A fast algorithm to calculate the critical coupling strength for synchronization in a chain of Kuramoto oscillators

Synchronization in a one-dimensional chain of Kuramoto oscillators with periodic boundary conditions is studied. An algorithm to rapidly calculate the critical coupling strength \(K_c\) for complete frequency synchronization is presented according to the mathematical constraint conditions and the periodic boundary conditions. By this new algorithm, we have checked the relation between \(\langle K_c\rangle \) and \(N\) , which is \(\langle K_c\rangle \sim \sqrt{N}\) , not only for small \(N\) , but also for large \(N\) . We also investigate the heavy-tailed distribution of \(K_c\) for random intrinsic frequencies, which is obtained by showing that the synchronization problem is equivalent to a discretization of Brownian motion. This theoretical result was checked by generating a large sample of \(K_c\) for large \(N\) from our algorithm to get the empirical density of \(K_c\) . Finally, we derive the permutation for the maximum coupling strength and its exact expression, which grows linearly with \(N\) and would provide the theoretical support for engineering applications.     

Some Aspects of Stability for Semigroup Actions and Control Systems

The present paper introduces both the notions of Lagrange and Poisson stabilities for semigroup actions. Let \(S\) be a semigroup acting on a topological space \(X\) with mapping \(\sigma :S\times X\rightarrow X\) , and let \(\mathcal {F}\) be a family of subsets of \(S\) . For \(x\in X\) the motion \(\sigma _{x}:S\rightarrow X\) is said to be forward Lagrange stable if the orbit \(Sx\) has compact closure in \(X\) . The point \(x\) is forward \(\mathcal {F}\) -Poisson stable if and only if it belongs to the limit set \(\omega \left( x,\mathcal {F}\right) \) . The concept of prolongational limit set is also introduced and used to describe nonwandering points. It is shown that a point \(x\) is \( \mathcal {F}\) -nonwandering if and only if \(x\) lies in its forward \(\mathcal {F} \) -prolongational limit set \(J\left( x,\mathcal {F}\right) \) . The paper contains applications to control systems.     

Stable disarrangement phases of elastic aggregates: a setting for the emergence of no-tension materials with non-linear response in compression

A recent field theory of elastic bodies undergoing non-smooth submacroscopic geometrical changes (disarrangements) provides a setting in which, for a given homogeneous macroscopic deformation \(F\) of the body, there are typically a number of different states \(G\) of smooth, submacroscopic deformation (disarrangement phases) available to the body. A tensorial consistency relation and the inequality \(\det G\le \det F\) that guarantees that \(F\) accommodates \(G\) determine the totality of disarrangement phases \(G\) corresponding to \(F\) , and it is natural to seek for a given \(F\) those disarrangement phases that minimize the Helmholtz free energy (stable disarrangement phases). We introduce these concepts in the particular context of continuous bodies comprised of many small elastic bodies (elastic aggregates) and in the context where disarrangements do not contribute to the Helmholtz free energy (purely dissipative disarrangements). In this setting, the Helmholtz free energy response \(G\longmapsto \varPsi (G)\) of the pieces of the aggregate determines the totality of disarrangement phases corresponding to \(F\) , which necessarily includes the phase \(G=F\) (compact phase) in which every piece of the aggregate undergoes the given macroscopic deformation \(F\) . When the response function \(\varPsi \) is isotropic and smooth, and when \(\varPsi \) possesses standard semiconvexity and growth properties, the body also admits phases of the form \(G=\zeta _{\min }R\) (loose phases) with \(R\) an arbitrary rotation, provided that \(\zeta _{\min }R \) satisfies the accommodation inequality \(\zeta _{\min }^{3}\le \det F\) . Loose phases, when available, achieve the global minimum \(\varPsi (\zeta _{\min }R)\) of the free energy and consequently are stable and stress-free. When \( \varPsi (G)\) has the specific form \(\varPsi _{\alpha \beta }(G)=(\alpha /2)(\det G)^{-2}+(\beta /2)tr(GG^{T})\) , with \(\alpha \) , \(\beta \) given elastic constants, we determine all of the disarrangement phases corresponding to \(F\) . These include not only the compact and loose phases, but also disarrangement phases \(G\) in which the stress \(D\varPsi (G)\) is uniaxial or planar. Our main result (“stability implies no-tension”) is the assertion that every stable disarrangement phase for \(\varPsi _{\alpha \beta }\) cannot support tensile tractions, and our treatment of elastic aggregates thus provides a natural setting for the emergence of no-tension materials whose response in compression is non-linear. Existing treatments of no-tension materials assume at the outset that the body cannot support tension and that the response in compression is linear.     

