Isotropy of Angular Frequencies and Weak Chimeras with Broken Symmetry

The notion of a weak chimeras provides a tractable definition for chimera states in networks of finitely many phase oscillators. Here, we generalize the definition of a weak chimera to a more general class of equivariant dynamical systems by characterizing solutions in terms of the isotropy of their angular frequency vector—for coupled phase oscillators the angular frequency vector is given by the average of the vector field along a trajectory. Symmetries of solutions automatically imply angular frequency synchronization. We show that the presence of such symmetries is not necessary by giving a result for the existence of weak chimeras without instantaneous or setwise symmetries for coupled phase oscillators. Moreover, we construct a coupling function that gives rise to chaotic weak chimeras without symmetry in weakly coupled populations of phase oscillators with generalized coupling.     

Dynamical Symmetry Breaking in Geometrodynamics

Using a first-order perturbative formulation, we analyze the local loss of symmetry when a source of electromagnetic and gravitational fields interacts with an agent that perturbs the original geometry associated with the source. We had proved that the local gauge groups are isomorphic to local groups of transformations of special tetrads. These tetrads define two orthogonal planes at every point in space–time such that every vector in these local planes is an eigenvector of the Einstein–Maxwell stress–energy tensor. Because the local gauge symmetry in Abelian or even non-Abelian field structures in four-dimensional Lorentzian space–times is manifested by the existence of local planes of symmetry, the loss of symmetry is manifested by a tilt of these planes under the influence of an external agent. In this strict sense, the original local symmetry is lost. We thus prove that the new planes at the same point after the tilting generated by the perturbation correspond to a new symmetry. Our goal here is to show that the geometric manifestation of local gauge symmetries is dynamical. Although the original local symmetries are lost, new symmetries arise. This is evidence for a dynamical evolution of local symmetries. We formulate a new theorem on dynamical symmetry evolution. The proposed new classical model can be useful for better understanding anomalies in quantum field theories.     

Quasi-Hamiltonian structure and Hojman construction

Given a smooth vector field Γ and assuming the knowledge of an infinitesimal symmetry X, Hojman [S. Hojman, The construction of a Poisson structure out of a symmetry and a conservation law of a dynamical system, J. Phys. A Math. Gen. 29 (1996) 667-674] proposed a method for finding both a Poisson tensor and a function H such that Γ is the corresponding Hamiltonian system. In this paper, we approach the problem from geometrical point of view. The geometrization leads to the clarification of several concepts and methods used in Hojman's paper. In particular, the relationship between the nonstandard Hamiltonian structure proposed by Hojman and the degenerate quasi-Hamiltonian structures introduced by Crampin and Sarlet [M. Crampin, W. Sarlet, Bi-quasi-Hamiltonian systems, J. Math. Phys. 43 (2002) 2505-2517] is unveiled in this paper. We also provide some applications of our construction.     

Conserved Quantities and Symmetries Related to Stochastic Dynamical Systems

The present article focuses on the three topics related to the notions of "conserved quantities" and "symmetries" in stochastic dynamical systems described by stochastic differential equations of Stratonovich type. The first topic is concerned with the relation between conserved quantities and symmetries in stochastic Hamilton dynamical systems, which is established in a way analogous to that in the deterministic Hamilton dynamical theory. In contrast with this, the second topic is devoted to investigate the procedures to derive conserved quantities from symmetries of stochastic dynamical systems without using either the Lagrangian or Hamiltonian structure. The results in these topics indicate that the notion of symmetries is useful for finding conserved quantities in various stochastic dynamical systems. As a further important application of symmetries, the third topic treats the similarity method to stochastic dynamical systems. That is, it is shown that the order of a stochastic system can be reduced, if the system admits symmetries. In each topic, some illustrative examples for stochastic dynamical systems and their conserved quantities and symmetries are given.     

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.     

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.     

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.     

Invariances and Conservation Laws of the Korteweg-de Vries Equation

Taking the Korteweg-de Vries equation as an example of a soliton system, a connection between conservation laws and infinitesimal transformations (symmetries) is investigated. It is shown that there exist an infinite number of form-invariant infinitesimal transformations. A one-to-one correspondence between conservation laws and infinitesimal transformations is established explicitly. Then it is concluded that the Korteweg-de Vries equation has an infinite number of conservation laws, each of which corresponds to the symmetry of the system. Two methods of weaving an infinitesimal transformation into a finite one are presented; one of them is the extension of the Lie transformation to the nonlinear case, and the other is a transformation reducible to the Backlund transformation. Furthermore, it is found that Hamiltonian formalism naturally leads to the quantization of the wave field. During the discussions, a number of conjectures and theorems which have been proposed in earlier works are confirmed by using the infinitesimal transformation extensively.     

Contact symmetries of the Helmholtz form

In this paper, vector fields which are symmetries of the contact ideal are studied. It is shown that contact symmetries of the Helmholtz form transform a dynamical form to a dynamical form which is variational (i.e. comes as the Euler-Lagrange form from a Lagrangian). The case of dynamical forms representing first-order classes in the variational sequence is analysed in detail, which means, by the variational sequence theory, that systems of ordinary differential equations of order ?3 are concerned.     

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.     

