Discrete data assimilation in the Lorenz and 2D Navier-Stokes equations

Consider a continuous dynamical system for which partial information about its current state is observed at a sequence of discrete times. Discrete data assimilation inserts these observational measurements of the reference dynamical system into an approximate solution by means of an impulsive forcing. In this way the approximating solution is coupled to the reference solution at a discrete sequence of points in time. This paper studies discrete data assimilation for the Lorenz equations and the incompressible two-dimensional Navier-Stokes equations. In both cases we obtain bounds on the time interval h between subsequent observations which guarantee the convergence of the approximating solution obtained by discrete data assimilation to the reference solution.     

Projecting low and extensive dimensional chaos from spatio-temporal dynamics

We review the spatio-temporal dynamical features of the Ananthakrishna model for the Portevin-Le Chatelier effect, a kind of plastic instability observed under constant strain rate deformation conditions. We then establish a qualitative correspondence between the spatio-temporal structures that evolve continuously in the instability domain and the nature of the irregularity of the scalar stress signal. Rest of the study is on quantifying the dynamical information contained in the stress signals about the spatio-temporal dynamics of the model. We show that at low applied strain rates, there is a one-to-one correspondence with the randomly nucleated isolated bursts of mobile dislocation density and the stress drops. We then show that the model equations are spatio-temporally chaotic by demonstrating the number of positive Lyapunov exponents and Lyapunov dimension scale with the system size at low and high strain rates. Using a modified algorithm for calculating correlation dimension density, we show that the stress-strain signals at low applied strain rates corresponding to spatially uncorrelated dislocation bands exhibit features of low dimensional chaos. This is made quantitative by demonstrating that the model equations can be approximately reduced to space independent model equations for the average dislocation densities, which is known to be low-dimensionally chaotic. However, the scaling regime for the correlation dimension shrinks with increasing applied strain rate due to increasing propensity for propagation of the dislocation bands. The stress signals in the partially propagating to fully propagating bands turn to have features of extensive chaos.     

Integrability and the variational formulation of non‐conservative mechanical systems

It is shown that one can obtain canonically‐defined dynamical equations for non‐conservative mechanical systems by starting with a first variation functional, instead of an action functional, and finding their zeroes. The kernel of the first variation functional, as an integral functional, is a 1‐form on the manifold of kinematical states, which then represents the dynamical state of the system. If the 1‐form is exact then the first variation functional is associated with the first variation of an action functional in the usual manner. The dynamical equations then follow from the vanishing of the dual of the Spencer operator that acts on the dynamical state. This operator, in turn, relates to the integrability of the kinematical states. The method is applied to the modeling of damped oscillators.     

On the general structure of mathematical models for physical systems

It is proposed that the mathematical models for any physical systems that are based in first principles, such as conservation laws or balance principles, have some common elements, namely, a space of kinematical states, a space of dynamical states, a constitutive law that associates dynamical states with kinematical states, as well as a duality principle. The equations of motion or statics then come about from, on the one hand, specifying the integrability of the kinematical state, and on the other hand, specifying a statement that is dual to it for the dynamical states. Examples are given from various fundamental physical systems.     

Evolution of host-parasitoid network through homeochaotic dynamics

Host-parasitoid systems with evolving mutation rates are studied. By increasing the growth rate of hosts, the diversity of both species is maintained dynamically. For the lower growth rate, diversity is brought about by mere parasitism. The average mutation rate for parasites is elevated to a high value, while that for hosts is suppressed at a low level. For the higher growth rate, the mutation rates for both hosts and parasites are elevated to form a symbiotic cluster connected by on-going mutation. This symbiotic state is sustained through a chaotic oscillation keeping some coherency among species. For a flat landscape for hosts, dynamical clustering of oscillation is observed. Lyapunov spectra of such oscillations show that high dimensional chaos with small positive exponents underlies in the symbiotic state. This weak high dimensional chaos, termed "homeochaos," is essential to the maintenance of symbiosis in ecosystems.     

