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MSC2010 34C23, 34D45, 65P20

Spiral chaos in Lotka-Volterra like models

Yu. V. Bakhanova1, A. O. Kazakov2, A. G. Korotkov3

AnnotationInvestigations of spiral chaos in generalized Lotka-Volterra systems and Rosenzweig-MacArthur systems that describe the interaction of three species are made in this work. It is shown that in systems under study the spiral chaos appears in agreement with Shilnikov's scenario, that is when changing a parameter in system a stable limit cycle and a saddle-focus born from stable equilibrium. Then the unstable invariant manifold of saddle-focus winds on the stable limit cycle and forms a whirlpool. For some parameter's value the unstable invariant manifold touches one-dimensional stable invariant manifold and forms homoclinic trajectory to saddle-focus. If in this case the limit cycle loses stability (for example, as result of sequence of period doubling bifurcations) and saddle value of saddle-focus is negative then strange attractor appears on base of homoclinic trajectory.
Keywordsspiral chaos, Lotka-Volterra like systems, strange attractor

1Yulia V. Bakhanova, student, IITMM, UNN (23 Prospekt Gagarina (Gagarin Avenue) BLDG 2, 2nd floor, 603950 Nizhni Novgorod, Russia); ORCID: 0000-0002-4067-1226, jul95-8@mail.ru

2Alexey O. Kazakov, senior researcher, Laboratory of topological methods in dynamics, National Research University Higher School of Economics (25/12 Bolshaja Pecherskaja Str., Nizhni Novgorod 603155, Russia); Lecturer, IITMM, UNN (23 Prospekt Gagarina (Gagarin Avenue) BLDG 2, 2nd floor, 603950 Nizhni Novgorod, Russia); PhD, ORCID: 0000-0003-0002-6553, kazakovdz@yandex.ru

3Alexander G. Korotkov, postgraduate, IITMM, UNN (23 Prospekt Gagarina (Gagarin Avenue) BLDG 2, 2nd floor, 603950 Nizhni Novgorod, Russia); ORCID: 0000-0003-0002-6553, koral81@bk.ru

Citation: Yu. V. Bakhanova, A. O. Kazakov, A. G. Korotkov, "[Spiral chaos in Lotka-Volterra like models]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,19:2 (2017) 13–24 (In Russian)

DOI 10.15507/2079-6900.19.201701.013-024