The dynamics of three three-dimensional repressilators globally coupled by a quorum sensing mechanism was numerically studied. This number (three) of coupled repressilators is sufficient to obtain such a set of self-consistent oscillation frequencies of signal molecules in the mean field that results in the appearance of self-organized quasiperiodicity and its complex evolution over wide areas of model parameters. Numerically analyzing the invariant curves as a function of coupling strength, we observed torus doubling, three torus arising via quasiperiodic Hopf bifurcation, the emergence of resonant cycles, and secondary Neimark-Sacker bifurcation. A gradual increase in the oscillation amplitude leads to chaotizations of the tori and to the birth of weak, but multidimensional chaos.We consider large networks of theta neurons on a ring, synaptically coupled with an asymmetric kernel. Such networks support stable "bumps" of activity, which move along the ring if the coupling kernel is asymmetric. We investigate the effects of the kernel asymmetry on the existence, stability, and speed of these moving bumps using continuum equations formally describing infinite networks. Depending on the level of heterogeneity within the network, we find complex sequences of bifurcations as the amount of asymmetry is varied, in strong contrast to the behavior of a classical neural field model.We extend the scope of the dynamical theory of extreme values to include phenomena that do not happen instantaneously but evolve over a finite, albeit unknown at the onset, time interval. We consider complex dynamical systems composed of many individual subsystems linked by a network of interactions. As a specific example of the general theory, a model of a neural network, previously introduced by other authors to describe the electrical activity of the cerebral cortex, is analyzed in detail. On the basis of this analysis, we propose a novel definition of a neuronal cascade, a physiological phenomenon of primary importance. We derive extreme value laws for the statistics of these cascades, both from the point of view of exceedances (that satisfy critical scaling theory in a certain regime) and of block maxima.We study how the dynamics of solitary wave (SW) interactions in integrable systems is different from that in nonintegrable systems in the context of crossing of two identical SWs in the (integrable) Toda and the (non-integrable) Hertz systems. We show that the collision process in the Toda system is perfectly symmetric about the collision point, whereas in the Hertz system, the collision process involves more complex dynamics. The symmetry in the Toda system forbids the formation of secondary SWs (SSWs), while the absence of symmetry in the Hertz system allows the generation of SSWs. We next show why the experimentally observed by-products of SW-SW interactions, the SSWs, must form in the Hertz system. We present quantitative estimations of the amount of energy that transfers from the SW after collision to the SSWs using (i) dynamical simulations, (ii) a phenomenological approach using energy and momentum conservation, and (iii) using an analytical solution introduced earlier to describe the SW in the Hertz system. We show that all three approaches lead to reliable estimations of the energy in the SSWs.Two-dimensional arrays of coupled waveguides or coupled microcavities allow us to confine and manipulate light. Based on a paradigmatic envelope equation, we show that these devices, subject to a coherent optical injection, support coexistence between a coherent and incoherent emission. In this regime, we show that two-dimensional chimera states can be generated. Depending on initial conditions, the system exhibits a family of two-dimensional chimera states and interaction between them. We characterize these two-dimensional structures by computing their Lyapunov spectrum and Yorke-Kaplan dimension. Finally, we show that two-dimensional chimera states are of spatiotemporal chaotic nature.Evolution and popularity are two keys of the Barabasi-Albert model, which generates a power law distribution of network degrees. Evolving network generation models are important as they offer an explanation of both how and why complex networks (and scale-free networks, in particular) are ubiquitous. We adopt the evolution principle and then propose a very simple and intuitive new model for network growth, which naturally evolves modular networks with multiple communities. The number and size of the communities evolve over time and are primarily subjected to a single free parameter. Surprisingly, under some circumstances, our framework can construct a tree-like network with clear community structures-branches and leaves of a tree. Results also show that new communities will absorb a link resource to weaken the degree growth of hub nodes.  https://www.selleckchem.com/products/shr0302.html  Our models have a common explanation for the community of regular and tree-like networks and also breaks the tyranny of the early adopter; unlike the standard popularity principle, newer nodes and communities will come to dominance over time. Importantly, our model can fit well with the construction of the SARS-Cov-2 haplotype evolutionary network.In this study, we investigate how specific micro-interaction structures (motifs) affect the occurrence of tipping cascades on networks of stylized tipping elements. We compare the properties of cascades in Erdős-Rényi networks and an exemplary moisture recycling network of the Amazon rainforest. Within these networks, decisive small-scale motifs are the feed forward loop, the secondary feed forward loop, the zero loop, and the neighboring loop. Of all motifs, the feed forward loop motif stands out in tipping cascades since it decreases the critical coupling strength necessary to initiate a cascade more than the other motifs. We find that for this motif, the reduction of critical coupling strength is 11% less than the critical coupling of a pair of tipping elements. For highly connected networks, our analysis reveals that coupled feed forward loops coincide with a strong 90% decrease in the critical coupling strength. For the highly clustered moisture recycling network in the Amazon, we observe regions of a very high motif occurrence for each of the four investigated motifs, suggesting that these regions are more vulnerable.