory providing precise tools for understanding the moments when equilibria split, merge, or give birth to oscillating solutions called limit cycles. In the planar setting — where trajectories live in two dimensions — researchers work with polynomial and piecewise linear vector fields to classify the number and arrangement of these cycles, a problem whose full resolution has been open since Hilbert posed it in 1900. Discontinuous and piecewise smooth systems have drawn growing attention because they model real switching phenomena in mechanics and biology, while techniques such as Abelian integrals and Darboux integrability offer algebraic handles on what are otherwise analytically intractable problems. Central open questions include sharp bounds on limit cycle counts near degenerate singularities like nilpotent equilibria, and a clearer picture of how Hopf bifurcations unfold when the smoothness assumptions classical theory relies on are dropped.

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53,852

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Keywords

BifurcationsPlanar SystemsPiecewise LinearLimit CyclesPolynomial Vector FieldsHopf Bifurcation