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The rapid spread of COVID-19 worldwide presents a great challenge to epidemic modelers. Model outcomes vary widely depending on the characteristics of a pathogen and the models. Here, we present a logistic model for the epidemic spread and divide the spread of the novel coronavirus into two phases the first phase is a natural exponential growth phase that occurs in the absence of intervention and the second phase is a regulated growth phase that is affected by enforcing social distancing and isolation. We apply the model to a number of pandemic centers. Our results are in good agreement with the data to date and show that social distancing significantly reduces the epidemic spread and flattens the curve. Predictions on the spreading trajectory including the total infections and peak time of new infections for a community of any size are made weeks ahead, providing the vital information and lead time needed to prepare for and mitigate the epidemic. The methodology presented here has immediate and far-reaching applications for ongoing outbreaks or similar future outbreaks of other emergent infectious diseases.Rössler had a brilliant and successful life as a scientist during which he published a benchmark dynamical system by using an electronic circuit interpreting chemical reactions. This is our contribution to honor his splendid erudite career. It is a hot topic to regulate a network behavior using the pinning control with respect to a small set of nodes in the network. Besides pinning to a small number of nodes, small perturbation to the node dynamics is also demanded. In this paper, the pinning synchronization of a coupled Rössler-network with time delay using univariate impulse control is investigated. Using the Lyapunov theory, a theorem is proved for the asymptotic stability of synchronization in the network. Simulation is given to validate the correctness of the analysis and the effectiveness of the proposed univariate impulse pinning controller.Cross-recurrence quantification analysis (CRQA), based on the cross-recurrence plot (CRP), is an effective method to characterize and quantify the nonlinear interrelationships between a pair of nonlinear time series. It allows the flexibility of reconstructing signals in the phase space and to identify different types of patterns at arbitrary positions between trajectories. These advantages make CRQA attractive for time series data mining tasks, which have been of recent interest in the literature. However, little has been done to exploit CRQA for pattern matching of multidimensional, especially spatiotemporal, physiological signals. In this paper, we present a novel methodology in which CRQA statistics serve as measures of dissimilarity between pairs of signals and are subsequently used to uncover clusters within the data. This methodology is evaluated on a real dataset consisting of 3D spatiotemporal vectorcardiogram (VCG) signals from healthy and diseased patients. Experimental results show that Lmax, the length of the longest diagonal line in the CRP, yields the best-performing clustering that almost exactly matches the ground truth diagnoses of patients. Results also show that our proposed measure, Rτmax, which characterizes the maximum similarity between signals over all pairwise time-delayed alignments, outperforms all other tested CRQA measures (in terms of matching the ground truth) when the VCG signals are rescaled to reduce the effects of signal amplitude.The theory of motion of edges of dispersive shock waves generated after wave breaking of simple waves is developed. It is shown that this motion obeys Hamiltonian mechanics complemented by a Hopf-like equation for evolution of the background flow, which interacts with the edge wave packets or the edge solitons. A conjecture about the existence of a certain symmetry between equations for the small-amplitude and soliton edges is formulated. In the case of localized simple-wave pulses propagating through a quiescent medium, this theory provided a new approach to derivation of an asymptotic formula for the number of solitons eventually produced from such a pulse.The simplest ring oscillator is made from three strongly nonlinear elements repressing each other unidirectionally, resulting in the emergence of a limit cycle. A popular implementation of this scheme uses repressor genes in bacteria, creating the synthetic genetic oscillator known as the Repressilator. We consider the main collective modes produced when two identical Repressilators are mean-field-coupled via the quorum-sensing mechanism. In-phase and anti-phase oscillations of the coupled oscillators emerge from two Andronov-Hopf bifurcations of the homogeneous steady state. Using the rate of the repressor's production and the value of coupling strength as the bifurcation parameters, we performed one-parameter continuations of limit cycles and two-parameter continuations of their bifurcations to show how bifurcations of the in-phase and anti-phase oscillations influence the dynamical behaviors for this system. Pitchfork bifurcation of the unstable in-phase cycle leads to the creation of novel inhomogeneous limit cycles with very different amplitudes, in contrast to the well-known asymmetrical limit cycles arising from oscillation death. The Neimark-Sacker bifurcation of the anti-phase cycle determines the border of an island in two-parameter space containing almost all the interesting regimes including the set of resonant limit cycles, the area with stable inhomogeneous cycle, and very large areas with chaotic regimes resulting from torus destruction and period doubling of resonant cycles and inhomogeneous cycles. We discuss the structure of the chaos skeleton to show the role of inhomogeneous cycles in its formation. Many regions of multistability and transitions between regimes are presented. These results provide new insights into the coupling-dependent mechanisms of multistability and collective regime symmetry breaking in populations of identical multidimensional oscillators.In this article, we construct a kind of three-dimensional piecewise linear (PWL) system with three switching manifolds and obtain four theorems with regard to the existence of a homoclinic orbit and a heteroclinic cycle in this class of PWL system. The first theorem studies the existence of a heteroclinic cycle connecting two saddle-foci. Selleck QX77 The existence of a homoclinic orbit connecting one saddle-focus is investigated in the second theorem, and the third theorem examines the existence of a homoclinic orbit connecting another saddle-focus. The last one proves the coexistence of the heteroclinic cycle and two homoclinic orbits for the same parameters. Numerical simulations are given as examples and the results are consistent with the predictions of theorems.
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