6 research outputs found
Heteroclinic cycles and chaos in a system of four identical phase oscillators with global biharmonic coupling
We study a system of four identical globally coupled phase oscillators with
biharmonic coupling function. Its dimension and the type of coupling make it
the minimal system of Kuramoto-type (both in the sense of the phase space's
dimension and the number of harmonics) that supports chaotic dynamics. However,
to the best of our knowledge, there is still no numerical evidence for the
existence of chaos in this system. The dynamics of such systems is tightly
connected with the action of the symmetry group on its phase space. The
presence of symmetries might lead to an emergence of chaos due to scenarios
involving specific heteroclinic cycles. We suggest an approach for searching
such heteroclinic cycles and showcase first examples of chaos in this system
found by using this approach.Comment: 18 pages, 8 figure
On the origin of chaotic attractors with two zero Lyapunov exponents in a system of five biharmonically coupled phase oscillators
We study chaotic dynamics in a system of four differential equations
describing the dynamics of five identical globally coupled phase oscillators
with biharmonic coupling. We show that this system exhibits strange spiral
attractors (Shilnikov attractors) with two zero (indistinguishable from zero in
numerics) Lyapunov exponents in a wide region of the parameter space. We
explain this phenomenon by means of bifurcation analysis of the
three-dimensional Poincar\'e map for the system under consideration. We show
that the chaotic dynamics develop here near a codimension three bifurcation,
when a periodic orbit (fixed point in the Poincar\'e map) has the triplet of multipliers. As it is known, the asymptotic flow normal form for this
bifurcation coincides with the three-dimensional
Arneodo-Coullet-Spiegel-Tresser (ACST) system in which spiral attractors exist.
Based on this, we conclude that the additional near-zero Lyapunov exponent for
orbits in the observed attractors appear due to the fact that the corresponding
three-dimensional Poincar\'e map is close to the time-shift map of the
three-dimensional ACST-system.Comment: 18 pages, 7 figure
2017 ESC Guidelines for the management of acute myocardial infarction in patients presenting with ST-segment elevation: The Task Force for the management of acute myocardial infarction in patients presenting with ST-segment elevation of the European Society of Cardiology (ESC).
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