Equivalence conditions for classes of linear and non-linear distributed parameter systems

Equivalence of certain classes of second-order non-linear distributed parameter systems and corresponding linear third-order systems is established through a differential transformation technique. As linear systems are amenable to analysis through existing techniques, this study is expected to offer a method of tackling certain classes of non-linear problems which may otherwise prove to be formidable in nature.     

Equivalent linear models for non-linear non-autonomous systems

Equivalence between a class of non-linear non-autonomous systems of second order and a linear model of lower order is established through a differential transformation relation. It is shown that this equivalence can be established only under a certain constraint on the non-linear functional parameters of the given system. The equivalence automatically leads to the first integral which then can be analyzed further to obtain the response of the system. The feasibility of obtaining closed form solutions through such analysis is illustrated by considering certain sub-classes of systems. Further, the practical value of the technique is demonstrated through an example.     

Conservation laws and hierarchies of potential symmetries for certain diffusion equations

We show that the so-called hidden potential symmetries considered in a recent paper [M.L. Gandarias, New potential symmetries for some evolution equations, Physica A 387 (2008) 2234-2242] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and collaborators [G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989; G.W. Bluman, G.J. Reid, S. Kumei, New classes of symmetries for partial differential equations, J. Math. Phys. 29 (1988) 806-811]. In fact, these are simplest potential symmetries associated with potential systems which are constructed with single conservation laws having no constant characteristics. Furthermore we classify the conservation laws for classes of porous medium equations, and then using the corresponding conserved (potential) systems we search for potential symmetries. This is the approach one needs to adopt in order to determine the complete list of potential symmetries. The provenance of potential symmetries is explained for the porous medium equations by using potential equivalence transformations. Point and potential equivalence transformations are also applied to deriving new results on potential symmetries and corresponding invariant solutions from known ones. In particular, in this way the potential systems, potential conservation laws and potential symmetries of linearizable equations from the classes of differential equations under consideration are exhaustively described. Infinite series of infinite-dimensional algebras of potential symmetries are constructed for such equations.     

SOME GENERAL EFFECTS OF STRONG HIGH-FREQUENCY EXCITATION: STIFFENING, BIASING AND SMOOTHENING

Mechanical high-frequency (HF) excitation provides a working principle behind many industrial and natural applications and phenomena. This paper concerns three particular effects of HF excitation, that may change the apparent characteristics of mechanical systems: (1) stiffening, by which the apparent linear stiffness associated with an equilibrium is changed, along with derived quantities such as stability and natural frequencies; (2) biasing by which the system is biased towards a particular state, static or dynamic, which does not exist or is unstable in the absence of the HF excitation; and (3) smoothening, referring to a tendency for discontinuities to be effectively “smeared out” by HF excitation. Illustrating first these effects for a few specific systems, analytical results are provided that quantify them for a quite general class of mechanical systems. This class covers systems that can be modelled by a finite number of second order ordinary differential equations, generally non-linear, with periodically oscillating excitation terms of high frequency and small amplitude. The results should be useful for understanding the effects in question in a broader perspective than is possible with specific systems, for calculating effects for specific systems using well-defined formulas, and for possibly designing systems that display prescribed characteristics in the presence of HF excitation.     

一种二阶混合有限体元格式的GAMG预条件子

针对一种含跳系数椭圆问题的二阶混合有限体元格式,讨论求解相应离散系统PGMRES法的预条件子构造问题.通过严格的理论分析,建立分层基下该二阶混合有限体元刚度矩阵和二次有限元刚度矩阵的谱等价关系,并利用关于二次有限元刚度矩阵的一种基于分层思想的GAMG预条件子,为二阶混合有限体元刚度矩阵设计一种高效GAMG预条件子.数值结果验证理论分析的正确性和新预条件子的高效性与稳定性.     

