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In this paper, we present a study on a network-based susceptible-infected-recovered (SIR) epidemic model with a saturated treatment function. It is well known that treatment can have a specific effect on the spread of epidemics, and due to the limited resources of treatment, the number of patients during severe disease outbreaks who need to be treated may exceed the treatment capacity. Consequently, the number of patients who receive treatment will reach a saturation level. Thus, we incorporated a saturated treatment function into the model to characterize such a phenomenon. The dynamics of the present model is discussed in this paper. We first obtained a threshold value R 0 , which determines the stability of a disease-free equilibrium. Furthermore, we investigated the bifurcation behavior at R 0 = 1. More specifically, we derived a condition that determines the direction of bifurcation at R 0 = 1. If the direction is backward, then a stable disease-free equilibrium concurrently exists with a stable endemic equilibrium even though R 0 < 1. Therefore, in this case, R 0 < 1 is not sufficient to eradicate the disease from the population. However, if the direction is forward, we find that for a range of parameters, multiple equilibria could exist to the left and right of R 0 = 1. In this case, the initial infectious invasion must be controlled to a lower level so that the disease dies out or approaches a lower endemic steady state. [ABSTRACT FROM AUTHOR]

This paper investigates feedback pinning control of synchronization behaviors aroused by epidemic spread on complex networks. Based on the quenched mean field theory, epidemic control synchronization models with the inhibition of contact behavior are constructed, combined with the epidemic transmission system and the adaptive dynamical network carrying active controllers. By the properties of convex functions and the Gerschgorin theorem, the epidemic threshold of the model is obtained, and the global stability of disease-free equilibrium is analyzed. For individual's infected situation, when an epidemic disease spreads, two types of feedback control strategies depending on the diseases' information are designed: the first one only adds controllers to infected individuals, and the other adds controllers to both infected and susceptible ones. By using the Lyapunov stability theory, under designed controllers, some criteria that guarantee the epidemic controlled synchronization system achieving behavior synchronization are also derived. Several numerical simulations are performed to show the effectiveness of our theoretical results. As far as we know, this is the first work to address the controlled behavioral synchronization induced by epidemic spread under the pinning feedback mechanism. It is hopeful that we may have deeper insights into the essence between the disease's spread and collective behavior under active control in complex dynamical networks. [ABSTRACT FROM AUTHOR]

To reduce parameter error caused by human factors and ensure the accuracy of the largest Lyapunov exponent (LLE) obtained from chaotic equations, this paper proposes a simple method based on two nearby pseudo-orbits. First, a point is selected from a solution trajectory of chaotic equation by the roundoff error. Second, the selected point is used as an initial condition to solve the same equation to obtain another solution trajectory. Third, the evolution distance of the two solution trajectories is calculated. Finally, the LLE is the slope of the linear region in the curve of the track distance of the natural algorithm. Our method has been successfully applied to simulate five well-known chaotic systems and some non-chaotic systems. The results show that, compared with other traditional methods, the proposed method is efficient, simple, and robust without reconstructing phase space and computing the Jacobian matrix. [ABSTRACT FROM AUTHOR]

