Opers and theta functions

We construct natural maps (the Klein and Wirtinger maps) from moduli spaces of semistable vector bundles over an algebraic curve X to affine spaces, as quotients of the nonabelian theta linear series. We prove a finiteness result for these maps over generalized Kummer varieties (moduli space of torus bundles), leading us to conjecture that the maps are finite in general. The conjecture provides canonical explicit coordinates on the moduli space. The finiteness results give low-dimensional parametrizations of Jacobians (in for generic curves), described by 2Θ functions or second logarithmic derivatives of theta.We interpret the Klein and Wirtinger maps in terms of opers on X. Opers are generalizations of projective structures, and can be considered as differential operators, kernel functions or special bundles with connection. The matrix opers (analogues of opers for matrix differential operators) combine the structures of flat vector bundle and projective connection, and map to opers via generalized Hitchin maps. For vector bundles off the theta divisor, the Szegö kernel gives a natural construction of matrix oper. The Wirtinger map from bundles off the theta divisor to the affine space of opers is then defined as the determinant of the Szegö kernel. This generalizes the Wirtinger projective connections associated to theta characteristics, and the associated Klein bidifferentials.     

Uniform Boundedness of Iterates of Analytic Mappings Implies Linearization: a Simple Proof and Extensions

A well-known result in complex dynamics shows that if the iterates of an analytic map are uniformly bounded in a complex domain, then the map is analytically conjugate to a linear map. We present a simple proof of this result in any dimension. We also present several generalizations and relations to other results in the literature.     

Polyhomogeneous maps and best local approximations of degree α

Contemporary investigations in the theory of nonlinear integral operators ([9], [10]) and in the differential calculus ([5], [11]) have led to generalizations of the notion of a polynomial map between two vector spaces. This article studies basic properties of such so-called polyhomogeneous maps. Our initial point of reference is the recent study of polynomial maps by Bochnak and Siciak [2]. Our examination of continuity properties leads to new characterizations of braked spaces and sequential spaces. Then, we turn to the polyhomogeneous approximating maps studied by Melamed and Perov [9] and Moore and Nashed [10]. We present some generalizations of the results of [9] and [10] and then go on to study permanence properties of such approximations.     

Chaos in the dynamics generated by sequences of maps, with applications to chaotic advection in flows with aperiodic time dependence

In this paper we develop analytical techniques for proving the existence of chaotic dynamics in systems where the dynamics is generated by infinite sequences of maps. These are generalizations of the Conley-Moser conditions that are used to show that a (single) map has an invariant Cantor set on which it is topologically conjugate to a subshift on the space of symbol sequences. The motivation for developing these methods is to apply them to the study of chaotic advection in fluid flows arising from velocity fields with aperiodic time dependence, and we show how dynamics generated by infinite sequences of maps arises naturally in that setting. Our methods do not require the existence of a homoclinic orbit in order to conclude the existence of chaotic dynamics. This is important for the class of fluid mechanical examples considered since one cannot readily identify a homoclinic orbit from the structure of the equations.¶We study three specific fluid mechanical examples: the Aref blinking vortex flow, Samelson's tidal advection model, and Min's rollup-merge map that models kinematics in the mixing layer. Each of these flows is modelled as a type of "blinking flow", which mathematically has the form of a linked twist map, or an infinite sequence of linked twist maps. We show that the nature of these blinking flows is such that it is possible to have a variety of "patches" of chaos in the flow corresponding to different length and time scales.     

Relative growth rate and contact Banach–Mazur distance

Geometriae Dedicata - Dilation surfaces are generalizations of translation surfaces where the geometric structure is modelled on the complex plane up to affine maps whose linear part is real. They...     

Entropy conjecture for continuous maps of nilmanifolds

In 1974 Michael Shub asked the following question [29]: When is the topological entropy of a continuous mapping of a compact manifold into itself is estimated from below by the logarithm of the spectral radius of the linear mapping induced in the cohomologies with real coefficients? This estimate has been called the Entropy Conjecture (EC). In 1977 the second author and Micha? Misiurewicz proved [23] that EC holds for all continuous mappings of tori. Here we prove EC for all continuous mappings of compact nilmanifolds. Also generalizations for maps of some solvmanifolds and another proof via Lefschetz and Nielsen numbers, under the assumption the map is not homotopic to a fixed points free map, are provided.     