Differential transform method for solving linear and non-linear systems of partial differential equations

In this Letter, we propose a reliable algorithm to develop exact and approximate solutions for the linear and non-linear systems of partial differential equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and non-linear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.     

Bifurcations to divergence and flutter in flow-induced oscillations: A finite dimensional analysis

The behaviours of a pipe conveying fluid and a fluid loaded panel are studied from the viewpoint of differentiable dynamics. Non-linear terms are included and it is shown how the partial differential equation of motion can be recast, by Galerkin's method and modal truncation, in the form of an ordinary differential equation in Euclidean n-space. This evolution equation is then analysed qualitatively, attention being paid to bifurcations which occur as the control parameters of axial force and flow velocity are varied. Bifurcations of fixed points occur when at least one of the eigenvalues of the linearized evolution equation crosses the imaginary axis in the complex plane. In this situation, centre manifold theory can be used to extract a low dimensional subsystem which completely captures the local bifurcational behaviour. Such essential models enable the onset of divergence and flutter to be analysed relatively simply and the inclusion of non-linear terms permits the global study of post-bifurcational behaviour. The general approach is illustrated by analysis of two mode models of a pipe and of a panel and some important omissions in previous treatments of linear and undamped systems are discussed.     

Parity and time inversion symmetries of electromagnetic systems

By forming the intersections of the parity and time reversal equivalence classes of physical entities that are represented by differential forms and differential form densities, a number of subsets of discrete symmetry classes for electromagnetic systems can be generated. Only one of these subsets is consistent with elementary thermodynamic arguments for dissipative systems and at the same time yields the notion that both charge and mass are spacetime scalars. This subset is not in correspondence with the two self-consistent presentations that now are implied in the literature.     

Investigation of Q-S synchronization in coupled chaotic incommensurate fractional order systems

Chaos and synchronization in fractional order systems have received increasing attention in recent years. In this paper, the problem of Q-S synchronization for different dimensional incommensurate fractional order chaotic systems is investigated. Based on Laplace transform and stability theory of linear integer order differential systems, some synchronization schemes are designed to achieve Q-S synchronization between n-D and m-D incommensurate fractional order chaotic systems. Test problems and numerical simulations are used to show the effectiveness of the proposed approach.     

TIME INTEGRATION OF NON-LINEAR DYNAMIC EQUATIONS BY MEANS OF A DIRECT VARIATIONAL METHOD

Non-linear dynamic problems governed by ordinary (ODE) or partial differential equations (PDE) are herein approached by means of an alternative methodology. A generalized solution named WEM by the authors and previously developed for boundary value problems, is applied to linear and non-linear equations. A simple transformation after selecting an arbitrary interval of interest T allows using WEM in initial conditions problems and others with both initial and boundary conditions. When dealing with the time variable, the methodology may be seen as a time integration scheme. The application of WEM leads to arbitrary precision results. It is shown that it lacks neither numerical damping nor chaos which were found to be present with the application of some of the time integration schemes most commonly used in standard finite element codes (e.g., methods of central difference, Newmark, Wilson-θ, and so on). Illustrations include the solution of two non-linear ODEs which govern the dynamics of a single-degree-of-freedom (s.d.o.f.) system of a mass and a spring with two different non-linear laws (cubic and hyperbolic tangent respectively). The third example is the application of WEM to the dynamic problem of a beam with non-linear springs at its ends and subjected to a dynamic load varying both in space and time, even with discontinuities, governed by a PDE. To handle systems of non-linear equations iterative algorithms are employed. The convergence of the iteration is achieved by takingn partitions of T. However, the values of T/n are, in general, several times larger than the usual Δt in other time integration techniques. The maximum error (measured as a percentage of the energy) is calculated for the first example and it is shown that WEM yields an acceptable level of errors even when rather large time steps are used.     