This paper considers a recently introduced D -dimensional generalized Kuramoto model for many (N ≫ 1) interacting agents, in which the agent states are D -dimensional unit vectors. It was previously shown that, for even (but not odd) D , similar to the original Kuramoto model (D = 2), there exists a continuous dynamical phase transition from incoherence to coherence of the time asymptotic attracting state (time t → ∞) as the coupling parameter K increases through a critical value which we denote K c (+) > 0. We consider this transition from the point of view of the stability of an incoherent state, where an incoherent state is defined as one for which the N → ∞ distribution function is time-independent and the macroscopic order parameter is zero. In contrast with D = 2 , for even D > 2 , there is an infinity of possible incoherent equilibria, each of which becomes unstable with increasing K at a different point K = K c. Although there are incoherent equilibria for which K c = K c (+) , there are also incoherent equilibria with a range of possible K c values below K c (+) , (K c (+) / 2) ≤ K c < K c (+) . How can the possible instability of incoherent states arising at K = K c < K c (+) be reconciled with the previous finding that, at large time (t → ∞) , the state is always incoherent unless K > K c (+) ? We find, for a given incoherent equilibrium, that, if K is rapidly increased from K < K c to K c < K < K c (+) , due to the instability, a short, macroscopic burst of coherence is observed, in which the coherence initially grows exponentially, but then reaches a maximum, past which it decays back into incoherence. Furthermore, after this decay, we observe that the equilibrium has been reset to a new equilibrium whose K c value exceeds that of the increased K. Thus, this process, which we call "Instability-Mediated Resetting," leads to an increase in the effective K c with continuously increasing K , until the equilibrium has been effectively set to one for which K c ≈ K c (+) . Thus, instability-mediated resetting leads to a unique critical point of the t → ∞ time asymptotic state (K = K c (+) ) in spite of the existence of an infinity of possible pretransition incoherent states. [ABSTRACT FROM AUTHOR]

In this paper, we present detailed analyses of the dynamics of a number of embodied neuromechanical systems of a class that has been shown to efficiently exploit chaos in the development and learning of motor behaviors for bodies of arbitrary morphology. This class of systems has been successfully used in robotics, as well as to model biological systems. At the heart of these systems are neural central pattern generating (CPG) units connected to actuators which return proprioceptive information via an adaptive homeostatic mechanism. Detailed dynamical analyses of example systems, using high resolution largest Lyapunov exponent maps, demonstrate the existence of chaotic regimes within a particular region of parameter space, as well as the striking similarity of the maps for systems of varying size. Thanks to the homeostatic sensory mechanisms, any single CPG "views" the whole of the rest of the system as if it was another CPG in a two coupled system, allowing a scale invariant conceptualization of such embodied neuromechanical systems. The analysis reveals chaos at all levels of the systems; the entire brain-body-environment system exhibits chaotic dynamics which can be exploited to power an exploration of possible motor behaviors. The crucial influence of the adaptive homeostatic mechanisms on the system dynamics is examined in detail, revealing chaotic behavior characterized by mixed mode oscillations (MMOs). An analysis of the mechanism of the MMO concludes that they stems from dynamic Hopf bifurcation, where a number of slow variables act as "moving" bifurcation parameters for the remaining part of the system. [ABSTRACT FROM AUTHOR]

In a network of high-dimensionality, it is not feasible to measure every single node. Thus, an important goal is to define the optimal choice of sensor nodes that provides a reliable state reconstruction of the network system state-space. This is an observability problem. In this paper, we propose a particle filtering (PF) framework as a way to assess observability properties of a dynamical network, where each node is composed of an individual dynamical system. The PF framework is applied to two benchmarks, networks of Kuramoto and Rössler oscillators, to investigate how the interplay between dynamics and topology impacts the network observability. Based on the numerical results, we conjecture that, when the network nodal dynamics are heterogeneous, better observability is conveyed for sets of sensor nodes that share some dynamical affinity to its neighbourhood. Moreover, we also investigate how the choice of an internal measured variable of a multidimensional sensor node affects the PF performance. The PF framework effectiveness as an observability measure is compared with a well-consolidated nonlinear observability metric for a small network case and some chaotic system benchmarks. [ABSTRACT FROM AUTHOR]

In this paper, the interactions of some given chaotic properties between a non-autonomous discrete system (X , f 0 , ∞) and its induced set-valued system (K (X) , f ¯ 0 , ∞) are obtained. It is proved that the specification property, property P, topological mixing, mild mixing, and topological exactness of (X , f 0 , ∞) are equivalent to those of (K (X) , f ¯ 0 , ∞) , respectively. It is shown that Robinson chaos (resp. Kato chaos) between (X , f 0 , ∞) and (K (X) , f ¯ 0 , ∞ ) are equivalent under certain conditions. Furthermore, Li-Yorke chaos and distributional chaos of (X , f 0 , ∞) imply those of (K (X) , f ¯ 0 , ∞) , respectively. Topological equi-conjugacy between two systems is proved to be preserved by their induced set-valued systems. Topological entropy of (X , f 0 , ∞) is guaranteed to be no larger than that of (K (X) , f ¯ 0 , ∞). Two examples are finally provided with computer simulations for illustration. [ABSTRACT FROM AUTHOR]

