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 $(1, 1, 1)$ 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).

The ESC Guidelines represent the views of the ESC and were produced after careful consideration of the scientific and medical knowledge and the evidence available at the time of their publication. The ESC is not responsible in the event of any contradiction, discrepancy and/or ambiguity between the ESC Guidelines and any other official recommendations or guidelines issued by the relevant public health authorities, in particular in relation to good use of healthcare or therapeutic strategies. Health professionals are encouraged to take the ESC Guidelines fully into account when exercising their clinical judgment, as well as in the determination and the implementation of preventive, diagnostic or therapeutic medical strategies; however, the ESC Guidelines do not override, in any way whatsoever, the individual responsibility of health professionals to make appropriate and accurate decisions in consideration of each patient's health condition and in consultation with that patient and, where appropriate and/or necessary, the patient's caregiver. Nor do the ESC Guidelines exempt health professionals from taking into full and careful consideration the relevant official updated recommendations or guidelines issued by the competent public health authorities, in order to manage each patient's case in light of the scientifically accepted data pursuant to their respective ethical and professional obligations. It is also the health professional's responsibility to verify the applicable rules and regulations relating to drugs and medical devices at the time of prescription