Effects of nonlinear damping suspension on nonperiodic motions of a flexible rotor in journal bearings

Nonlinear damping suspension is a promising method to be used in a rotor-bearing system for vibration isolation between the bearing and environment. However, the nonlinearity of the suspension may influence the stability of the rotor-bearing system. In this paper, the motions of a flexible rotor in short journal bearings with nonlinear damping suspension are studied. A computational method is used to solve the equations of motion, and the bifurcation diagrams, orbits, Poincaré maps, and amplitude spectra are used to display the motions. The results show that the effect of the nonlinear damping suspension on the motions of the rotor-bearing system depends on the speed of rotor: (a) For low speeds, the rotor- bearing system presents the same motion pattern under the nonlinear damping ( \(p=0.5, 2, 3\) ) suspension as for the linear damping ( \(p=1\) ) suspension; (b) For high speeds, the effect of nonlinear damping depends on a combination of the damping exponent and damping coefficient. The square root damping model ( \(p=0.5\) ) shows a wider stable speed range than the linear damping for large damping coefficients. The quadratic damping ( \(p=2\) ) shows similar results to linear damping with some special damping coefficients. The cubic damping ( \(p=3\) ) shows more stable response than the linear damping in general.     

Robust mixed H_2 /H_\infty active control for offshore steel jacket platform

This paper presents a robust mixed \(H_2 /H_\infty \) control method for wave-excited offshore jacket platforms. Its objective was to design a controller that minimizes the upper bound of the \(H_2 \) performance measure on platform dynamics satisfying some \(H_\infty \) norm bound constraint simultaneously. Based on mixed \(H_2 /H_\infty \) control theory and linear matrix inequality techniques, a novel approach to stabilize offshore platform vibration with constrained \(H_2 /H_\infty \) performances is proposed. Uncertainties of the wave excitation are considered in dynamic performance analysis of offshore platforms. A reduced mode offshore platform structure under wave excitation is analyzed, and simulations are used to verify the effectiveness of the proposed approach. Compared with existing \(H_\infty \) control methods, the proposed approach makes a significant improvement for dynamic performances of offshore platforms under random wave excitation.     

Inverse Approach in Ordinary Differential Equations: Applications to Lagrangian and Hamiltonian Mechanics

This paper is on the so called inverse problem of ordinary differential equations, i.e. the problem of determining the differential system satisfying a set of given properties. More precisely we characterize under very general assumptions the ordinary differential equations in \(\mathbb {R}^N\) which have a given set of either \(M\) partial integrals, or \(M first integral, or \(M partial and first integrals. Moreover, for such systems we determine the necessary and sufficient conditions for the existence of \(N-1\) independent first integrals. We give two relevant applications of the solutions of these inverse problem to constrained Lagrangian and Hamiltonian systems respectively. Additionally we provide the general solution of the inverse problem in dynamics.     

Standard decomposition of expansive ergodically supported dynamics

In this work, we introduce the notion of weak quasigroups, which are quasigroup operations defined almost everywhere on some set. Then, we prove that the topological entropy and the ergodic period of an invertible expansive ergodically supported dynamical system \((X,T)\) with the shadowing property establish a sufficient criterion for the existence of quasigroup operations defined almost everywhere outside of universally null sets and for which \(T\) is an automorphism. Furthermore, we find a decomposition of the dynamics of \(T\) in terms of \(T\) -invariant weak topological subquasigroups.     

