Reversible maps and their symmetry lines

In this paper, we present some methods to determine whether a planar map is reversible. Using these methods, we show that four automorphisms are reversible including Cremona map, cubic Hénon map, Knuth map and McMillan map. Some of them are not polynomial automorphism. We give the recurrence formulas of their symmetry lines, draw their phase portraits and symmetry lines with MATLAB software. Some special properties of their symmetry lines are explained and their beauties are also visually displayed.     

Conserving algorithms for the dynamics of Hamiltonian systems on lie groups

Summary Three conservation laws are associated with the dynamics of Hamiltonian systems with symmetry: The total energy, the momentum map associated with the symmetry group, and the symplectic structure are invariant under the flow. Discrete time approximations of Hamiltonian flows typically do not share these properties unless specifically designed to do so. We develop explicit conservation conditions for a general class of algorithms on Lie groups. For the rigid body these conditions lead to a single-step algorithm that exactly preserves the energy, spatial momentum, and symplectic form. For homogeneous nonlinear elasticity, we find algorithms that conserve angular momentum and either the energy or the symplectic form.     

Gradient map of isoparametric polynomial and its application to Ginzburg-Landau system

In this note, we study properties of the gradient map of the isoparametric polynomial. For a given isoparametric hypersurface in sphere, we calculate explicitly the gradient map of its isoparametric polynomial which turns out many interesting phenomenons and applications. We find that it should map not only the focal submanifolds to focal submanifolds, isoparametric hypersurfaces to isoparametric hypersurfaces, but also map isoparametric hypersurfaces to focal submanifolds. In particular, it turns out to be a homogeneous polynomial automorphism on certain isoparametric hypersurface. As an immediate consequence, we get the Brouwer degree of the gradient map which was firstly obtained by Peng and Tang with moving frame method. Following Farina's construction, another immediate consequence is a counterexample of the Brézis question about the symmetry for the Ginzburg-Landau system in dimension 6, which gives a partial answer toward the Open problem 2 raised by Farina.     

Normal form theory for volume preserving maps

We discuss the theory of normal forms for volume preserving maps in the non-resonant case. The concept of normal forms is introduced by using the commutative properties with respect to some symmetry groups related to the resonances of the linear eigenvalues. The existence of formal solutions to the conjugation equation, which reduces a map to normal form, is discussed. We also analyze the asymptotic character of the normal forms series and we prove a general theorem which shows that the error between the normal form dynamics and the true dynamics can be exponentially small as a function of the radius of the chosen polydisk centered at the fixed point. As a consequence it is possible to construct analytic manifolds which are approximately invariant under the action of the initial map up to an error which is exponentially small. The connection with a possible KAM theory for volume-preserving maps is suggested. We also show that Noether's theorem can be generalized to volume preserving maps.     

Blow up and grazing collision in viscous fluid solid interaction systems

We investigate qualitative properties of strong solutions to a classical system describing the fall of a rigid ball under the action of gravity inside a bounded cavity filled with a viscous incompressible fluid. We prove contact between the ball and the boundary of the cavity implies blow up of strong solutions and such a contact has to occur in finite time under symmetry assumptions on the initial data.     

Random partition masking model for censored and masked competing risks data

We consider the parametric estimation with right-censored competing risks data and with masked failure cause. We propose a new model, called the random partition masking (RPM) model. The existing model based on the so called symmetry assumption, but the RPM model does not need the symmetry assumption. We propose a wide class of parametric distribution families of the failure time and cause, which does not need the assumption of independence between the components of the system. We also study the asymptotic properties of the maximum likelihood estimator under the new model, and apply our procedure to a medical and an industrial data sets.     

Decomposing a new nonlinear differential-difference system under a Bargmann implicit symmetry constraint

Firstly, a hierarchy of integrable lattice equations and its bi-Hamilt-onian structures are established by applying the discrete trace identity. Secondly, under an implicit Bargmann symmetry constraint, every lattice equation in the nonlinear differential-difference system is decomposed by an completely integrable symplectic map and a finite-dimensional Hamiltonian system. Finally, the spatial part and the temporal part of the Lax pairs and adjoint Lax pairs are all constrained as finite dimensional Liouville integrable Hamiltonian systems.     

Inner reflectors and non-orientable regular maps

Regular maps on non-orientable surfaces are considered with particular reference to the properties of inner reflectors, corresponding to symmetries of the 2-fold smooth orientable covering which project onto local reflections of the map itself. An example is given where no inner reflector is induced by an involution, and the existence of such involutions is related to questions of symmetry of coset diagrams for the symmetry group of the map.     

Regular reduction of controlled Hamiltonian system with symplectic structure and symmetry

In this paper, our goal is to study the regular reduction theory of regular controlled Hamiltonian (RCH) systems with symplectic structure and symmetry, and this reduction is an extension of regular symplectic reduction theory of Hamiltonian systems under regular controlled Hamiltonian equivalence conditions. Thus, in order to describe uniformly RCH systems defined on a cotangent bundle and on the regular reduced spaces, we first define a kind of RCH systems on a symplectic fiber bundle. Then we introduce regular point and regular orbit reducible RCH systems with symmetry by using momentum map and the associated reduced symplectic forms. Moreover, we give regular point and regular orbit reduction theorems for RCH systems to explain the relationships between RpCH-equivalence, RoCH-equivalence for reducible RCH systems with symmetry and RCH-equivalence for associated reduced RCH systems. Finally, as an application we regard rigid body and heavy top as well as them with internal rotors as the regular point reducible RCH systems on the rotation group SO(3) and on the Euclidean group SE(3), as well as on their generalizations, respectively, and discuss their RCH-equivalence. We also describe the RCH system and RCH-equivalence from the viewpoint of port Hamiltonian system with a symplectic structure.     