Quantum superchemistry: dynamical quantum effects in coupled atomic and molecular Bose-Einstein condensates

We show that in certain parameter regimes there is a macroscopic dynamical breakdown of the Gross-Pitaevskii equation. Stochastic field equations for coupled atomic and molecular condensates are derived using the functional positive- P representation. These equations describe the full quantum state of the coupled condensates and include the commonly used Gross-Pitaevskii equation as the noiseless limit. The full quantum theory includes the spontaneous processes which will become significant when the atomic population is low. The experimental signature of the quantum effects will be the time scale of the revival of the atomic population after a near total conversion to the molecular condensate.     

General Dynamical Equations for Free Particles and?Their Galilean Invariance

Dynamical equations describing evolution of state functions in space-time of a given metric are important components of physical theories of particles. A method based on a group of the metric is used to obtain an infinite set of general dynamical equations for a scalar and analytical function representing free and spinless particles. It is shown that this set of equations is the same for any group of the metric that consists of an invariant Abelian subgroup of translations in time and space. For Galilean space-time, such group is the extended Galilei group. Using this group, it is proved that the infinite set of equations has only one subset of Galilean invariant dynamical equations, and that the equations of this subset are Schr?dinger-like equations.     

Non-linear Relaxation of the Photo-induced High Spin State in Spin-Crossover Solids: Effect of Correlations

We investigate a microscopic model based on the pair approximation, for the dynamical properties of the photo-induced high-spin state in spin-crossover solids at low temperature. The model uses the Ising-like hamiltonian and combines long- and short-range interactions. The stochastic treatment of the latter provides a set of two coupled differential equations for the macroscopic short- and long-range order parameters, which reproduce successfully the main features of the experimental relaxation curves.     

Cr4+∶YAG晶体的激发态吸收研究

对Cr4 + ∶YAG晶体作为 10 6 4nm激光的可饱和吸收体进行了研究。利用速率方程 ,描述了强激光下晶体的激发态吸收 (ESA)的动力学过程。用实验验证了所导出的光强与透过率的理论曲线 ,并因此论证了可以利用晶体的激发态吸收进行激光调Q ,并对其进行优化。     

原子-异核-三聚物分子转化系统暗态的动力学不稳定性

研究了受激拉曼绝热过程中原子-异核-三聚物分子转化系统暗态的动力学稳定性.通过将量子哈密顿对应到经典哈密顿,并求解和分析线性化经典运动方程后得到的哈密顿-雅克比矩阵本征值,解析地得到了原子-三聚物暗态的动力学不稳定性发生的条件.并以异核原子⁸⁷Rb和⁴¹K混合凝聚体为例,数值地给出了系统发生动力学不稳定性的区域.研究发现,这种动力学不稳定性是由粒子之间的相互作用带来的.此外,还发现系统动力学不稳定性的发生不仅与哈密顿-雅克比矩阵是否出现实数或复数的本征值有关,还 关键词： 原子-异核-三聚物分子转化系统 暗态 受激拉曼绝热过程 动力学不稳定性     

非重正交的李雅普诺夫指数谱的计算方法

推导了一种快速、有效的计算动力系统李雅普诺夫(Lyapunov)指数谱的方法.该方法避免了 一般算法的频繁的重正交过程;且在维数不高(n<5)的情况下,所需求解的方程数也较一般 方法更少.该算法即适用于连续又适用于离散系统,当指数出现简并时同样有效.对以Lorenz 动力系统为主的数值计算,验证了该算法的快速性及其稳定性. 关键词： 混沌 李雅普诺夫指数 复合矩阵 特征值     

Entangled State in Quantization of Magnetic Flux Qubits with Mutual Inductance Coupling

Via the Hamilton dynamical approach we have constructed Hamiltonian for the mutual inductance coupling magnetic flux qubits. The entangled state representation is used to propose Cooper-pair number-phase quantization and the Hamiltonian operator for the whole system. The dynamical evolution of the phase difference operator and the Cooper-pairs number operator is investigated by virtue of Heisenberg equations. Project 10574060 supported by the National Natural Science Foundation of China and project X071045 supported by the Science Foundation of Liaocheng University.     

Relativistic Dynamics of Vector Bosons in the Field of Gravitational Radiation

We consider a model of the state evolution of relativistic vector bosons, which includes both the dynamical equations for the particle four-velocity and the equations for the polarization four-vector evolution in the field of a nonlinear plane gravitational wave. In addition to the gravitational minimal coupling, tidal forces linear in curvature tensor are suggested to drive the particle state evolution. The exact solutions of the evolutionary equations are obtained. Birefringence and tidal deviations from the geodesic motion are discussed.     

