Whether the synchrony of intrinsic alpha rhythms in the occipital cortex can be entrained by transcranial magnetic stimulation (TMS) is an open question. We also apply the Kaplan-Yorke formula to study the fractal structure of the extensively chaotic attractors.Parieto-occipital alpha rhythms (8–12 Hz) underlie cortical excitability and influence visual performance. This behavior is used to reveal the mechanism of the explosive transition. Furthermore, examining the clumped state, as the system approaches the explosive transition to extensive chaos, we find that the oscillator population distribution between the clumps continually evolves so that the clumped state is always marginally stable. We observe explosive (i.e., discontinuous) transitions between the clumped states (which correspond to low dimensional dynamics) and the extensively chaotic states. An important focus of this paper is the transition between clumped states and extensive chaos as the system is subjected to slow adiabatic parameter change. We argue that this second type of behavior is $^$ in the sense that the chaotic attractor in the full phase space of the system has a fractal dimension that scales linearly with $N$ and that the number of positive Lyapunov exponents of the attractor also scales with linearly $N$. Previously, it has been observed that this type of system can exhibit a variety of different dynamical behaviors including clumped states in which each oscillator is in one of a small number of groups for which all oscillators in each group have the same state which is different from group to group, as well as situations in which all oscillators have different states and the macroscopic dynamics of the mean field is chaotic. In this paper, we study dynamical systems in which a large number $N$ of identical Landau-Stuart oscillators are globally coupled via a mean-field. Our consideration is related to the power-law long-range interaction that makes it possible to find new regimes and to establish a link of the corresponding equations of motion to the equations with fractional derivatives. The long-range interaction between oscillators leads to a new qualitative dynamics and thermodynamics. A chain of nonlinear interacting oscillators is a model of the widespread investigation of different physical phenomena such as synchronized behavior of the system, bi-furcations to different regimes, spatiotemporal turbulent or chaotic dynamics, different instabilities, appearance of defects, and many others. We study different spatiotemporal patterns of the dynamics depending on and show transitions from synchronization of the motion to broad-spectrum oscillations and to chaos. Such a system has a new parameter that is responsible for the complexity of the medium and that strongly influences possible regimes of the dynamics, especially near = 2 and =1. In the continuous limit, the system's dynamics is described by a fractional generalization of the Ginzburg-Landau equation with complex coefficients. We consider a chain of nonlinear oscillators with long-range interaction of the type 1 / l 1+, where l is a distance between oscillators and 0 2.
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