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.      

Harnack inequalities,exponential separation,and perturbations of principal Floquet bundles for linear parabolic equations

We consider the Dirichlet problem for linear nonautonomous second order parabolic equations with bounded measurable coefficients on bounded Lipschitz domains. Using a new Harnack-type inequality for quotients of positive solutions, we show that each positive solution exponentially dominates any solution which changes sign for all times. We then examine continuity and robustness properties of a principal Floquet bundle and the associated exponential separation under perturbations of the coefficients and the spatial domain.     

Reverse bifurcation and fractal of the compound logistic map

The nature of the fixed points of the compound logistic map is researched and the boundary equation of the first bifurcation of the map in the parameter space is given out. Using the quantitative criterion and rule of chaotic system, the paper reveal the general features of the compound logistic map transforming from regularity to chaos, the following conclusions are shown: (1) chaotic patterns of the map may emerge out of double-periodic bifurcation and (2) the chaotic crisis phenomena and the reverse bifurcation are found. At the same time, we analyze the orbit of critical point of the compound logistic map and put forward the definition of Mandelbrot–Julia set of compound logistic map. We generalize the Welstead and Cromer’s periodic scanning technology and using this technology construct a series of Mandelbrot–Julia sets of compound logistic map. We investigate the symmetry of Mandelbrot–Julia set and study the topological inflexibility of distributing of period region in the Mandelbrot set, and finds that Mandelbrot set contain abundant information of structure of Julia sets by founding the whole portray of Julia sets based on Mandelbrot set qualitatively.     

（E0，E）型渐近光滑映射和耗散波方程

本文在无穷维空间引入（Ｅ_０，Ｅ）型渐近光滑映射的概念，研究了其基本性质和变为Ｅ中渐近光滑映射的条件，我们证明了（Ｅ_０，Ｅ）型吸引子存在性定理和（Ｅ_０，Ｅ）型吸引子转化为Ｅ中吸引子的条件定理，所有结果都应用于一类耗散波方程渐近性态的研究。映射，吸引子，耗散波     

Creation of twistless circles in a model of stellar pulsations

In the present paper, we study the Poincaré map associated to a periodic perturbation, both in space and time, of a linear Hamiltonian system. The dynamical system embodies the essential physics of stellar pulsations and provides a global and qualitative explanation of the chaotic oscillations observed in some stars. We show that this map is an area preserving one with an oscillating rotation number function. The nonmonotonic property of the rotation number function induced by the triplication of the elliptic fixed point is superposed on the nonmonotonic character due to the oscillating perturbation. This superposition leads to the co-manifestation of generic phenomena such as reconnection and meandering, with the nongeneric scenario of creation of vortices. The nonmonotonic property due to the triplication bifurcation is shown to be different from that exhibited by the cubic Hénon map, which can be considered as the prototype of area preserving maps which undergo a triplication followed by the twistless bifurcation. Our study exploits the reversibility property of the initial system, which induces the time-reversal symmetry of the Poincaré map.     

On rotating black holes in equilibrium in general relativity

We study asymptotically flat axially symmetric stationary solutions of the Einstein vacuum equations. These represent rotating black holes in equilibrium. The equations reduce outside the axis of symmetry to a harmonic map problem into the hyperbolic plane, with prescribed rates of blow-up for the map on the axis and at infinity as boundary conditions. We prove existence and uniqueness of solutions in the case of zero total angular momentum.     

A family of integrable differential-difference equations and its Bargmann symmetry constraint

A family of integrable differential-difference equations is constructed through discrete zero curvature equation. The Hamiltonian structures of the resulting differential-difference equations are established by the discrete trace identity. The Bargmann symmetry constraint of the resulting family is presented. Under this symmetry constraint, every differential-difference equation in the resulting family is factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense.     

Model Dirac-Maxwell equations with pseudounitary symmetry

We introduce a new system of equations called a model system of Dirac-Maxwell equations, reproducing the main properties of the standard system. At the same time, the model system of equations differs from the standard system in several ways; in particular, it is a tensor system and has a new symmetry with respect to the pseudounitary group. We also propose a version of the model system of Dirac-Maxwell equations with local (gauge) pseudounitary symmetry. We show that any spinor solution of the standard system of Dirac-Maxwell equations can be obtained from the corresponding tensor solution of the model system. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 425–435, December, 2008.     

Symmetry breaking in a bull and bear financial market model

We investigate bifurcation structures in the parameter space of a one-dimensional piecewise linear map with two discontinuity points. This map describes endogenous bull and bear market dynamics arising from a simple asset-pricing model. An important feature of our model is that some speculators only enter the market if the price is sufficiently distant to its fundamental value. Our analysis starts with the investigation of a particular case in which the map is symmetric with respect to the origin, associated with equal market entry thresholds in the bull and bear market. We then generalize our analysis by exploring how novel bifurcation structures may emerge when the map’s symmetry is broken.     

Cyclic invariant sets for one class of maps

We consider the dynamical system that is determined by a multidimensional map with scalar type nonlinearity and a nonnegative matrix of special form. For this map we establish the bifurcation character for the location of cyclic invariant sets in the phase space of the system, determine their location and periods depending on the properties of the matrix.     

Dimension reduction for semidefinite programs via Jordan algebras

Mathematical Programming - We propose a new method for simplifying semidefinite programs (SDP) inspired by symmetry reduction. Specifically, we show if an orthogonal projection map satisfies...