Robust output synchronization of second-order systems

A technique to synchronize a network of dynamical systems described by second-order ordinary differential equations is presented. Each system can be driven by a coupling control signal, which is synthesized such that, at steady-state, the outputs of two given systems, say y _i and y _j , i ≠ j, satisfy a specified ratio, that is, y _i /y _j = α _i /α _j , α _i ≠ 0 ≠ α _j . Among others, this includes the cases where the outputs are synchronized in-phase or anti-phase. The proposed synchronization technique is robust; this means that a small synchronization error is preserved at steady-state, even if the systems were perturbed by external disturbances. Some level of parameter uncertainty can also be tolerated. The coupling control signals are synthesized based on a classical controller and a robust observer that estimates the generalized velocities and provides an estimation of the perturbation terms. Some experimental results, showing the performance of the proposed synchronization technique, are included.     

Synchronization, multistability and basin crisis in coupled pendula

The synchronization dynamics of two linearly coupled pendula is studied in this paper. Based on the Lyapunov stability theory and Linear matrix inequality (LMI); some necessary and sufficient conditions for global asymptotic synchronization are derived from which an estimated threshold coupling k_th, for the on-set of full synchronization is obtained. The numerical value of k_th determined from the average energies of the systems is in good agreement with theoretical analysis. Prior to the on-set of synchronization, the boundary crisis of the chaotic attractor is identified. In the bistable states, where two asymmetric periodic attractors co-exist, it is shown that the coupled pendula can attain multistable states via a new dynamical transition—the basin crisis that occur prior to the on-set of stable synchronization. The essential feature of basin crisis is that the two co-existing attractors are destroyed while new three or more co-existing attractors of the same or different periodicity are created. In addition, the linear perturbation technique and the Routh-Hurwitz criteria are employed to investigate the stability of steady states, and clearly identify the different types of bifurcations likely to be encountered. Finally, two-parameter phase plots, show various regions of chaos, hyperchaos and periodicity.     

Pinning-controlled synchronization of complex networks with bounded or unbounded synchronized regions

This paper studies pinning-controlled synchronization of complex networks with bounded or unbounded synchronized regions. To study a state-feedback pinning-controlled network with N nodes, it first converts the controlled network to an extended network of N+1 nodes without controls. It is shown that the controlled synchronizability of the given network is determined by the real part of the smallest nonzero eigenvalue of the coupling matrix of its extended network when the synchronized region is unbounded; but it is determined by the ratio of the real parts of the largest and the smallest nonzero eigenvalues of the coupling matrix when the synchronized region is bounded. Both theoretical analysis and numerical simulation show that the portion of controlled nodes has no critical values when the synchronized region is unbounded, but it has a critical value when the synchronized region is bounded. In the former case, therefore, it is possible to control the network to achieve synchronization by pinning only one node. In the latter case, the network can achieve controlled synchronization only when the portion of controlled nodes is larger than the critical value.     

Bounded perturbations of the P(ϕ)2 theory

As a perturbation to the P(?)₂ theory we consider interaction densities of the form V(?(x)), where ?(x) is a scalar hermitian boson field and V(α) is a bounded real continuous function. It is proved that the asymptotic fields exist and are equal to the asymptotic fields of the P(?)₂ theory. The connection with non-polynomial theories of rational type is indicated. Furthermore the consequences of a bounded perturbation for the S-matrix and the spectral properties are discussed.     

Quantum phase transition in XY spin chain with three-site interaction studied in terms of Loschmidt echo and Berry phase

By means of the Loschmidt Echo (LE) and Berry Phase (BP) calculations, quantum phase transition (QPT) of an XY spin chain with three-site interaction (α) in a transverse magnetic field (λ) is studied. Both the LE and BP?s λ derivative present anomaly behaviors at the critical regions λ₁,λ₂ and λ₃. The model is in the Ferromagnetic phase as λ>λ₁=1+α and in the Spin Liquid I phase as −1+α<λ<1+α. λ₁ and λ₂ are independent on the anisotropy parameter γ. But, the anisotropy interaction can shift the critical line λ₃ between the Spin Liquid II phase and the Ferromagnetic phase. The present work suggests that QPT of the XY spin chain with three-site interaction can be characterized by exploring the dynamical behaviors of the LE and BP.     

Synchronization rate of synchronized coupled systems

Synchronization phenomena, an emergent property in networks of interacting dynamical elements, are widely observed in nature, and have become the subject of intense research. Here we will investigate the synchronization rate problem in coupled limit-cycle and chaotic oscillators. Based on the mode decomposition method and Gershgörin's discs theorem, some sufficient conditions for synchronization of coupled systems are obtained, and a synchronization rate is then derived. Such a synchronization rate indicates that the error functions between state variables of underlying individual systems tend to zero in the exponential form as time tends to the infinity. Several numerical examples are also given to validate the theoretical results.     