Gas Sorption and the Consequent Volumetric and Permeability Change of Coal II: Numerical Modeling

The permeability of coalbed methane reservoirs may evolve during the recovery of methane and injection of gas, due to the change of effective stress and gas adsorption and desorption. Experimental and numerical studies were conducted to investigate the sorption-induced permeability change of coal. This paper presents the numerical modeling part of the work. It was found that adsorption of pure gases on coal was well represented by parametric adsorption isotherm models in the literature. Based on the experimental data of this study, adsorption of pure \(\hbox {N}_2\) was modeled using the Langmuir equation, and adsorption of pure \(\hbox {CO}_2\) was well represented by the N-Layer BET equation. For the modeling of CO \(_2\) & N \(_2\) binary mixture adsorption, the ideal adsorbed solution (IAS) model and the real adsorbed solution (RAS) model were used. The IAS model estimated the total amount of mixture adsorption and the composition of the adsorbed phase based on the pure adsorption isotherms. The estimated total adsorption and adsorbed-phase composition were very different from the experimental results, indicating nonideality of the CO \(_2\) –N \(_2\) –Coal-adsorption system. The measured sorption-induced strain was linearly proportional to the total amount of adsorption despite the species of the adsorbed gas. Permeability reduction followed a linear correlation with the volumetric strain with the adsorption of pure \(\hbox {N}_2\) and the tested CO \(_2\) & N \(_2\) binary mixtures, and an exponential correlation with the adsorption of pure \(\hbox {CO}_2\) .     

Integrability with symbolic computation on the Bogoyavlensky–Konoplechenko model: Bell-polynomial manipulation,bilinear representation,and Wronskian solution

With symbolic computation, this paper investigates some integrable properties of a two-dimensional generalization of the Korteweg-de Vries equation, i.e., the Bogoyavlensky–Konoplechenko model, which can govern the interaction of a Riemann wave propagating along the \(y\) -axis and a long wave propagating along the \(x\) -axis. Within the framework of Bell-polynomial manipulations, Bell-polynomial expressions are firstly given, which then are cast into bilinear forms. The \(N\) -soliton solutions in the form of an \(N\) th-order polynomial in the \(N\) exponentials and in terms of the Wronskian determinant are, respectively, constructed with the Hirota bilinear method and Wronskian technique. Bilinear Bäcklund transformation is also derived with the achievement of a family of explicit solutions.     

Expansivity and Cone-fields in Metric Spaces

Due to the results of Lewowicz and Tolosa expansivity can be characterized with the aid of Lyapunov function. In this paper we study a similar problem for uniform expansivity and show that it can be described using generalized cone-fields on metric spaces. We say that a function \(f:X\rightarrow X\) is uniformly expansive on a set \(\varLambda \subset X\) if there exist \(\varepsilon >0\) and \(\alpha \in (0,1)\) such that for any two orbits \(\hbox {x}:\{-N,\ldots ,N\} \rightarrow \varLambda \) , \(\hbox {v}:\{-N,\ldots ,N\} \rightarrow X\) of \(f\) we have $$\begin{aligned} \sup _{-N\le n\le N}d(\hbox {x}_n,\hbox {v}_n) \le \varepsilon \implies d(\hbox {x}_0,\hbox {v}_0) \le \alpha \sup _{-N\le n\le N}d(\hbox {x}_n,\hbox {v}_n). \end{aligned}$$ It occurs that a function is uniformly expansive iff there exists a generalized cone-field on \(X\) such that \(f\) is cone-hyperbolic.     

Output feedback \varvec{\mathcal {H}}_{{\varvec{\infty }}} control of stochastic nonlinear time-delay systems with state and disturbance-dependent noises

This paper is concerned with the output feedback \(\mathcal {H}_\infty \) control problem for a class of stochastic nonlinear systems with time-varying state delays; the system dynamics is governed by the stochastic time-delay It \(\hat{o}\) -type differential equation with state and disturbance contaminated by white noises. The design of the output feedback \(\mathcal {H}_\infty \) control is based on the stochastic dissipative theory. By establishing the stochastic dissipation of the closed-loop system, the delay-dependent and delay-independent approaches are proposed for designing the output feedback \(\mathcal {H}_\infty \) controller. It is shown that the output feedback \(\mathcal {H}_\infty \) control problem for the stochastic nonlinear time-delay systems can be solved by two delay-involved Hamilton–Jacobi inequalities. A numerical example is provided to illustrate the effectiveness of the proposed methods.     