相对论Birkhoff系统的形式不变性与Noether守恒量

研究相对论Birkhoff系统的形式不变性,寻求系统的守恒量。在群的无限小变换下,给出相对论Birkhoff系统的形式不变性的定义和判剧。基于相对论Pfaff-Birkhoff-D'Alembert原理在群的无限小变换下的变形形式,建立相对论Birkhoff系统的Noether对称性理论。通过研究形式不变性与Noether对称性之间的关系,得到相对论Birkhoff系统的守恒量。研究结果表明:在一定的条件下,相对论Birkhoff系统的形式不变性导致Noether对称性的守恒量。     

Estimating Stochastic Dynamical Systems Driven by a Continuous-Time Jump Markov Process

We discuss the use of a continuous-time jump Markov process as the driving process in stochastic differential systems. Results are given on the estimation of the infinitesimal generator of the jump Markov process, when considering sample paths on random time intervals. These results are then applied within the framework of stochastic dynamical systems modeling and estimation. Numerical examples are given to illustrate both consistency and asymptotic normality of the estimator of the infinitesimal generator of the driving process. We apply these results to fatigue crack growth modeling as an example of a complex dynamical system, with applications to reliability analysis.      

Computation of local symmetries of second-order ordinary differential equations by the Cartan equivalence method

The Cartan equivalence method is used to find out if a given equation has a nontrivial Lie group of point symmetries. In particular, we compute invariants that permit one to recognize equations with a three-dimensional symmetry group. An effective method to transform the Lie system (the system of partial differential equations to be satisfied by the infinitesimal point symmetries) into a formally integrable form is given. For equations with a three-dimensional symmetry group, the formally integrable form of the Lie system is found explicitly. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 75–91, July, 1996.     

一类微分-差分方程组的内禀对称和等价群变换

研究一类微分-差分方程组的对称和等价群变换.采取内禀的无穷小算子方法,给出了方程组的内禀对称和等价群变换.为结合抽象Lie代数结构,给方程完全分类提供了理论基础.     

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。     

Symmetries, Conservation Laws, and Cohomology of Maxwell's Equations Using Potentials

New nonlocal symmetries and conservation laws are derived for Maxwell's equations in 3 + 1 dimensional Minkowski space using a covariant system of joint vector potentials for the electromagnetic tensor field and its dual. A key property of this system, as well as of this class of new symmetries and conservation laws, is their invariance under the duality transformation that exchanges the electromagnetic field with its dual. (In contrast the standard potential system using a single vector potential is not duality-invariant.) The nonlocal symmetries of Maxwell's equations come from an explicit classification of all symmetries of a certain natural geometric form admitted by the joint potential system in Lorentz gauge. In addition to scaling and duality-rotation symmetries, and the well-known Poincaré and dilation symmetries which involve homothetic Killing vectors, the classification yields new geometric symmetries involving Killing–Yano tensors related to rotations/boosts and inversions. The nonlocal conservation laws of Maxwell's equations are constructed from these geometric symmetries by applying a conserved current formula that uses the joint potentials and directly generates conservation laws from any (local or nonlocal) symmetries of Maxwell's equations. This formula is shown to arise through a series of mappings that relate, respectively, symmetries/adjoint-symmetries of the joint potential system and adjoint-symmetries/symmetries of Maxwell's equations. The mappings are derived as by-products of the study of cohomology of closed one-forms and two-forms locally constructed from the electromagnetic field and its derivatives to any finite order for all solutions of Maxwell's equations. In particular it is shown that the only nontrivial cohomology consists of the electromagnetic field (two-form) itself as well as its dual (two-form), and that this two-form cohomology is killed by the introduction of corresponding potentials.     

Parametric Representation of Infinitesimal Generators on the Polydisk

In this paper, analogues of the Berkson–Porta formula for the infinitesimal generators of one-parameter semigroup of holomorphic self-maps on the polydisk are obtained. We give a necessary and sufficient condition for a holomorphic vector field to be an infinitesimal generator which improves the theorem given by Contreras, de Fabritiis and Díaz-Madrigal.     

Symmetries of PDE systems and correspondences between jet spaces

Let M be a manifold. A PDE system can be prolonged to another one R^?⊆T^?M (Jiménez et al. (2005) [10]). In analogy with the higher-order symmetries, symmetries of R^? will be called higher-dimensional symmetries of R. For a broad class of PDE systems we prove that every (infinitesimal or finite) symmetry of R comes from another one of R^?. We show that R^? does not have internal (infinitesimal) symmetries (modulo trivial symmetries). This fact allows us, in the infinitesimal case, to compute the internal symmetries of R as external symmetries of R^?. We also give an algorithmic method to obtain solutions of R invariant by a given internal symmetry.     

Affine infinitesimal generators of semigroups on the polydisk

This paper concerns affine infinitesimal generators of continuous semigroups of holomorphic functions on the polydisk. An easy-to-check necessary and sufficiently condition for an affine vector field to be an infinitesimal generator is given.     

Solving Optimal Control Problems by Exploiting Inherent Dynamical Systems Structures

Computing globally efficient solutions is a major challenge in optimal control of nonlinear dynamical systems. This work proposes a method combining local optimization and motion planning techniques based on exploiting inherent dynamical systems structures, such as symmetries and invariant manifolds. Prior to the optimal control, the dynamical system is analyzed for structural properties that can be used to compute pieces of trajectories that are stored in a motion planning library. In the context of mechanical systems, these motion planning candidates, termed primitives, are given by relative equilibria induced by symmetries and motions on stable or unstable manifolds of e.g. fixed points in the natural dynamics. The existence of controlled relative equilibria is studied through Lagrangian mechanics and symmetry reduction techniques. The proposed framework can be used to solve boundary value problems by performing a search in the space of sequences of motion primitives connected using optimized maneuvers. The optimal sequence can be used as an admissible initial guess for a post-optimization. The approach is illustrated by two numerical examples, the single and the double spherical pendula, which demonstrates its benefit compared to standard local optimization techniques.