A scale-free network with limiting on vertices

We propose and analyze a random graph model which explains a phenomena in the economic company network in which company may not expand its business at some time due to the limiting of money and capacity. The random graph process is defined as follows: at any time-step t, (i) with probability α(k) and independently of other time-step, each vertex is inactive which means it cannot be connected by more edges, where k is the degree of v_i at the time-step t; (ii) a new vertex v_t is added along with m edges incident with v_t at one time and its neighbors are chosen in the manner of preferential attachment. We prove that the degree distribution P(k) of this random graph process satisfies if α(⋅) is a constant α₀; and P(k)∝C₂k⁻³ if α(?)↓0 as ?↑∞, where C₁,C₂ are two positive constants. The analytical result is found to be in good agreement with that obtained by numerical simulations. Furthermore, we get the degree distributions in this model with m-varying functions by simulation.     

Fuzzy model based adaptive synchronization of uncertain chaotic systems: Robust tracking control approach

This Letter presents fuzzy model-based robust tracking control for the adaptive synchronization of uncertain chaotic systems. Fuzzy model and adaptive algorithm are employed to present the unknown chaotic systems. H^∞ and sliding mode control are combined to construct a robust tracking controller. The incorporated H^∞ controller can attenuate the external disturbance and approximation error to any prescribed level. The proposed scheme guarantees that all the variables are bounded and the tracking error is compensated.     

Quantum electrodynamic corrections to the 23S1 → 11S0 + one photon transition rate in helium and helium-like lons

The order (Zα)⁵, α(Zα)⁴, and α(Zα)⁴ log(Zα) energy denominator and recoil corrections to the leading order (Zα)⁴ decay amplitude for the 2³S₁ → 1¹S₀ + one photon transition are considered. It is found that order α(Zα)⁴ terms arise in the individual diagrams but exactly cancel upon summation. In view of the results of this and other recent theoretical work it is concluded that the rates predicted by pure quantum electrodynamics are known to considerable precision. The recent discrepancy between theory and experiment in the decay rate for several helium-like ions appears to have been resolved by placing the detector farther downstream from the foil where the metastable states are produced. At present, all the experimental values are in good agreement with the theory.     

Boundedness and synchronization of y-coupled Lorenz systems with or without controllers

The purpose of this paper is to investigate the boundedness and synchronization of y-coupled Lorenz systems. When the coupling term is only added to the second variable, we call them y-coupled Lorenz systems. In this paper, we first prove the boundedness of y-coupled Lorenz systems, which ensures the existence and uniqueness of the solution when t→∞. Based on the boundedness, we prove that for y-coupled Lorenz systems, QUAD condition is satisfied. It should be pointed out that QUAD condition plays an essential role in the discussion of synchronization. Based on the boundedness and QUAD condition, we prove that if the coupling is strong enough, the y-coupled Lorenz systems can achieve the complete synchronization globally and exponentially.     

Ergodicity for the Randomly Forced 2D Navier–Stokes Equations

We study space-periodic 2D Navier–Stokes equations perturbed by an unbounded random kick-force. It is assumed that Fourier coefficients of the kicks are independent random variables all of whose moments are bounded and that the distributions of the first N ₀ coefficients (where N ₀ is a sufficiently large integer) have positive densities against the Lebesgue measure. We treat the equation as a random dynamical system in the space of square integrable divergence-free vector fields. We prove that this dynamical system has a unique stationary measure and study its ergodic properties.     

H∞ synchronization of chaotic systems using output feedback control design

This article investigates the H_∞ synchronization problem for a general class of chaotic systems. Based on Lyapunov theory, linear matrix inequality (LMI) and linear matrix equality (LME) formulation, the output feedback controller is established to not only guarantee stable synchronization of both master and slave systems but also reduce the effect of external disturbance to an H_∞-norm constraint. Two illustrative examples are provided to demonstrate the effectiveness of the developed theoretical results.     

Inter-relationship between optical dielectric constant and interatomic separation of mixed crystals in ionic and covalent binary compound families

An ion dependent dielectric model is developed for mixed binary crystals. The interatomic separation (R) and optical dielectric constant (ε_∞) of the mixed crystals are computed from the measured values of R and ε_∞ of pure crystals. An empirical relation between ε_∞ and R is therefore found for mixed crystals by using the computed values. It is found that the dielectric behaviour of ionic mixed crystals is cation dependent while that of covalent mixed crystals is anion dependent. The prescribed theory can therefore be used to form different mixed crystals with particular values of ε_∞ required in any specific opto-electronic devices.     