Dynamically driven renormalization group

We present a detailed discussion of a novel dynamical renormalization group scheme: the dynamically driven renormalization group (DDRG). This is a general renormalization method developed for dynamical systems with non-equilibrium critical steady state. The method is based on a real-space renormalization scheme driven by a dynamical steady-state condition which acts as a feedback on the transformation equations. This approach has been applied to open nonlinear systems such as self-organized critical phenomena, and it allows the analytical evaluation of scalling dimensions and critical exponents. Equilibrium models at the critical point can also be considered. The explicit application to some models and the corresponding results are discussed.     

Nonequilibrium Phase Transition in Spin-S Ising Ferromagnet Driven by Propagating and Standing Magnetic Field Wave

The dynamical response of spin-S(S=1, 3/2, 2, 3) Ising ferromagnet to the plane propagating wave, standing magnetic field wave and uniformly oscillating field with constant frequency are studied separately in two dimensions by extensive Monte Carlo simulation. Depending upon the strength of the magnetic field and the value of the spin state of the Ising spin lattice two different dynamical phases are observed. For a fixed value of S and the amplitude of the propagating magnetic field wave the system undergoes a dynamical phase transition from propagating phase to pinned phase as the temperature of the system is cooled down. Similarly in case with standing magnetic wave the system undergoes dynamical phase transition from high temperature phase where spins oscillate coherently in alternate bands of half wavelength of the standing magnetic wave to the low temperature pinned or spin frozen phase. For a fixed value of the amplitude of magnetic field oscillation the transition temperature is observed to decrease to a limiting value as the value of spin S is increased. The time averaged magnetisation over a full cycle of the magnetic field oscillation plays the role of the dynamic order parameter. A comprehensive phase boundary is drawn in the plane of magnetic field amplitude and dynamic transition temperature. It is found that the phase boundary shrinks inwards for high value of spin state S.Also in the low temperature(and high field) region the phase boundaries are closely spaced.     

The dominant “heating” mode: bending excitation of water molecules by low-energy positron impact

We report a quantum dynamical treatment of the vibrational excitation of the bending mode of water molecules by collision with low energy positrons in the energy regions close to threshold openings. The exact vibrationally coupled-channel equations derived for the total e+-H2O system are solved in a Body-Fixed-Vibrational-Coupled-Channels (BF-VCC) reference frame, using a single-center expansion of the total wavefunction and of the interaction potential. The vibrationally inelastic cross-sections for transitions from the ground to the lowest excited state of the bending mode clearly show the bending excitation channel to be the dominant inelastic process at low collision energies. Comparisons with our earlier calculations for the other modes and for the excited processes induced by electron impact are also presented and analysed.     

Relaxation of Excited States in C60

The static study has shown that the excited state with electron-hole pair in C₆₀ can cause the distortion of the bond structure to form a polaron-like exciton with symmetry D_5d. This paper further reveals the relaxation process from the initial electron-hole pair state to the polaron-like exciton by solving the dynamical equations. The relaxation time of this dynamical process can be determined from the time-dependent bond distortion with time step 4 femtoseconds.     

Effective actions for statistical data assimilation

Data assimilation is a problem in estimating the fixed parameters and state of a model of an observed dynamical system as it receives inputs from measurements passing information to the model. Using methods developed in statistical physics, we present effective actions and equations of motion for the mean orbits associated with the temporal development of a dynamical model when it has errors, there is uncertainty in its initial state, and it receives information from noisy measurements. If there are statistical dependences among errors in the measurements they can be included in this approach.     

Excitonic-vibronic coupled dimers: A dynamic approach

The dynamical properties of exciton transfer coupled to polarization vibrations in a two site system are investigated in detail. A fixed point analysis of the full system of Bloch-oscillator equations representing the coupled excitonic-vibronic flow is performed. For overcritical polarization a bifurcation converting the stable bonding ground state to a hyperbolic unstable state which is basic to the dynamical properties of the model is obtained. The phase space of the system is generally of a mixed type: Above bifurcation chaos develops starting from the region of the hyperbolic state and spreading with increasing energy over the Bloch sphere leaving only islands of regular dynamics. The behaviour of the polarization oscillator accordingly changes from regular to chaotic.