We study symmetry properties and the possibility of exact integration of Klein–Gordon equations in external electromagnetic fields on 3D de Sitter background dS3. We present an algorithm for constructing the first-order symmetry algebra and describe its structure in terms of Lie algebra extensions. Based on the well-known classification of the subalgebras of the algebra s o (1 , 3) , we classify all electromagnetic fields on dS3 for which the corresponding Klein–Gordon equations admit first-order symmetry algebras. Then, we select the integrable cases, and for each of them, we construct exact solutions using the noncommutative integration method developed by Shapovalov and Shirokov [Theor. Math. Phys. 104, 921–934 (1995)]. We also propose an original algebraic method for constructing the special local coordinates on de Sitter space dS3, in which basis vector fields for subalgebras of the Lie algebra s o (1 , 3) have the simplest form. [ABSTRACT FROM AUTHOR]

Continuing the work of Monnier [Israel J. Math. 129, 189–207 (2002)], we determine the Gerstenhaber algebra structure over the Poisson cohomology groups for a large class of Poisson structures with isolated singularities over the plane. It reveals that there exists a GAGA-type phenomenon. We give an explicit description via generators and relations for the case of simple singularities. [ABSTRACT FROM AUTHOR]

We consider the problem of variable separation for the classical integrable Hamiltonian systems possessing gl(n)-valued Lax matrices satisfying quadratic Poisson brackets of Freidel and Maillet [Phys. Lett. B 262(2-3), 278 (1991)] with spectral-parameter dependent a-b-c-d tensors. We formulate, in terms of the corresponding a-b-c-d tensors, a sufficient condition that guarantees that the separating functions A(u), B(u) of Sklyanin [Commun. Math. Phys. 150(1), 181 (1992)], Scott [e-print arXiv:hep-th 940303 (1994)], and Gekhtman [Commun. Math. Phys. 167, 593 (1995)] produce canonical coordinates. We consider an important subcase of a-b-c-d algebras, namely, the case of classical reflection equation algebras [E. Sklyanin, J. Phys. A: Math. Gen. 21(10), 2357 (1988)], and formulate, in terms of the corresponding r-s-matrices, the analogous sufficient condition that guarantees the canonicity of the constructed coordinates. For the case s = 0, we recover our previous results on the variable separation for quadratic Sklyanin brackets [B. Dubrovin and T. Skrypnyk, J. Math. Phys. 60, 093506 (2019)]. We consider an example of gl(n) ⊗ gl(n)-valued trigonometric a-b-c-d tensors that satisfy the considered condition and find a class of Lax matrices for them for which the obtained set of canonical coordinates is complete. [ABSTRACT FROM AUTHOR]

We defined the notion of the quantum double inclusion [S. De, J. Math. Phys. 60, 071701 (2019)] associated with a finite-index and finite-depth subfactor, which is closely related to that of Ocneanu's asymptotic inclusion, and studied the quantum double inclusion associated with the Kac algebra subfactor RH ⊂ R, where H is a finite-dimensional Kac algebra acting outerly on the hyperfinite II1 factor R and RH denotes the fixed-point subalgebra. In this article, we analyze quantum double inclusions associated with the family of Kac algebra subfactors given by { R H ⊂ R ⋊ H ⋊ H * ⋊ ⋯ ︸ m times : m ≥ 1 }. For each m > 2, we construct a model N m ⊂ M for the quantum double inclusion of R H ⊂ R ⋊ H ⋊ H * ⋊ ⋯ ︸ m − 2 times , where N m = ((⋯ ⋊ H − 2 ⋊ H − 1 ) ⊗ ( H m ⋊ H m + 1 ⋊ ⋯ )) ′ ′ , M = (⋯ ⋊ H − 1 ⋊ H 0 ⋊ H 1 ⋊ ⋯ ) ′ ′ , and for any integer i, the notation Hi stands for H or H* according as i is odd or even. In this article, we give an explicit description of the subfactor planar algebra associated with N m ⊂ M (m > 2) which turns out to be a planar subalgebra of P * (m) ( H m ) (the adjoint of the m-cabling of the planar algebra of Hm). We then show that for each m > 2, the depth of N m ⊂ M is always two. Observing that N m ⊂ M is reducible for all m > 2, we study in great detail the weak Kac algebra structure of the relative commutant ( N m ) ′ ∩ M 2 . [ABSTRACT FROM AUTHOR]

