A technique to construct symmetric variable-stepsize linear multistep methods for second-order systems

Some previous works show that symmetric fixed- and variable-stepsize linear multistep methods for second-order systems which do not have any parasitic root in their first characteristic polynomial give rise to a slow error growth with time when integrating reversible systems. In this paper, we give a technique to construct variable-stepsize symmetric methods from their fixed-stepsize counterparts, in such a way that the former have the same order as the latter. The order and symmetry of the integrators obtained is proved independently of the order of the underlying fixed-stepsize integrators. As this technique looks for efficiency, we concentrate on explicit linear multistep methods, which just make one function evaluation per step, and we offer some numerical comparisons with other one-step adaptive methods which also show a good long-term behaviour.

     

CHARACTERIZATIONS OF SYMMETRIC MULTISTEP RUNGE-KUTTA METHODS

Some characterizations for symmetric multistep Runge-Kutta(RK) methods are obtained. Symmetric two-step RK methods with one and two-stages are presented. Numerical examples show that symmetry of multistep RK methods alone is not sufficient for long time integration for reversible Hamiltonian systems. This is an important difference between one-step and multistep symmetric RK methods.     

The reversibility and the center problem

In this work we study the narrow relation between reversibility and the center problem and also between reversibility and the integrability problem. It is well known that an analytic system having either a non-degenerate or nilpotent center at the origin is analytically reversible or orbitally analytically reversible, respectively. In this paper we prove the existence of a smooth map that transforms an analytic system having a degenerate center at the origin with either an analytic first integral or a C^∞ inverse integrating factor into a reversible linear system (after rescaling the time). Moreover, if the degenerate center has an analytic or a C^∞ reversing symmetry, then the transformed system by the map also has a reversing symmetry. From the knowledge of a first integral near the center we give a procedure to detect reversing symmetries.     

Analysis of variable-stepsize linear multistep methods with special emphasis on symmetric ones

In this paper we deal with several issues concerning variable-stepsize linear multistep methods. First, we prove their stability when their fixed-stepsize counterparts are stable and under mild conditions on the stepsizes and the variable coefficients. Then we prove asymptotic expansions on the considered tolerance for the global error committed. Using them, we study the growth of error with time when integrating periodic orbits. We consider strongly and weakly stable linear multistep methods for the integration of first-order differential systems as well as those designed to integrate special second-order ones. We place special emphasis on the latter which are also symmetric because of their suitability when integrating moderately eccentric orbits of reversible systems. For these types of methods, we give a characterization for symmetry of the coefficients, which allows their construction, and provide some numerical results for them.

     

Symmetry structure of the hyperbolic bifurcation without reflection of periodic orbits in the standard map

For the area preserving maps, the linearized tangent map determines the stability of the fixed point. When the trace of the tangent map is less than −2, the fixed point is inversion hyperbolic, thus the subsequent points of mapping alternate across the destabilized fixed point. That is to say, the fixed point undergoes periodic doubling bifurcation. While for the trace of the tangent map is larger than +2, the fixed point undergoes the hyperbolic bifurcation without reflection. Here, the processes of the hyperbolic bifurcation without reflection in the standard map have been examined in terms of the higher order symmetry in the momentum inversion. It is shown that the higher order symmetry lines approach asymptotically to the separatrix of the hyperbolic fixed point, and the existing symmetry lines cannot determine the structure of the periodic islands born after the hyperbolic bifurcation without reflection.     

Classical Lie point symmetry analysis of nonlinear diffusion equations describing thermal energy storage

In this paper, we employed the linear transformation group approach to time dependent nonlinear diffusion equations describing thermal energy storage problem. Symmetry analysis of the governing equation resulted in admitted large Lie symmetry algebras for some special cases of the arbitrary constants and the source term. Some transformations that lead to equations with fewer arbitrary parameters are applied and classical Lie point symmetry methods are employed to analyze the transformed equations. Some symmetry reductions are performed and wherever possible the reduced ordinary differential equations are completely solved subject to realistic boundary conditions.     

