Approximate symmetries and conservation laws for Itô and Stratonovich dynamical systems

A new definition for the approximate symmetries of Itô dynamical system is given. Determining systems of approximate symmetries for Itô and Stratonovich dynamical systems have been obtained. It has been shown that approximate conservation laws can be found from the approximate symmetries of stochastic dynamical systems which do not arise from a Hamiltonian. The results have been applied to an example.     

On the connections between symmetries and conservation rules of dynamical systems

The strict connection between Lie point‐symmetries of a dynamical system and its constants of motion is discussed and emphasized through old and new results. It is shown in particular how the knowledge of the symmetry of a dynamical system can allow us to obtain conserved quantities that are invariant under the symmetry. In the case of Hamiltonian dynamical systems, it is shown that if the system admits a symmetry of a ‘weaker’ type (specifically, a λ or a Λ‐symmetry), then the generating function of the symmetry is not a conserved quantity, but the deviation from the exact conservation is ‘controlled’ in a well‐defined way. Several examples illustrate the various aspects. Copyright © 2012 John Wiley & Sons, Ltd.     

A Method to Generate First Integrals from Infinitesimal Symmetries

We propose a method to construct first integrals of a dynamical system, starting with a given set of linearly independent infinitesimal symmetries. In the case of two infinitesimal symmetries, a rank two Poisson structure on the ambient space it is found, such that the vector field that generates the dynamical system, becomes a Poisson vector field. Moreover, the symplectic leaves and the Casimir functions of the associated Poisson manifold are characterized. Explicit conditions that guarantee Hamilton–Poisson realizations of the dynamical system are also given.     

Symmetries in (1+1)-dimensional Integrable System with Their Algebraic Structures and Conserved Quantities

In this paper we offer a general method or constructing symmetries and conserved quantities in the (1 + 1)-dimensional integrable system, prove the algebraic relations between symmetries, and what is more, give applications of this method in many integrable systems with physical significance.     

变质量完整力学系统的Lie对称与守恒量

研究变质量完整系统的Lie对称和守恒量。利用常微分方程在无限小变换下的不变性建立系统Lie对称的确定方程。给出结构方程和守恒量。举例说明结果的应用。     

准坐标下非完整力学系统的Lie对称性和守恒量

研究准坐标下非完整系统的Ｌｉｅ对称性,首先,对准坐标下非完整力学系统定义无限小变换生成元,由微分方程在无限小变换下的不变性,建立Ｌｉｅ对称性的确定方程,得到结构方程并求出守恒量;其次,研究上述问题的逆问题;根据已知积分求相应的Ｌｉｅ对称性,举例说明结果的应用。     

转动相对论系统的Lie对称性和守恒量

研究转动相对论性完整与非完整力学系统的Lie对称性和守恒量.定义转动相对论力学系统的无限小变换生成元,利用微分方程在无限小变换下的不变性,建立转动相对论性力学系统的Lie对称确定方程,得到结构方程和守恒量的形式,并给出应用实例.     

汽车车体振动系统的对称性与守恒量研究

用Lie群方法研究汽车车体振动系统的对称性,寻找其存在的守恒量.以汽车车体做上下垂直振动和绕其质心的前后俯仰振动,采用Lagrange函数的方法,构建汽车车体振动系统.以此系统为对象,引入Lie群方法,给出该振动系统的Noether对称性理论与Lie对称性理论;由此推导该汽车系统存在的Noether对称性与Lie对称性,并得到系统相应的的守恒量.该方法对车体振动问题提出了新的对称性解法,同时扩大了Lie群方法的应用范围.     

时滞Lagrange系统的Lie对称性与守恒量研究

研究了位形间中含单时滞参数的非保守力学系统的Lie对称性和守恒量。首先,利用含时滞的动力学Hamilton原理,建立了含时滞的非保守系统的分段Lagrange运动方程;其次,利用微分方程容许Lie群理论,得到系统的Lie对称确定方程;然后,根据对称性与守恒量之间的关系,通过构造结构方程,得到含时滞的非保守系统的Lie定理;最后,给出了两个具体的算例说明了方法的应用。结果表明:时滞参数的存在使非保守系统的Lagrange方程呈现分段特性,相应的Lie对称性确定方程的个数应是自由度数目的2倍,这对生成元函数提出了更高的限制,同时,守恒量呈现依赖速度项的分段表达。     

相空间中单面非Chataev型非完整系统的Lie对称性与守恒量

研究相空间中单面非Ｃｈｅｔａｅｖ型非完整系统的Ｌｉｅ对称性与守恒量．首先根据微分方程在无限小变换下的不变性建立Ｌｉｅ对称性所满足的确定方程和限制方程,给出结构方程和守恒量;其次讨论系统的Ｌｉｅ对称性逆问题;最后举一实例说明结果的应用．     

Birkhoff系统的一类新型守恒量

给出了Birkhoff系统的一类新型守恒量。首先,建立了Birkhoff系统的运动方程及其Mei对称性的定义和判据;其次,给出了系统的一类新型守恒量的存在定理,并导出了用于确定无限小生成元的广义Killing方程;最后,建立了守恒定理的逆定理     

Birkhoff系统的一类新型守恒量

给出了Birkhoff系统的一类新型守恒量。首先，建立了Birkhoff系统的运动方程及其Mei对称性的定义和判据；其次，给出了系统的一类新型守恒量的存在定理，并导出了用于确定无限小生成元的广义Killing方程；最后，建立了守恒定理的逆定理     