We consider the Kuramoto-Sakaguchi model of identical coupled phase oscillators with a common noisy forcing. While common noise always tends to synchronize the oscillators, a strong repulsive coupling prevents the fully synchronous state and leads to a nontrivial distribution of oscillator phases. In previous numerical simulations, the formation of stable multicluster states has been observed in this regime. However, we argue here that because identical phase oscillators in the Kuramoto-Sakaguchi model form a partially integrable system according to the Watanabe-Strogatz theory, the formation of clusters is impossible. Integrating with various time steps reveals that clustering is a numerical artifact, explained by the existence of higher order Fourier terms in the errors of the employed numerical integration schemes. By monitoring the induced change in certain integrals of motion, we quantify these errors. We support these observations by showing, on the basis of the analysis of the corresponding Fokker-Planck equation, that two-cluster states are non-attractive. On the other hand, in ensembles of general limit cycle oscillators, such as Van der Pol oscillators, due to an anharmonic phase response function as well as additional amplitude dynamics, multiclusters can occur naturally. [ABSTRACT FROM AUTHOR]

In theory, groups of enzyme-loaded particles producing an acid or base may show complex behavior including dynamical quorum sensing, the appearance of synchronized oscillations above a critical number or density of particles. Here, experiments were performed with the enzyme urease loaded into mm-sized agarose beads and placed in a solution of urea, resulting in an increase in pH. This behavior was found to be dependent upon the number of beads present in the array; however, reaction-induced convection occurred and plumes of high pH developed that extended to the walls of the reactor. The convection resulted in the motion of the mm-sized particles and conversion of the solution to high pH. Simulations in a simple model of the beads demonstrated the suppression of dynamical quorum sensing in the presence of flow. [ABSTRACT FROM AUTHOR]

We develop a new algorithm for the estimation of rare event probabilities associated with the steady-state of a Markov stochastic process with continuous state space R d and discrete time steps (i.e., a discrete-time R d -valued Markov chain). The algorithm, which we coin Recurrent Multilevel Splitting (RMS), relies on the Markov chain's underlying recurrent structure, in combination with the Multilevel Splitting method. Extensive simulation experiments are performed, including experiments with a nonlinear stochastic model that has some characteristics of complex climate models. The numerical experiments show that RMS can boost the computational efficiency by several orders of magnitude compared to the Monte Carlo method. [ABSTRACT FROM AUTHOR]

We present a model-free forecast algorithm that dynamically combines multiple forecasts using multivariate time series data. The underlying principle is based on the fact that forecast performance depends on the position in the state space. This property is exploited to weight multiple forecasts via a local loss function. Specifically, additional weights are assigned to appropriate forecasts depending on their positions in a state space reconstructed via delay coordinates. The function evaluates the forecast error discounted by the distance in the space, whereas most existing methods discount the error in relation to time. In addition, forecasts are selected with the function for each time step based on the existing multiview embedding approach; by contrast, the original multiview embedding selects the reconstructions in advance before starting the forecast. The proposed prediction method has the smallest mean squared error among conventional ensemble methods for the Rössler and the Lorenz'96I models. The results of comparison of the proposed method with conventional machine-learning methods using a flood forecast example indicate that the proposed method yields superior accuracy. We also demonstrate that the proposed method might even correctly forecast the maximum water level of rivers without any prior knowledge. [ABSTRACT FROM AUTHOR]