Anisotropic Estimates for the Two-Dimensional Kuramoto–Sivashinsky Equation

We address the global solvability of the Kuramoto–Sivashinsky equation in a rectangular domain \([0,L_1]\times [0,L_2]\) . We give sufficient conditions on the width \(L_2\) of the domain, depending on the length \(L_1\) , so that the obtained solutions are global. Our proofs are based on anisotropic estimates.     

On the neutralization of bacterial spores in post-detonation flows

In multiple operational scenarios, explosive charges are used to neutralize confined or unconfined stores of bacterial spores. The spore destruction is achieved by post-detonation combustion and mixing of hot detonation product gases with the ambient flow and spore clouds. In this work, blast wave interaction with bacterial spore clouds and the effect of post-detonation combustion on spore neutralization are investigated using numerical simulations. Spherical explosive charges (radius, \(R_\mathrm{C}\) = 5.9 cm) comprising of nitromethane are modeled in the vicinity of a spore cloud, and the spore kill in the post-detonation flow is quantified. The effect of the mass of the spores and the initial distance, \(d^0\) , of the spore cloud from the explosive charge on the percentage of spores neutralized is investigated. When the spores are initially placed within a distance of 3.0 \(R_\mathrm{C}\) , within 0.1 ms after detonation of the charge, all the spores are neutralized by the blast wave and the hot detonation product gases. In contrast, almost all the spores survived the explosion when \(d^0\) is greater than 8.0 \(R_\mathrm{C}\) . The percentage of intact spores varied from 0 to 100 for 3.0 \(R_\mathrm{C}\) \( 8.0 \(R_\mathrm{C}\) with spore neutralization dependent on time spent by the spores in the post-detonation mixing/combustion zone.     

Robust {\varvec{L}}_{\varvec{2}}-gain control for a class of cascade switched nonlinear systems

In this paper, we study the robust finite \(L_2 \) -gain control for a class of cascade switched nonlinear systems with parameter uncertainty. Each subsystem of the switched system under consideration is composed of a zero-input asymptotically stable nonlinear part which is a lower dimension switched system, and of a linearizable part. The uncertainty appears in the control channel of each subsystem. We give sufficient conditions under which the nonlinear feedback controllers are derived to guarantee that the \(L_2 \) -gain of the closed-loop switched system is less than a prespecified value for all admissible uncertainty under arbitrary switching. Moreover, we also develop the \(L_2\) -gain controllers for the switched systems with nonminimum phase case.     

Fractional models of anomalous relaxation based on the Kilbas and Saigo function

We revisit the Kilbas and Saigo functions of the Mittag-Leffler type of a real variable \(t\) , with two independent real order-parameters. These functions, subjected to the requirement to be completely monotone for \(t>0\) , can provide suitable models for the responses and for the corresponding spectral distributions in anomalous (non–Debye) relaxation processes, found e.g. in dielectrics. Our analysis includes as particular cases the classical models referred to as Cole–Cole (the one-parameter Mittag-Leffler function) and to as Kohlrausch (the stretched exponential function). After some remarks on the Kilbas and Saigo functions, we discuss a class of fractional differential equations of order \(\alpha \in (0,1]\) with a characteristic coefficient varying in time according to a power law of exponent \(\beta \) , whose solutions will be presented in terms of these functions. We show 2D plots of the solutions and, for a few of them, the corresponding spectral distributions, keeping fixed one of the two order-parameters. The numerical results confirm the complete monotonicity of the solutions via the non-negativity of the spectral distributions, provided that the parameters satisfy the additional condition \(0<\alpha +\beta \le 1\) , assumed by us.     

Random Attractors for Degenerate Stochastic Partial Differential Equations

We prove the existence of random attractors for a large class of degenerate stochastic partial differential equations (SPDE) perturbed by joint additive Wiener noise and real, linear multiplicative Brownian noise, assuming only the standard assumptions of the variational approach to SPDE with compact embeddings in the associated Gelfand triple. This allows spatially much rougher noise than in known results. The approach is based on a construction of strictly stationary solutions to related strongly monotone SPDE. Applications include stochastic generalized porous media equations, stochastic generalized degenerate $p$ -Laplace equations and stochastic reaction diffusion equations. For perturbed, degenerate $p$ -Laplace equations we prove that the deterministic, $\infty $ -dimensional attractor collapses to a single random point if enough noise is added.     