Exchange of conserved quantities,shock loci and Riemann problems

Systems of conservation laws admitting extensions, such as entropy density/flux functions, generate related systems obtained by exchanging the extension with one of the constituent equations. Often if not always, the smooth solutions of the two systems coincide, and weak solutions of one system containing only small discontinuities are approximate weak solutions of the other. The adiabatic approximation for the Euler system illustrates the utility of this procedure. Such an exchange of conserved quantities preserves hyperbolicity and genuine non‐linearity in the sense of Lax. On the other hand, the topological structure of the shock locus of a point in phase space and the solvability of Riemann problems in the large can be strongly affected. A discussion of when and how this occurs is given here. In this paper the exchange of conserved quantities is conveniently described by a simple homotopy in an extended version of the usual ‘symmetric variables’. A dynamical system in phase space is constructed, the trajectories of which describe the Hugoniot locus of a fixed point in phase space at each state of the homotopy. The appearance of critical points for this dynamical system is identified with the alteration of the topological structure of the Hugoniot locus by the exchange of conserved quantities. Copyright © 2001 John Wiley & Sons, Ltd.     

Symmetric spaces and Lie triple systems in numerical analysis of differential equations

A remarkable number of different numerical algorithms can be understood and analyzed using the concepts of symmetric spaces and Lie triple systems, which are well known in differential geometry from the study of spaces of constant curvature and their tangents. This theory can be used to unify a range of different topics, such as polar-type matrix decompositions, splitting methods for computation of the matrix exponential, composition of selfadjoint numerical integrators and dynamical systems with symmetries and reversing symmetries. The thread of this paper is the following: involutive automorphisms on groups induce a factorization at a group level, and a splitting at the algebra level. In this paper we will give an introduction to the mathematical theory behind these constructions, and review recent results. Furthermore, we present a new Yoshida-like technique, for self-adjoint numerical schemes, that allows to increase the order of preservation of symmetries by two units. The proposed techniques has the property that all the time-steps are positive.     

变质量非Четаев型非完整系统在相空间的Lie对称性与守恒量

在相空间引入无限小群变换,研究变质量非Четаев型非完整系统的Lie对称和守恒量.利用系统运动微分方程在无限小群变换下的不变性建立Lie对称的确定方程和限制方程,得到Lie对称的结构方程和守恒量,并举例说明结果的应用.     

二阶非完整力学系统的Lie对称性与守恒量

研究二阶非完整力学系统的Lie对称与守恒量.首先利用系统运动微分方程在无限小变换下的不变性建立Lie对称的确定方程和限制方程,得到Lie对称的结构方程和守恒量;其次研究上述问题的逆问题;最后举例说明结果的应用.     

Parareal Algorithms Applied to Stochastic Differential Equations with Conserved Quantities

In this paper, we couple the parareal algorithm with projection methods of the trajectory on a specific manifold, defined by the preservation of some conserved quantities of stochastic differential equations. First, projection methods are introduced as the coarse and fine propagators. Second, we apply the projection methods for systems with conserved quantities in the correction step of original parareal algorithm. Finally, three numerical experiments are performed by different kinds of algorithms to show the property of convergence in iteration, and preservation in conserved quantities of model systems.     

具有单面非完整约束的力学系统的Lie对称性与守恒量

研究具有单面非完整约束的力学系统的Ｌｉｅ对称性。给出由Ｌｉｅ对称性得到系统守恒量的条件和守恒量的形式,并研究上述问题的逆问题,即根据系统的已知积分来求相应的Ｌｉｅ对称性,最后举例说明结果的应用。     

Conservation laws and symmetries of semilinear radial wave equations

Classifications of symmetries and conservation laws are presented for a variety of physically and analytically interesting wave equations with power nonlinearities in n spatial dimensions: a radial hyperbolic equation, a radial Schrödinger equation and its derivative variant, and two proposed radial generalizations of modified Korteweg-de Vries equations, as well as Hamiltonian variants. The mains results classify all admitted local point symmetries and all admitted local conserved densities depending on up to first order spatial derivatives, including any that exist only for special powers or dimensions. All such cases for which these wave equations admit, in particular, dilational energies or conformal energies and inversion symmetries are determined. In addition, potential systems arising from the classified conservation laws are used to determine nonlocal symmetries and nonlocal conserved quantities admitted by these equations. As illustrative applications, a discussion is given of energy norms, conserved Hs norms, critical powers for blow-up solutions, and one-dimensional optimal symmetry groups for invariant solutions.     

The extremal index,hitting time statistics and periodicity

The extremal index appears as a parameter in Extreme Value Laws for stochastic processes, characterising the clustering of extreme events. We apply this idea in a dynamical systems context to analyse the possible Extreme Value Laws for the stochastic process generated by observations taken along dynamical orbits with respect to various measures. We derive new, easily checkable, conditions which identify Extreme Value Laws with particular extremal indices. In the dynamical context we prove that the extremal index is associated with periodic behaviour. The analogy of these laws in the context of hitting time statistics, as studied in the authors’ previous works on this topic, is explained and exploited extensively allowing us to prove, for the first time, the existence of hitting time statistics for balls around periodic points. Moreover, for very well behaved systems (uniformly expanding) we completely characterise the extremal behaviour by proving that either we have an extremal index less than 1 at periodic points or equal to 1 at any other point. This theory then also applies directly to general stochastic processes, adding both useful tools to identify the extremal index and giving deeper insight into the periodic behaviour it suggests.