Numerous well-known processes of complex systems such as spreading and cascading are mainly affected by a small number of critical nodes. Identifying influential nodes that lead to broad spreading in complex networks is of great theoretical and practical importance. Since the identification of vital nodes is closely related to propagation dynamics, a novel method DynamicRank that employs the probability model to measure the ranking scores of nodes is suggested. The influence of a node can be denoted by the sum of probability scores of its i order neighboring nodes. This simple yet effective method provides a new idea to understand the identification of vital nodes in propagation dynamics. Experimental studies on both Susceptible-Infected-Recovered and Susceptible-Infected-Susceptible models in real networks demonstrate that it outperforms existing methods such as Coreness, H-index, LocalRank, Betweenness, and Spreading Probability in terms of the Kendall τ coefficient. The linear time complexity enables it to be applied to real large-scale networks with tens of thousands of nodes and edges in a short time. [ABSTRACT FROM AUTHOR]

Many systems are dynamic and time-varying in the real world. Discovering the vital nodes in temporal networks is more challenging than that in static networks. In this study, we proposed a temporal information gathering (TIG) process for temporal networks. The TIG-process, as a node's importance metric, can be used to do the node ranking. As a framework, the TIG-process can be applied to explore the impact of temporal information on the significance of the nodes. The key point of the TIG-process is that nodes' importance relies on the importance of its neighborhood. There are four variables: temporal information gathering depth n , temporal distance matrix D , initial information c , and weighting function f. We observed that the TIG-process can degenerate to classic metrics by a proper combination of these four variables. Furthermore, the fastest arrival distance based TIG-process (fad-tig) is performed optimally in quantifying nodes' efficiency and nodes' spreading influence. Moreover, for the fad-tig process, we can find an optimal gathering depth n that makes the TIG-process perform optimally when n is small. [ABSTRACT FROM AUTHOR]

We reinvestigate the two different q-Witt algebras and construct their n-algebras. In one case, the super version is also presented. Moreover we investigate the central extensions and present the (super) q-Virasoro n-algebras for the n even case. We study a toy model for the q-Virasoro constraints. A q-Witt n-algebra is discussed in this model. [ABSTRACT FROM AUTHOR]

Six fine gradings on the real form e 6 , − 14 are described, precisely those ones coming from fine gradings on the complexified algebra. The universal grading groups are Z 2 3 × Z 3 2 , Z 2 6 , Z × Z 2 4 , Z 2 7 , Z × Z 2 5 , and Z 2 × Z 2 3 . [ABSTRACT FROM AUTHOR]

Over the years, coronal heating has been the most fascinating question among the scientific community. In the present article, a heating mechanism has been proposed based on the wave-wave interaction. Under this wave-wave interaction, the high frequency kinetic Alfvén wave interacts with the low frequency ion acoustic wave. These waves are three dimensionally propagating and nonlinearly coupled through ponderomotive nonlinearity. A numerical code based on pseudo-spectral technique has been developed for solving these normalized dynamical equations. Localization of kinetic Alfvén wave field has been examined, and magnetic power spectrum has also been analyzed which shows the cascading of energy to higher wavenumbers, and this cascading has been found to have Kolmogorov scaling, i.e., k-5/3. A breakpoint appears after Kolmogorov scaling and next to this spectral break; a steeper scaling has been obtained. The presented nonlinear interaction for coronal loops plasmas is suggested to generate turbulent spectrum having Kolmogorov scaling in the inertial range and steepened scaling in the dissipation range. Since Kolmogorov turbulence is considered as the main source for coronal heating; therefore, the suggested mechanism will be a useful tool to understand the mystery of coronal loop heating through Kolmogorov turbulence and dissipation. [ABSTRACT FROM AUTHOR]