Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear partial differential evolution equations of dynamical systems

Using functional derivative technique in quantum field theory, the algebraic dynamics approach for solution of ordinary differential evolution equations was generalized to treat partial differential evolution equations. The partial differential evolution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynamics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new numerical algorithm—algebraic dynamics algorithm was proposed for partial differential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experiments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically. Supported by the National Natural Science Foundation of China (Grant Nos. 10375039, 10775100 and 90503008), the Doctoral Program Foundation of the Ministry of Education of China, and the Center of Nuclear Physics of HIRFL of China     

New C-integrable and S-integrable systems of nonlinear partial differential equations

A technique to identify new C-integrable and S-integrable systems of nonlinear partial differential equations is reported, with two representative examples displayed and tersely discussed.     

Solving systems of fractional differential equations by homotopy-perturbation method

In this Letter, approximate analytical solutions of systems of Fractional Differential Equations (FDEs) are derived by the Homotopy-Perturbation Method (HPM). The fractional derivatives are described in the Caputo sense. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms. Numerical results reveal that HPM is very effective and simple for obtaining approximate solutions of nonlinear systems of FDEs.     

On the exact solutions of nonlinear diffusion-reaction equations with quadratic and cubic nonlinearities

Attempts have been made to look for the soliton content in the solutions of the recently studied nonlinear diffusion-reaction equations [R S Kaushal, J. Phys. 38, 3897 (2005)] involving quadratic or cubic nonlinearities in addition to the convective flux term which renders the system nonconservative and the corresponding Hamiltonian non-Hermitian.     

一类动力学方程的Mei对称性

研究一类动力学方程的Mei对称性的定义和判据，由Mei对称性通过Noether对称性可找到Noether守恒量.由Mei对称性通过Lie对称性可找到Hojman守恒量.同时，也可找到一类新型守恒量.     

Variational principle and dynamical equations of discrete nonconservative holonomic systems

By analogue with the methods and processes in continuous mechanics, a Lagrangian formulation and a Hamiltonian formulation of discrete mechanics are obtained. The dynamical equations including Euler--Lagrange equations and Hamilton's canonical equations of the discrete nonconservative holonomic systems are derived on a discrete variational principle. Some illustrative examples are also given.     

Tangent response in coupled dynamical systems

We construct new unidirectional coupling schemes for autonomous and nonautonomous drive systems, respectively. Each of these schemes makes the state of the response system asymptotically approach the first-order derivative of the state of the driver. From the point of view of geometry, the first-order derivative of the state of the driver can be viewed as a tangent vector of the trajectory of the driver, so the proposed schemes are named tangent response schemes. Numerical simulations of the Lorenz system and the forced Duffing oscillator verify the validity of the tangent response schemes. We further point out that the tangent response can be interpreted as a special kind of generalised synchronisation, thereby explaining why the response system can exhibit rich geometrical structures in its state space.     

Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems

For any quantity of interest in a system governed by ordinary differential equations, it is natural to seek the largest (or smallest) long-time average among solution trajectories, as well as the extremal trajectories themselves. Upper bounds on time averages can be proved a priori using auxiliary functions, the optimal choice of which is a convex optimization problem. We prove that the problems of finding maximal trajectories and minimal auxiliary functions are strongly dual. Thus, auxiliary functions provide arbitrarily sharp upper bounds on time averages. Moreover, any nearly minimal auxiliary function provides phase space volumes in which all nearly maximal trajectories are guaranteed to lie. For polynomial equations, auxiliary functions can be constructed by semidefinite programming, which we illustrate using the Lorenz system.     

On information-processing abilities of chaotic dynamical systems

A mechanism is suggested to explain the information processing abilities of simple natural brains, which, by experimental evidence, display behavior like chaotic dynamical systems while at rest. The Lorenz system of equations is dealt with as a case study, and a comparison of the suggested mechanism with the standard theory of neural networks is made.     

Normal form transforms separate slow and fast modes in stochastic dynamical systems

Modelling stochastic systems has many important applications. Normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from insignificant detailed microscale dynamics. We explore such coordinate transforms of stochastic differential systems when the dynamics have both slow modes and quickly decaying modes. The thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems. Thus we not only must reduce the dimensionality of the dynamics, but also endeavour to separate all slow processes from all fast time processes, both deterministic and stochastic. Quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes. Some examples demonstrate that the coordinate transform may be only locally valid or may be globally valid depending upon the dynamical system. The results will help us accurately model, interpret and simulate multiscale stochastic systems.     

Impulsive synchronization of networked nonlinear dynamical systems

In this Letter, we investigate the problem of impulsive synchronization of networked multi-agent systems, where each agent can be modeled as an identical nonlinear dynamical system. Firstly, an impulsive control protocol is designed for network with fixed topology based on the local information of agents. Then sufficient conditions are given to guarantee the synchronization of the networked nonlinear dynamical system by using algebraic graph theory and impulsive control theory. Furthermore, how to select the discrete instants and impulsive constants is discussed. The case that the topologies of the networks are switching is also considered. Numerical simulations show the effectiveness of our theoretical results.     

