Application of the Matcont Package for Constructing Bifurcation Diagrams of Homoclinic Loops of Equilibrium States

Student: Burdygina Tatiana

Faculty: Faculty of Informatics, Mathematics, and Computer Science (HSE Nizhny Novgorod)

Educational Programme: Mathematics (Bachelor)

Final Grade: 10

Year of Graduation: 2021

For systems showing spiral Shilnikov attractors, the structure of bifurcation curves, which correspond to the appearance of a homoclinic loop of the saddle-focus equilibrium state, is studied. Using the example of the classical Rössler and Arneodo-Coullet-Tresser systems, it is shown that the corresponding bifurcation curve has the shape of "saw teeth", and between each pair of teeth there exist bifurcation curves which correspond to secondary homoclinic saddle-focus loops. Using the example of the Arneodo-Coullet-Tresser system, it is shown that the "saw teeth" accumulate to a point of codimension two, at which the corresponding equilibrium state has zero and a pair of purely imaginary eigenvalues. The corresponding two-parameter bifurcation analysis was performed numerically using the MatCont package.

Full text (added May 28, 2021)

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