Invariant noncanonical mappings of singular theories. II

In a paper of the same title published in Physical Revview Dit was shown that in singular theories (i.e. theories incorporating constraints) non-canonical infinitesimal mappings that map equivalence classes intact on each other lead to the divergence of a vector field associated with the mapping constant throughout the equivalence class. The infinitesimal mappings form the germ of the group of finite mappings of equivalence classes on each other that change the form of the symplectic tensor field. Any non-canonical coordinate system on phase space thus obtained defines a scalar density field that is constant over an equivalence class. The constant of the motion obtained earlier represents the infinitesimal deviation of this new field from unity.     

How particular is the physics of the free energy principle?

The free energy principle (FEP) states that any dynamical system can be interpreted as performing Bayesian inference upon its surrounding environment. Although, in theory, the FEP applies to a wide variety of systems, there has been almost no direct exploration or demonstration of the principle in concrete systems. In this work, we examine in depth the assumptions required to derive the FEP in the simplest possible set of systems – weakly-coupled non-equilibrium linear stochastic systems. Specifically, we explore (i) how general the requirements imposed on the statistical structure of a system are and (ii) how informative the FEP is about the behaviour of such systems. We discover that two requirements of the FEP – the Markov blanket condition (i.e. a statistical boundary precluding direct coupling between internal and external states) and stringent restrictions on its solenoidal flows (i.e. tendencies driving a system out of equilibrium) – are only valid for a very narrow space of parameters. Suitable systems require an absence of perception-action asymmetries that is highly unusual for living systems interacting with an environment. More importantly, we observe that a mathematically central step in the argument, connecting the behaviour of a system to variational inference, relies on an implicit equivalence between the dynamics of the average states of a system with the average of the dynamics of those states. This equivalence does not hold in general even for linear stochastic systems, since it requires an effective decoupling from the system's history of interactions. These observations are critical for evaluating the generality and applicability of the FEP and indicate the existence of significant problems of the theory in its current form. These issues make the FEP, as it stands, not straightforwardly applicable to the simple linear systems studied here and suggest that more development is needed before the theory could be applied to the kind of complex systems that describe living and cognitive processes.     

Periodic solution finder for an impact oscillator with a drift

In this paper, an efficient semi-analytical method is developed to compute periodic solutions for a new model of an impact oscillator with a drift, which explains the progression mechanism in vibro-impact systems and can be used to optimize their performance. The method constructs a periodic response assuming that each period is comprised of a sequence of distinct phases for which analytical solutions are known. For example, a period may consist of the following sequential phases: (I) contact with progression, (II) contact without progression, (III) no contact and (IV) contact without progression. Using this information, a system of four piecewise linear first order differential equations is transformed to a system of non-linear algebraic equations. The method allows one to accurately predict a range of control parameters for which the best progression rates are obtained.     

Three-loop vacuum integral with four-propagators using hypergeometry

A hypergeometric function is proposed to calculate the scalar integrals of Feynman diagrams.In this study,we verify the equivalence between the Feynman parametrization and the hypergeometric technique for the scalar integral of the three-loop vacuum diagram with four propagators.The result can be described in terms of generalized hypergeometric functions of triple variables.Based on the triple hypergeometric functions,we establish the systems of homogeneous linear partial differential equations(PDEs)satisfied by the scalar integral of three-loop vacuum diagram with four propagators.The continuation of the scalar integral from its convergent regions to entire kinematic domains can be achieved numerically through homogeneous linear PDEs by applying the element method.     

The Hamilton-Cartan formalism for higher order conserved currents: 1. Regular first order Lagrangians

The Hamilton–Cartan formalism for regular first order Lagrangian field theories is extended to deal with conserved currents which depend on higher order derivatives of the field variables. These conserved currents are characterized. Exterior differential systems I^{(k + 1)} and I equivalent to the k-th and infinite prolongations of the Euler-Lagrange equations are defined. It is shown that to each conserved current is associated an equivalence class of infinitesimal symmetries of I. Conserved charges are defined and a Poisson bracket is constructed by analogy with the usual definition. The sine-Gordon equation is treated briefly as an application of the formalism.     

