Matrix formulation and singular-value decomposition algorithm for structured varimax rotation in multivariate singular spectrum analysis

We show that multivariate singular spectrum analysis (M-SSA) greatly helps study phase synchronization in a large system of coupled oscillators and in the presence of high observational noise levels. With no need for detailed knowledge of individual subsystems nor any a priori phase definition for each of them, we demonstrate that M-SSA can automatically identify multiple oscillatory modes and detect whether these modes are shared by clusters of phase- and frequency-locked oscillators. As an essential modification of M-SSA, here we introduce variance-maximization (varimax) rotation of the M-SSA eigenvectors to optimally identify synchronized-oscillator clustering.

The transition to phase synchronization in systems consisting of a large number (N) of coupled nonlinear oscillators via the route of phase clustering (phase synchronization among subsets of oscillators) is investigated. We elucidate the mechanism for the merger of phase clusters and find an algebraic scaling between the critical coupling parameter required for phase synchronization and N. Our result implies that, in realistic situations, phase clustering may be more prevalent than full phase synchronization.

A study is performed of intermittent behavior near the boundary of phase synchronization in systems with complex topology of the attractor. Statistical characteristics of laminar phase durations are calculated to establish that eyelet intermittency occurs at the phase synchronization boundary when there is relatively weak disorder between systems.

The paper evaluates various approaches to construction of high-precision control equipment designed for testing high-speed integrated circuits of high level of integration, operating in wide frequency range. Automated test and diagnostics equipment significantly reduce the number of operations, which are being performed during testing and reveal most of the failures and flaws. Methods for improving the systems under development are proposed in this paper, these methods are based on structural optimization and allow improving performance, providing precision setting of the test signals levels and increasing rate of generating test sequences. The paper also analyses a method of control based on the phase synchronization systems.

The strongly coupled phase of the quark-gluon plasma (QGP) is studied here by resorting to a $T$-matrix formulation in which the medium is seen as a nonideal gas of quasiparticles (quarks, antiquarks, and gluons) interacting nonpertubatively. In the temperature range under study, (1--5)${T}_{c}$, where ${T}_{c}$ is the temperature of deconfinement, the interactions are expected to be strong enough to generate bound states. The dissociation temperature of such binary bound states is thus computed here. The more the quasiparticles involved in the binary system are heavy, the more the bound state is likely to survive significantly above ${T}_{c}$. Then the QGP equations of state at zero and small baryonic potential are computed for ${N}_{f}=2$ and ${N}_{f}=2+1$ by resorting to the Dashen, Ma, and Bernstein formulation of statistical mechanics. Comparisons with current lattice QCD data are presented.

Many systems, arising in electrical and electronic engineering may be represented as an interconnection of LTI system and a periodic nonlinear block, as exemplified by phase-locked loops (PLL) and more general systems, based on controlled phase synchronization of several periodic processes (“phase synchronization” systems, or PSS). Typically such systems are featured by the gradient-like behavior, i.e. the system has infinite sequence of equilibria points, and any solution converges to one of them (which may be interpreted as the phase locking). This property however says nothing about the transient behavior of the system, whose important qualitative index is the maximal phase error. Before the phase is locked, the error may increase up to several periods, which phenomenon is known as cycle slipping and was introduced by J. Stoker for the model of mathematical pendulum. Since the cycle slipping is considered to be undesired behavior of PLLs, it is important to find efficient estimates for the number of slipped cycles. In the present paper, we address the problem of cycle-slipping for phase synchronization systems with infinite-dimensional linear part. New effective estimates for a number of slipped cycles are obtained by means of Popov's method of “a priori integral indices”, which was originally developed for proving absolute stability of nonlinear systems.

The weighted reproducing kernel collocation method exhibits high accuracy and efficiency in solving inverse problems as compared with traditional mesh-based methods. Nevertheless, it is known that computing higher order reproducing kernel (RK) shape functions is generally an expensive process. Computational cost may dramatically increase, especially when dealing with strong-from equations where high-order derivative operators are required as compared to weak-form approaches for obtaining results with promising levels of accuracy. Under the framework of gradient approximation, the derivatives of reproducing kernel shape functions can be constructed synchronically, thereby alleviating the complexity in computation. In view of this, the present work first introduces the weighted high-order gradient reproducing kernel collocation method in the inverse analysis. The convergence of the method is examined through the weights imposed on the boundary conditions. Then, several configurations of multiply connected domains are provided to numerically investigate the stability and efficiency of the method. To reach the desired accuracy in detecting the outer boundary for two special cases, special treatments including allocation of points and use of ghost points are adopted as the solution strategy. From four benchmark examples, the efficacy of the method in detecting the unknown boundary is demonstrated.