Geodesic flow on three-dimensional ellipsoids with equal semi-axes

Following on from our previous study of the geodesic flow on three dimensional ellipsoid with equal middle semi-axes, here we study the remaining cases: Ellipsoids with two sets of equal semi-axes with SO(2) × SO(2) symmetry, ellipsoids with equal larger or smaller semiaxes with SO(2) symmetry, and ellipsoids with three semi-axes coinciding with SO(3) symmetry. All of these cases are Liouville-integrable, and reduction of the symmetry leads to singular reduced systems on lower-dimensional ellipsoids. The critical values of the energy-momentum maps and their singular fibers are completely classified. In the cases with SO(2) symmetry there are corank 1 degenerate critical points; all other critical points are non-degenreate. We show that in the case with SO(2) × SO(2) symmetry three global action variables exist and the image of the energy surface under the energy-momentum map is a convex polyhedron. The case with SO(3) symmetry is non-commutatively integrable, and we show that the fibers over regular points of the energy-casimir map are T ² bundles over S ².      

Hamilton–Pontryagin Integrators on Lie Groups Part I: Introduction and Structure-Preserving Properties

In this paper, structure-preserving time-integrators for rigid body-type mechanical systems are derived from a discrete Hamilton–Pontryagin variational principle. From this principle, one can derive a novel class of variational partitioned Runge–Kutta methods on Lie groups. Included among these integrators are generalizations of symplectic Euler and Störmer–Verlet integrators from flat spaces to Lie groups. Because of their variational design, these integrators preserve a discrete momentum map (in the presence of symmetry) and a symplectic form. In a companion paper, we perform a numerical analysis of these methods and report on numerical experiments on the rigid body and chaotic dynamics of an underwater vehicle. The numerics reveal that these variational integrators possess structure-preserving properties that methods designed to preserve momentum (using the coadjoint action of the Lie group) and energy (for example, by projection) lack.     

Fibonacci迹映射的不变锥面

该文利用［１］中关于可反转映射的ＫＡＭ定理,证明一定条件下Ｒ~３上的可反转映射在下动点附近存在不变锥面．并验证Ｆｉｂｏｎａｃｃｉ迹映射中存在这种不变锥面．     

Two novel methods for vibration diagnosis to characterize non-linear response

Poincaré maps have been proved to be a valuable tool in the analysis of non-linear dynamical systems, which usually reduce a continuous phase flow into a two-dimensional discrete map. However, they may be inconvenient for reflecting some characteristics of the system response. In this paper, two novel methods, using the period sampling peak-to-peak value (PSP) diagram and the modified Poincaré map, are presented for characterizing different types of non-linear response. These two methods take advantage of some parameters of the response, such as the peak-to-peak value within an exterior excitation period and the mean value of the displacement. In the PSP diagram method, a two-dimensional graph is plotted by taking the peak-to-peak value as ordinate and the sequential periodically sampling number as abscissa. On the other hand, the modified Poincaré map takes the mean value of the velocity within an exterior excitation period as ordinate and the relevant mean value of the displacement as abscissa. The non-linear responses of a Duffing system, a pendulum with circular motion support and an oscillating circuit are studied by these methods. We also studied the intermittent chaos of the Lorenz system by the PSP diagram method. The PSP diagram is a set of mapping points, which form: a straight line for a one-period response; multi-straight lines for a multi-period response; orderly periodic curves for a quasi-period response; long lines interrupted by transitoriness confusion points for intermittent chaos; and totally out-of-order points for chaos. The figures for the modified Poincaré maps for the period, multi-period, quasi-period responses and chaos are almost identical to those for the Poincaré maps, but the modified maps take more sampling points and can reflect the mean values of the responses. Some numerical results are given based on these methods to show their efficiency in distinguishing different non-linear responses.     

On the structure of additive systems of integers

A sum-and-distance system is a collection of finite sets of integers such that the sums and differences formed by taking one element from each set generate a prescribed arithmetic progression. Such systems, with two component sets, arise naturally in the study of matrices with symmetry properties and consecutive integer entries. Sum systems are an analogous concept where only sums of elements are considered. We establish a bijection between sum systems and sum-and-distance systems of corresponding size, and show that sum systems are equivalent to principal reversible cuboids, which are tensors with integer entries and a symmetry of ‘reversible square’ type. We prove a structure theorem for principal reversible cuboids, which gives rise to an explicit construction formula for all sum systems in terms of joint ordered factorisations of their component set cardinalities.

