Elementary bifurcations for a simple dynamical system under non-Gaussian Levy noises

Nonlinear dynamical systems are sometimes under the influence of random fluctuations. It is desirable to examine possible bifurcations for stochastic dynamical systems when a parameter varies. A computational analysis is conducted to investigate bifurcations of a simple dynamical system under non-Gaussian {\alpha}-stable Levy motions, by examining the changes in stationary probability density functions for the solution orbits of this stochastic system. The stationary probability density functions are obtained by numerically solving a non local Fokker-Planck equation. This allows numerically investigating phenomenological bifurcation, or P-bifurcation, for stochastic differential equations with non-Gaussian Levy noises.Comment: priprin

Slow foliation of a slow-fast stochastic evolutionary system

This work is concerned with the dynamics of a slow-fast stochastic evolutionary system quantified with a scale parameter. An invariant foliation decomposes the state space into geometric regions of different dynamical regimes, and thus helps understand dynamics. A slow invariant foliation is established for this system. It is shown that the slow foliation converges to a critical foliation (i.e., the scale parameter is zero) in probability distribution, as the scale parameter tends to zero. The approximation of slow foliation is also constructed with error estimate in distribution. Furthermore, the geometric structure of the slow foliation is investigated: every fiber of the slow foliation parallels each other, with the slow manifold as a special fiber. In fact, when an arbitrarily chosen point of a fiber falls in the slow manifold, the fiber must be the slow manifold itself

Highly Accurate Nystr\"{o}m Volume Integral Equation Method for the Maxwell equations for 3-D Scatters

In this paper, we develop highly accurate Nystr\"{o}m methods for the volume integral equation (VIE) of the Maxwell equation for 3-D scatters. The method is based on a formulation of the VIE equation where the Cauchy principal value of the dyadic Green's function can be computed accurately for a finite size exclusion volume with some explicit corrective integrals of removable singularities. Then, an effective interpolated quadrature formula for tensor product Gauss quadrature nodes in a cube is proposed to handle the hyper-singularity of integrals of the dyadic Green's function. The proposed high order Nystr\"{o}m VIE method is shown to have high accuracy and demonstrates $p$-convergence for computing the electromagnetic scattering of cubes in $R^3$

Cost and Effects of Pinning Control for Network Synchronization

In this paper, the problem of pinning control for synchronization of complex dynamical networks is discussed. A cost function of the controlled network is defined by the feedback gain and the coupling strength of the network. An interesting result is that lower cost is achieved by the control scheme of pinning nodes with smaller degrees. Some rigorous mathematical analysis is presented for achieving lower cost in the synchronization of different star-shaped networks. Numerical simulations on some non-regular complex networks generated by the Barabasi-Albert model and various star-shaped networks are shown for verification and illustration.Comment: 12 pages, 18 figure

X-ray afterglow of GRB 050712: Multiple energy injections into the external shock

As indicated by the observed X-ray flares, a great amount of energy could be intermediately released from the postburst central engine of gamma-ray bursts (GRBs). As a natural consequence, the GRB external shock could be energized over and over. With such a multiple energy injection model, we explore the unique X-ray afterglow light curve of GRB 050712, which exists four apparent shallow decay plateaus. Together with three early X-ray flares, the central engine of GRB 050712 is supposed to release energy at least seven times after the burst. Furthermore we find that the energy released during four plateaus are all on the same order of magnitude, but the luminosity decreases with time significantly. These results may provide some interesting implications for the GRB central engine.Comment: 7 pages, two figures, Research in Astronomy and Astrophysics (RAA), 2014, in pres

Thermal engineering in low-dimensional quantum devices: a tutorial review of nonequilibrium Green's function methods

Thermal engineering of quantum devices has attracted much attention since the discovery of quantized thermal conductance of phonons. Although easily submerged in numerous excitations in macro-systems, quantum behaviors of phonons manifest in nanoscale low-dimensional systems even at room temperature. Especially in nano transport devices, phonons move quasi-ballistically when the transport length is smaller than their bulk mean free paths. It has been shown that phonon nonequilibrium Green's function method (NEGF) is effective for the investigation of nanoscale quantum transport of phonons. In this tutorial review two aspects of thermal engineering of quantum devices are discussed using NEGF methods. One covers transport properties of pure phonons; the other concerns the caloritronic effects, which manipulate other degrees of freedom, such as charge, spin, and valley, via the temperature gradient. For each part, we outline basic theoretical formalisms first, then provide a survey on related investigations on models or realistic materials. Particular attention is given to phonon topologies and a generalized phonon NEGF method. Finally, we conclude our review and summarize with an outlook.Comment: 52 pages, 10 figure

Optimal Perfect Distinguishability between Unitaries and Quantum Operations

We study optimal perfect distinguishability between a unitary and a general quantum operation. In 2-dimensional case we provide a simple sufficient and necessary condition for sequential perfect distinguishability and an analytical formula of optimal query time. We extend the sequential condition to general d-dimensional case. Meanwhile, we provide an upper bound and a lower bound for optimal sequential query time. In the process a new iterative method is given, the most notable innovation of which is its independence to auxiliary systems or entanglement. Following the idea, we further obtain an upper bound and a lower bound of (entanglement-assisted) q-maximal fidelities between a unitary and a quantum operation. Thus by the recursion in [1] an upper bound and a lower bound for optimal general perfect discrimination are achieved. Finally our lower bound result can be extended to the case of arbitrary two quantum operations.Comment: 11 pages, 0 figures. Comments are welcom

Analysis and control of network synchronizability

In this paper, the investigation is first motivated by showing two examples of simple regular symmetrical graphs, which have the same structural parameters, such as average distance, degree distribution and node betweenness centrality, but have very different synchronizabilities. This demonstrates the complexity of the network synchronizability problem. For a given network with identical node dynamics, it is further shown that two key factors influencing the network synchronizability are the network inner linking matrix and the eigenvalues of the network topological matrix. Several examples are then provided to show that adding new edges to a network can either increase or decrease the network synchronizability. In searching for conditions under which the network synchronizability may be increased by adding edges, it is found that for networks with disconnected complementary graphs, adding edges never decreases their synchronizability. This implies that better understanding and careful manipulation of the complementary graphs are important and useful for enhancing the network synchronizability. Moreover, it is found that an unbounded synchronized region is always easier to analyze than a bounded synchronized region. Therefore, to effectively enhance the network synchronizability, a design method is finally presented for the inner linking matrix of rank 1 such that the resultant network has an unbounded synchronized region, for the case where the synchronous state is an equilibrium point of the network.Comment: 12 pages, 8 figure

Disconnected synchronized regions of complex dynamical networks

This paper addresses the synchronized region problem, which is reduced to a matrix stability problem, for complex dynamical networks. For any natural number $n$, the existence of a network which has $n$ disconnected synchronized regions is theoretically demonstrated. This shows the complexity in network synchronization. Convexity characteristic of stability for matrix pencils is further discussed. Smooth and generalized smooth Chua's circuit networks are finally discussed as examples for illustration.Comment: 13 pages, 2 figure

Evolution systems of measures for stochastic flows

A new concept of {\em an evolution system of measures for stochastic flows} is considered. It corresponds to the notion of an invariant measure for random dynamical systems (or cocycles). The existence of evolution systems of measures for asymptotically compact stochastic flows is obtained. For a white noise stochastic flow, there exists a one to one correspondence between evolution systems of measures for a stochastic flow \emph{and} evolution systems of measures for the associated Markov transition semigroup. As an application, an alternative approach for evolution systems of measures of 2D stochastic Navier-Stokes equations with a time-periodic forcing term is presented.Comment: 14 page