The 56Fe(α, γ)60Ni reaction at excitation energies in the region of the giant dipole resonance

Excitation functions for the ⁵⁶Fe(α, γ₀)⁶⁰Ni and ⁵⁶Fe(α, γ₁)⁶⁰Ni reactions have been measured at 90° to the beam direction over the bombarding energy range 8.0–17.6 MeV. Gamma-ray angular distributions were measured at ten bombarding energies. Excitation functions for the ⁵⁹Co(p, γ₀)⁶⁰Ni and ⁵⁹Co(p, γ₁)⁶⁰Ni reactions were measured over the range E_x = 16.58–16.92 MeV and compared with the (α, γ) data. The angular distribution data indicate that the (α, γ₀) and (α,γ₁) reactions proceed through 1^?, and 1^? and 3^? states, respectively. The (α, γ) excitation functions are discussed with respect to isospin splitting of the ⁶⁰Ni giant dipole resonance. The fine structure observed in the excitation functions is shown to be most probably due to Ericson fluctuations. The gross (α, γ) cross sections are shown to be in reasonable agreement with the results of calculations made using the theory of Hauser and Feshbach assuming excitation of the giant dipole resonance.     

Relationship between the thermal expansion coefficient, plasmon energy, and bond length of ternary chalcopyrite semiconductors

A simple relation between the average bond length and plasmon energy, that is, d=15.30(?ω_p)^−2/3, has been proposed for A^IIB^IVC^V₂ and A^IB^IIIC^VI₂ chalcopyrite semiconductors. The average linear thermal expansion coefficient (α_L) of these semiconductors has been calculated using plasmon energy data. The linear expansion coefficients (α_a) and (α_c) of the lattice parameters a and c, respectively, have also been calculated. The calculated values of d, α_L, α_a and α_c have been compared with the experimental values and the values reported by different workers. A fairly good agreement has been obtained between them.     

Entanglement dynamics for two interacting spins

We study the dynamical generation of entanglement for a very simple system: a pair of interacting spins, s₁ and s₂, in a constant magnetic field. Two different situations are considered: (a) s₁ → ∞, s₂ = 1/2 and (b) s₁ = s₂ → ∞, corresponding, respectively, to a quantum degree of freedom coupled to a semiclassical one (a qubit in contact with an environment) and a fully semiclassical system, which in this case displays chaotic behavior. Relations between quantum entanglement and classical dynamics are investigated.     

Synchronization of networks

We study the synchronization of coupled dynamical systems on networks. The dynamics is governed by a local nonlinear oscillator for each node of the network and interactions connecting different nodes via the links of the network. We consider existence and stability conditions for both single- and multi-cluster synchronization. For networks with time-varying topology we compare the synchronization properties of these networks with the corresponding time-average network. We find that if the different coupling matrices corresponding to the time-varying networks commute with each other then the stability of the synchronized state for both the time-varying and the time-average topologies are approximately the same. On the other hand, for non-commuting coupling matrices the stability of the synchronized state for the time-varying topology is in general better than the time-average topology.      

A new route to Explosive Percolation

The biased link occupation rule in the Achlioptas process (AP) discourages the large clusters from growing much ahead of others and encourages faster growth of clusters which lag behind. In this paper we propose a model where this tendency is sharply reflected in the Gamma distribution of the cluster sizes, unlike the power law distribution in AP. In this model single edges between pairs of clusters of sizes s_i and s_j are occupied with a probability ∝(s_is_j)^α. The parameter α is continuously tunable over the entire real axis. Numerical studies indicate that for α<α_c the transition is first order, α_c=0 for a square lattice and α_c=−1/2 for random graphs. In the limits of α=−∞,+∞ this model coincides with models well established in the literature.     

Current-voltage characteristics of type-II superconductors

For a wide range of samples and conditions the I-V characteristic of type-II superconductors fits the expression V ∞ (I-I_p)² (I-αI_p)⁻¹. The initial non-linearity is attributed to the fact that the fluxons do not move with uniform velocity.     

Compositional effects on the optical and thermal properties of sodium borophosphate glasses

A series of sodium borophosphate glasses of the composition (1−x)NaPO₃−xB₂O₃ have been synthesised from Na₂CO₃, B₂O₃ and P₂O₅ and their optical and thermal properties investigated. The results show that refractive index (n) and glass transition temperature (T_g) show a maximum at about B/(B+P)=0.6 while thermal expansion coefficient (α) and thermo-optic coefficient (dn/dT) change monotonically with the B/(B+P) ratio. These observations can be interpreted based on the incorporation of BO₃ and BO₄ units into the glass structural network.