     

Numerical solution of singular regular boundary value problems by pole detection with qd-algorithm

In this paper, we present a method using qd-algorithm combined with finite difference method for numerically solving singular boundary value problems for certain ordinary differential equation having singular coefficients. (We can also use qd-algorithm combined with other numerical methods.)These problems arise when reducing partial differential equation to ordinary differential equation by physical symmetry. Some illustrative examples are given to demonstrate the efficiency and high accuracy of the proposed method and compare it to the numerical results made with other methods.     

Decompositions of analytic 1-manifolds

In an author’s previous work, analytic 1-submanifolds had been classified w.r.t. their symmetry under a given regular and separately analytic Lie group action on an analytic manifold. It was shown that such an analytic 1-submanifold is either free or (via the exponential map) analytically diffeomorphic to the unit circle or an interval. In this paper, we show that each free analytic 1-submanifold is discretely generated by the symmetry group, i.e., naturally decomposes into countably many symmetry free segments that are mutually and uniquely related by the Lie group action. This is shown under the same assumptions that were used in the author’s previous work to prove analogous decomposition results for analytic immersive curves. Together with the results obtained there, this completely classifies 1-dimensional analytic objects (analytic curves and analytic 1-submanifolds) w.r.t. their symmetry under a given regular and separately analytic Lie group action.     

热方程的非古典势对称群与不变解

主要研究了热方程与波方程的非古典势对称群生成元及相应的群不变解．研究表明对于守恒形式的偏微分方程,可通过其伴随系统求得的非古典势对称群生成元来构造其显式解．这些显式解不能由方程本身的Lie对称群生成元或Lie-B?cklund对称群生成元构造得到．     

Higher-order optimality conditions for strict and weak efficient solutions in set-valued optimization

In this paper, we introduce a notion of higher-order Studniarski epiderivative of a set-valued map and study its properties. Then, we discuss their applications to optimality conditions in set-valued optimization. Higher-order optimality conditions for strict and weak efficient solutions of a constrained set-valued optimization problem are established. Some remarks on the existing results in the literature are given from our results.     

A novel image encryption algorithm using chaos and reversible cellular automata

In this paper, a novel image encryption scheme is proposed based on reversible cellular automata (RCA) combining chaos. In this algorithm, an intertwining logistic map with complex behavior and periodic boundary reversible cellular automata are used. We split each pixel of image into units of 4 bits, then adopt pseudorandom key stream generated by the intertwining logistic map to permute these units in confusion stage. And in diffusion stage, two-dimensional reversible cellular automata which are discrete dynamical systems are applied to iterate many rounds to achieve diffusion on bit-level, in which we only consider the higher 4 bits in a pixel because the higher 4 bits carry almost the information of an image. Theoretical analysis and experimental results demonstrate the proposed algorithm achieves a high security level and processes good performance against common attacks like differential attack and statistical attack. This algorithm belongs to the class of symmetric systems.     

Two Limit Cycles in Liénard Piecewise Linear Differential Systems

Some techniques for studying the existence of limit cycles for smooth differential systems are extended to continuous piecewise linear differential systems. Rigorous new results are provided on the existence of two limit cycles surrounding the equilibrium point at the origin for systems with three zones separated by two parallel straight lines without symmetry. As a relevant application, it is shown the existence of bistable regimes in an asymmetric memristor-based electronic oscillator.

     

TheR-function method in boundary-value problems with geometric and physical symmetry

This paper is devoted to developing constructive methods of the theory of R-functions. It gives the first discussion of methods of constructing the equations of loci with symmetry of translation type for duplicated domains not separated by lines and with point symmetry of cyclic type for regions both possessing and lacking axial symmetry. The possible applications are illustrated by numerous examples carried out using the POLE system, including solutions of boundary-value problems involving the influence of cyclically located circular and star-shaped insulators on an electric field. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 146–159.     

图象的自反与渐近自反

本文对B（X,Y）的自反子空间及渐近自反子空间上的映射（或映射组）,分别给出了一些判别它们的图象（或联合图象）仍为自反及渐近自反的方法。     

The hartman-grobman theorem for reversible systems on banach spaces

Summary In this note we give a version of the Hartman-Grobman Theorem for reversible systems defined on a Banach space. We prove that the homeomorphism that reduces the nonlinear system to the linearized one preserves the symmetry. Applications to coupled nonlinear second-order ordinary differential equations and to coupled nonlinear wave equations are discussed.