The evolution and homogeneity of EU economies (with an econometric approach)

We consider a model for the evolutions of \(m>2\) economies \(y_{i}(t),\) where we assume that their interaction is based on the differences of the values of their evolution status. Since the economies have structures which cause delays, we introduce in the equations a mathematical memory formalism represented by a derivative of fractional order which leads to a system of integro-differential equations. The solution is obtained from a set of \(m\) linear equations in the Laplace Transform (LT) of the \(y_{i}(t)\) . Differently from a previous note Caputo (Non Linear Dyn Econ 16:2–22, 2012), in the present one each economy is affected by a different memory. Is found that the asymptotic values of the state of evolutions of the economies are all equal to the initial value of the economy which has the memory represented by a fractional derivative of order smaller than the others. A method for the definition and estimate of the measure of economies and their comparison is also presented and applied to study the homogeneities of 5 EU economies with an estimate of fields, where they are inhomogeneous.     

On solutions of two coupled fractional time derivative Hirota equations

We consider the well-known nonlinear Hirota equation (NLH) with fractional time derivative and derive its periodic wave solution and approximate analytic solitary wave solution using the homotopy analysis method (HAM). We also apply HAM to two coupled time fractional NLHs and construct their periodic wave solution and approximate solitary wave solution. We observe that the obtained periodic wave solution in both cases can be written in terms of the Mittag–Leffler function when the convergence control parameter \({c}_0\) equals \(-1\) . Convergence of the obtained solution is discussed. The derived approximate analytic solution and the effect of time-fractional order \(\alpha \) are shown graphically.     

Diversity evolution and parameter matching of periodic-impact motions of a periodically forced system with a clearance

A two-degree-of-freedom periodically forced system with a clearance is considered. The correlative relationship and matching law between dynamics and system parameters are analyzed by the co-simulation analysis of multi-parameter and multi-performance. Key parameters of the system, such as the exciting frequency, clearance value and constraint stiffness, are emphasized to analyze the influence of the main factors on its soft impact characteristics and reveal diversity, evolution and distribution regions of periodic-impact motions. The quantity of the fundamental group of \(p/1\) impact motions, which have the excitation period and differ by the number \(p\) of impacts, is basically determined by the constraint stiffness. A series of grazing bifurcations occur with decreasing the exciting frequency so that the number \(p\) of impacts of the fundamental group of motions correspondingly increases one by one. As the constraint stiffness is very large, the impact number \(p\) of the fundamental group of motions becomes also big enough in low exciting frequency range. Consequently, the system possibly exhibits chattering-impact characteristics and the relative impact velocities successively attenuate in an excitation period. There exist a series of singular points between any two adjacent ones of the fundamental group of motions as the damping ratio or the damping constant of the viscous dashpot connecting two masses is very small, i.e., real-grazing and bare-grazing bifurcation boundaries of one of them, saddle-node and period-doubling bifurcation boundaries of the other mutually cross themselves at the point of intersection and create two types of transition regions: hysteresis and tongue-shaped regions. A series of zones of regular and complex subharmonic impact motions are found to dominate in the tongue-shaped regions. The dimensionless parameters have been designed technically, under which large mass ratio or small supporting stiffness ratio leads to diversity and complexity of periodic-impact motions of the system. Based on the sampling ranges of parameters, the influence of system parameters on relative impact velocities, existence regions and correlative distribution of different types of periodic-impact motions of the system is emphatically studied.     

Microsize and initial pressure effects on shock wave propagation in a tube

In this paper, the shock wave propagation in a channel with a micrometric hydraulic diameter is numerically simulated for an initial Mach number \(M_{s}=2.61\) . The obtained values of the Mach number along the tube are compared to experimental and numerical data given in the literature. The microscale effects on the flow behavior, such as shock wave attenuation and pressure increase behind the shock wave, are amplified by further reducing the scaling ratio (or Reynolds number) of the flow. This reduction is obtained by either decreasing the hydraulic diameter \(D_\mathrm{H}\) or the initial driven gas pressure \(P_1\) . Under these conditions, the flow behavior changes drastically.