In this paper we study teleseismic shear-wave splitting due to multiple anisotropic layers. The effects can be described in terms of a multilayered splitting operator that may be calculated using a matrix formulation. As a special case, we consider smooth variations of anisotropy with depth. The model is characterized by the (horizontal)fast-axis directions at the top (φt) and bottom (φb) of the anisotropic region, and the total delay time (Δt). Numerical results show that the particle motion at relatively long periods (T/Δtnbb 5) is clearly elliptical and is a function of the polarization direction of the incident wave. At shorter periods (T/Δtnab1); however, the seismogram is characterized by two distinctly separated arrivals corresponding to the fast- and slow-axis directions at the top of the anisotropic region. Relative amplitudes of the fast and slow arrivals depend on the orientation of the initial polarization with respect to the axis directions at the bottom. Only the high-frequency results agree with predictions based on ray theory. At fixed initial polarization, the longer-period results can be described in terms of an apparent fast-polarization direction (φa) and an apparent delay time (δta). The apparent splitting parameters exhibit a π/2-periodicity as functions of initial polarization (or backazimuth), which is similar to the two-layer case considered by Silver & Savage (1994). Assuming that φt−φb < 45 °, we find variations of φa and δta to be smaller than 10 per cent over most (≈ 2/3) of the backazimuth range. In this case, φa≃ (φt+φb)/2 and δta≃Δt. The calculated apparent splitting parameters agree well with direct measurements using synthetic waveforms and may thus be applied to the inversion of observations Long-period apparent splitting parameters for different types of depth variations cannot be resolved uniquely without further constraints from the deformation history at a given station. For example, we find that recently observed azimuthal variations of splitting parameters (Özalaybey & Savage 1994; Brechner et al. 1998) can be modelled equally well in terms of two distinct layers (a four-parameter model) or smooth variations of fast-axis directions (a three-parameter model). Depth-dependent models with randomly chosen fast-axis directions and delay times in each layer display a characteristic decrease of the most probable apparent delay time, δta (Pmax), with increasing number of layers N. We find that δta (Pmax)∼ 1/√N. The fact that most measurements yield values of δta≃ 1 s (Silver 1996) suggests that the majority of observations can be explained in terms of one or two anisotropic layers, provided the fast-axis directions in different layers are independent.

Papers published in Hydrology and Earth System Sciences Discussions are under open-access review for the journal Hydrology and Earth System Sciences Evolutionary geomorphology: thresholds and nonlinearity in landform response to

Cumulative author index of volumes 55–65

This paper studies the analytical and dynamical behavior of a generalized (3+1)-dimensional extended Kadomtsev-Petviashvili (eKP) equation, which includes higher-order nonlinear and dispersive effects. Using the exp$ (-\varphi(\xi)) $-expansion method, exact traveling wave solutions are obtained successfully. The solutions exhibit rich nonlinear wave structures, including localized solitons, periodic waves, and breathing patterns. To further explore the qualitative behavior of the system, a reduction of traveling waves is used, resulting in a planar dynamical system. A bifurcation analysis is carried out in detail by investigating the equilibrium points, Jacobian structures, and their relations to system parameters. The phase portraits show topological transitions such as centers, saddles, and cusp points. Moreover, a perturbed form of the reduced system is considered in order to investigate chaotic dynamics under periodic external forcing. The existence of chaos is verified using bifurcation diagrams, sensitivity analysis, Poincaré maps, time series, and both 2D and 3D phase portraits. These numerical simulations confirm the theoretical predictions and demonstrate the system's complex behavior across varying perturbation parameters. Overall, the outcomes provide a complete picture of the soliton structures and chaotic regimes that the eKP model may exhibit in physico-nonlinear wave propagation in multidimensional contexts.

Mapping a triangulated surface to 2D space (or a tetrahedral mesh to 3D space) is an important problem in geometry processing. In computational physics, untangling plays an important role in mesh generation: it takes a mesh as an input, and moves the vertices to get rid of foldovers. In fact, mesh untangling can be considered as a special case of mapping where the geometry of the object is to be defined in the map space and the geometric domain is not explicit, supposing that each element is regular. In this paper, we propose a mapping method inspired by the untangling problem and compare its performance to the state of the art. The main advantage of our method is that the untangling aims at producing locally injective maps, which is the major challenge of mapping. In practice, our method produces locally injective maps in very difficult settings, both in 2D and 3D. We demonstrate it on a large reference database as well as on more difficult stress tests. For a better reproducibility, we publish the code in Python for a basic evaluation, and in C++ for more advanced applications.

Heat transfer and phase change in multiphase media involve complex behavior such as multi-physics coupling, unsteady interfaces, and boundary generation/disappearance. The smoothed particle hydrodynamics (SPH) method is used to simulate multicomponent problems (initially solid material surrounded by a fluid) characterized by heat transfer, solid–liquid phase change, and multiphase flow. First, the interface behavior of the melted liquids and ambient fluids are simulated by a multiphase SPH model, where corrected algorithms and techniques are adopted to provide a stable simulation for the interface with a large density ratio. The wetting effect of the solid surface by melted liquids is considered by extending the continuum surface force model to three phases including solid, melted liquid and ambient fluid. Second, a thermal dynamics model is incorporated with the multiphase SPH model. The solid–liquid phase change occurs in a small temperature interval in which the latent heat is incorporated using the equivalent heat capacity. Third, the numerical model is validated using several benchmark examples. Five specially configured cases are simulated to demonstrate the robustness, adaptability, and stability of the model. The intriguing phenomena of evolution of the fluid interface of melted liquids, accumulation of thin-layer liquids, and melted droplet formation are reproduced.

A partly matrix-free solver for the gyrokinetic field equation in three-dimensional geometry

A numerical algorithm for analysis of disturbance rejection for nonlinear systems is proposed. The algorithm seeks solutions to Euler-Lagrange equations and is similar to power algorithms for mu-analysis lower bounds, beating conventional optimization on speed and global convergence properties. This is a special case of earlier work and focuses on a problem for which a simple proof of global convergence is available for the case of linear dynamics. Thus we can focus our attention on the convergence problems and lack of convexity created by nonlinearities, and benchmark examples are used to explore these issues.