Boolean delay equations. II. Periodic and aperiodic solutions

Boolean delay equations (BDEs) areevolution equations for a vector of discrete variables x(t). The value of each componentX _i (t), 0 or 1. depends on previous values of all componentsx _j (t– t _ij ), x _i (t)=f _i (x₁(t–t _i1),...,x _n (t –t _in )). BDEs model the evolution of biological and physical systems with threshold behavior and nonlinear feedbacks. The delays model distinct interaction times between pairs of variables. In this paper, BDEs are studied by algebraic, analytic, and numerical methods. It is shown that solutions depend continuously on the initial data and on the delays. BDEs are classified intoconservative anddissipative. All BDEs with rational delays only haveperiodic solutions only. But conservative BDEs with rationally unrelated delays haveaperiodic solutions of increasing complexity. These solutions can be approximated arbitrarily well by periodic solutions of increasing period.Self-similarity andintermittency of aperiodic solutions is studied as a function of delay values, and certain number-theoretic questions related toresonances and diophantine approximation are raised. Period length is shown to be a lower semicontinuous function of the delays for a given BDE, and can be evaluated explicitly for linear equations. We prove that a BDE isstructurable stable if and only if it has eventually periodic solutions of bounded period, and if the length of initial transients is bounded. It is shown that, for dissipative BDEs, asymptotic solution behavior is typically governed by areduced BDE. Applications toclimate dynamics and other problems are outlined.     

Fractal characteristics of critical and localized states in incommensurate quantum systems

Incommensurate quantum systems with two competing periodicities exhibit metallic (with Bloch-type extended wave functions), insulating (with exponentially localized wave functions) as well ascritical (with fractal wave functions) phases. An exact renormalization method, which takes into account the inherent incommensurability, is used to obtain the phase diagram of various quantum models for the two-dimensional electron gas and for quantum spin chains in a magnetic field. In this approach, the scaling properties of the fractal eigenstates are characterized by a fixed point or a strange invariant set of the renormalization flow. One of our novel results is the existence of self-similar fluctuations in the localized states once the exponentially decaying envelope is factorized out. In almost all cases under investigation here, the universality classes can be broadly classified as those of the nearest-neighbor square or triangular lattices.     

The dynamics of coupled nonlinear model Boltzmann equations

A multispecies gas described by coupled nonlinear Boltzmann equations is studied as a dynamical system. Properties are determined of theN coupled nonlinear ODEs for the number densities obtained from the Boltzmann equations for the spatially uniform system ofN species undergoing binary scattering, removal, and regeneration in the presence of an external force field and a reservoir of background gas. The physically realizable setQ, the nonnegative cone in theN-dimensional phase space of species number densities, is established as invariant under the flow. The fixed-point equations for the ODEs are shown to be equivalent to 2 ^N linear systems, and conditions for the stability and instability of the fixed points are then established. Stable fixed points are demonstrated to exist inQ by showing that they enter via a sequence of transcritical bifurcations as physical parameters are varied. For the two-species case the typical global structure of the solutions is established. Various particular cases are described including one which possesses an infinite family of periodic solutions and one that depends delicately upon initial conditions due to a separatrix that separatesQ into two invariant sets.     

Probabilistic models for stochastic elliptic partial differential equations

Mathematical requirements that the random coefficients of stochastic elliptical partial differential equations must satisfy such that they have unique solutions have been studied extensively. Yet, additional constraints that these coefficients must satisfy to provide realistic representations for physical quantities, referred to as physical requirements, have not been examined systematically.     

Lie symmetries and conserved quantities of controllable nonholonomic dynamical systems

This paper concentrates on studying the Lie symmetries and conserved quantities of controllable nonholonomicdynamicM systems. Based on the infinitesimal transformation, we establish the Lie symmetric determining equationsand restrictive equations and give three definitions of Lie symmetries before the structure equations and conservedquantities of tile Lie symmetries are obtained. Then we make a study of the inverse problems. Finally, an example ispresented for illustrating the results.     

Stability of nonstationary states of classical,many-body dynamical systems

We summarize recent arguments which show that for a broad class of classical, many-body dynamical model systems with short-range interactions (such as coupled maps, cellular automata, or partial differential equations), collectively chaotic states—nonstationary states wherein some Fourier amplitude varies chaotically in time—cannot occur generically. While chaos occurs ubiquitously on alocal level in such systems, the macroscopic state of the system typically remains periodic or stationary. This implies that the dimensionD of chaotic (strange) attractors must diverge with the linear sizeL of the system likeD(L/C)^d ind space dimensions, where (<) is the spatial coherence length. We also summarize recent work which demonstrates that in spatially isotropic systems that have short-range interactions and evolve (like coupled maps) in discrete time, periodic states are never stable under generic conditions. In spatially anisotropic systems, however, short-range interactions that exploit the anisotropy and so allow for the stabilization of periodic states do exist.     

On diffusion equations for dynamical systems driven by noise

A unified formalism is presented to study Hamiltonian linear systems driven by noise. With this formalism, the phase averaging approximation, valid at weak noise, is easily performed. Already known results are straightforwardly recovered and new ones are obtained. After introducing this formalism on the exactly solvable one-degree-of-freedom problem with uncorrelated noise, one studies the corresponding exponentially correlated case. The validity of the approximate results thus obtained is considered by investigating the systematic weak-disorder expansion beyond the quasilinear approximation. In particular, it is argued that this expansion behaves uniformly for weak and large correlation time. The two-degrees-of-freedom problem is completely solved at the low-disorder approximation and this result is applied to the two-channel Anderson localization problem. The invariant measure and the two positive Lyapunov exponents are computed at all coupling between the channels. For systems withn degree of freedom the phase averaging leads to a Fokker-Planck equation for the measure in action space describing the system. However, it is argued that it is not solvable except in a special case which is explicitly displayed and solved. Nevertheless, in the large-n limit, it is possible to compute the largest Lyapunov exponent. Moreover, generalized Lyapunov exponents are calculated in this limit, and they do not exhibit a dispersion: in particular, log/log1, where is the energy of the system and where the brackets denote averaging over the noise. On the other hand, it is possible to compute at weak noise the sum of all the positive Lyapunov exponents. Taking into account all these results allows more insight on the whole spectrum of Lyapunov exponents.