EFFECTS OF BASE POINTS AND NORMALIZATION SCHEMES ON THE NON-LINEAR NORMAL MODES OF CONSERVATIVE SYSTEMS

In this paper, two factors that affect the behaviors of the non-linear normal modes (NNMs) of conservative vibratory systems are investigated. The first factor is the base points (which are equivalent to Taylor series expanding points) of the non-linear normal modes and the second one is the normalization schemes of the corresponding linear modes. For non-linear systems, in general only the approximated NNM manifolds are obtainable in practice, so different base points may lead to different forms of NNM oscillators and different normalization schemes lead to different forward and backward transformations which in turn affect the numerical computation errors. Three different kinds of base points and two different normalization schemes are adopted for comparison respectively. Two examples of non-linear systems with two and three degrees of freedom, respectively, are given as illustration. Simulations for various cases are made. The analysis and the simulation results indicated that, the best base points are the abstract base points determined via the linear normal mode, which would eliminate the third order terms containing velocity (for cubic systems) or quadratic terms (for quadratic systems) in equations of the NNM oscillators. A better invariance of the NNMs would also be maintained with such base points. The best scheme of normalization is the norm-one scheme that would minimize the numerical errors.     

Quasiclassical symmetry properties II

Proofs are given that the quasiclassical approach proposed previously is able to work in conjunction with the 1/N method. Accordingly, the expansion parameter k = 2l + N − a of the shifted 1/N method should be chosen, order by order, such that the sum of corrections to the zeroth-order result vanishes. This interconnection criterion leads to order-dependent algebraic equations for the parameter k. In turn, the underlying phase-space quantum becomes just half the parameter k implied in this way. Alternative fixing conditions, based on the selection of a dominant potential term, can also be proposed. Quasiclassical symmetry transformations leaving corresponding equivalence classes of Hamiltonian forms invariant are established. Then energy levels and couplings characterizing such Hamiltonians become subject to mutual conversions. Scaling properties of the phase-space quantum are discussed. In addition, this quantum exhibits a covariance behaviour with respect to the quasiclassical symmetry transformations mentioned above. Critical coupling for several short-range potentials are given to first order. Generalizations to resonances and nonlinear quantum-mechanical potentials are also made. Except for dyons, we restrict ourselves to spherically symmetric nonrelativistic Hamiltonians.     

Classification of Dark Modified KdV Equation

The dark Korteweg-de Vries(KdV) systems are defined and classified by Kupershmidt sixteen years ago. However, there is no other classifications for other kinds of nonlinear systems. In this paper, a complete scalar classification for dark modified KdV(MKdV) systems is obtained by requiring the existence of higher order differential polynomial symmetries. Different to the nine classes of the dark KdV case, there exist twelve independent classes of the dark MKdV equations. Furthermore, for the every class of dark MKdV system, there is a free parameter. Only for a fixed parameter, the dark MKdV can be related to dark KdV via suitable Miura transformation. The recursion operators of two classes of dark MKdV systems are also given.     

Analysis of the faster-than-Nyquist optimal linear multicarrier system

Faster-than-Nyquist signalization enables a better spectral efficiency at the expense of an increased computational complexity. Regarding multicarrier communications, previous work mainly relied on the study of non-linear systems exploiting coding and/or equalization techniques, with no particular optimization of the linear part of the system. In this article, we analyze the performance of the optimal linear multicarrier system when used together with non-linear receiving structures (iterative decoding and direct feedback equalization), or in a standalone fashion. We also investigate the limits of the normality assumption of the interference, used for implementing such non-linear systems. The use of this optimal linear system leads to a closed-form expression of the bit-error probability that can be used to predict the performance and help the design of coded systems. Our work also highlights the great performance/complexity trade-off offered by decision feedback equalization in a faster-than-Nyquist context.