A proof of Perko's conjectures for the Bogdanov-Takens system
Armengol Gasull, Hector Giacomini, Set Perez-Gonzalez, Joan Torregrosa
TL;DR
This paper proves Perko's conjectures regarding the saddle-loop bifurcation curve in the Bogdanov-Takens system, establishing its properties and providing precise algebraic bounds.
Contribution
It offers the first rigorous proof of Perko's conjectures and derives sharp algebraic bounds for the bifurcation curve in the system.
Findings
Confirmed Perko's conjectures about the bifurcation curve.
Derived sharp algebraic upper and lower bounds.
Established analytic properties of the saddle-loop bifurcation.
Abstract
The Bogdanov-Takens system has at most one limit cycle and, in the parameter space, it exists between a Hopf and a saddle-loop bifurcation curves. The aim of this paper is to prove the Perko's conjectures about some analytic properties of the saddle-loop bifurcation curve. Moreover, we provide sharp piecewise algebraic upper and lower bounds for this curve.
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Mathematical Dynamics and Fractals · Quantum chaos and dynamical systems