Wave propagation properties in oscillatory chains with cubic nonlinearities via nonlinear map approach

Free wave propagation properties in one-dimensional chains of nonlinear oscillators are investigated by means of nonlinear maps. In this realm, the governing difference equations are regarded as symplectic nonlinear transformations relating the amplitudes in adjacent chain sites (n, n + 1) thereby considering a dynamical system where the location index n plays the role of the discrete time. Thus, wave propagation becomes synonymous of stability: finding regions of propagating wave solutions is equivalent to finding regions of linearly stable map solutions. Mechanical models of chains of linearly coupled nonlinear oscillators are investigated. Pass- and stop-band regions of the mono-coupled periodic system are analytically determined for period-q orbits as they are governed by the eigenvalues of the linearized 2D map arising from linear stability analysis of periodic orbits. Then, equivalent chains of nonlinear oscillators in complex domain are tackled. Also in this case, where a 4D real map governs the wave transmission, the nonlinear pass- and stop-bands for periodic orbits are analytically determined by extending the 2D map analysis. The analytical findings concerning the propagation properties are then compared with numerical results obtained through nonlinear map iteration.     

Existence of nonzero positive solutions of systems of second order

Existence of nonzero positive solutions  of systems of second order elliptic boundary value problems under sublinear conditions is obtained. The methodology is to establish a new result on existence of nonzero positive solutions of a fixed point equation in real Banach spaces by using the well-known theory of fixed point index for compact maps defined on cones, where the fixed point equation involves composition of a compact linear operator and a continuous nonlinear map. The conditions imposed on the nonlinear maps involve the spectral radii of the compact linear operators. Moreover, the nonlinear maps are not required to be increasing in  ordered Banach spaces.     

Erratum to: A Nonlinear Generalization of Singular Value Decomposition and Its Applications to Mathematical Modeling and Chaotic Cryptanalysis

Singular Value Decomposition (SVD) is a powerful tool in linear algebra and has been extensively applied to Signal Processing, Statistical Analysis and Mathematical Modeling. We propose an extension of SVD for both the qualitative detection and quantitative determination of nonlinearity in a time series. The method is to augment the embedding matrix with additional nonlinear columns derived from the initial embedding vectors and extract the nonlinear relationship using SVD. The paper demonstrates an application of nonlinear SVD to identify parameters when the signal is generated by a nonlinear transformation. Examples of maps (Logistic map and Henon map) and flows (Van der Pol oscillator and Duffing oscillator) are used to illustrate the method of nonlinear SVD to identify parameters. The paper presents the recovery of parameters in the following scenarios: (i) data generated by maps and flows, (ii) comparison of the method for both noisy and noise-free data, (iii) surrogate data analysis for both the noisy and noise-free cases. The paper includes two applications of the method: (i) Mathematical Modeling and (ii) Chaotic Cryptanalysis.     

Bounds on nonlinear operators in finite-dimensional banach spaces

Summary We consider Lipschitz-continuous nonlinear maps in finite-dimensional Banach and Hilbert spaces. Boundedness and monotonicity of the operator are characterized quantitatively in terms of certain functionals. These functionals are used to assess qualitative properties such as invertibility, and also enable a generalization of some well-known matrix results directly to nonlinear operators. Closely related to the numerical range of a matrix, the Gerschgorin domain is introduced for nonlinear operators. This point set in the complex plane is always convex and contains the spectrum of the operator's Jacobian matrices. Finally, we focus on nonlinear operators in Hilbert space and hint at some generalizations of the von Neumann spectral theory.     

Nonlinear Perron–Frobenius theory in finite dimensions

A nonlinear Perron–Frobenius theory in finite dimensions is described which applies to positively homogeneous and increasing maps. Sufficient conditions for the existence and uniqueness of eigenvectors in the interior of a cone are developed even when eigenvectors at the boundary of the cone exist. Several ways to benefit from nonlinearities are pointed out in order to show that the classical, linear, theory truly deserves nonlinear generalization. The main technical novelties are: a link between Collatz–Wielandt numbers and Gâteaux derivatives, and the almost one-dimensional dynamics of the nonlinear map.     

Discrete and continuous coupled nonlinear integrable systems via the dressing method

A discrete analog of the dressing method is presented and used to derive integrable nonlinear evolution equations, including two infinite families of novel continuous and discrete coupled integrable systems of equations of nonlinear Schrödinger type. First, a demonstration is given of how discrete nonlinear integrable equations can be derived starting from their linear counterparts. Then, starting from two uncoupled, discrete one‐directional linear wave equations, an appropriate matrix Riemann‐Hilbert problem is constructed, and a discrete matrix nonlinear Schrödinger system of equations is derived, together with its Lax pair. The corresponding compatible vector reductions admitted by these systems are also discussed, as well as their continuum limits. Finally, by increasing the size of the problem, three‐component discrete and continuous integrable discrete systems are derived, as well as their generalizations to systems with an arbitrary number of components.     

A comparative study on the co-existing attractors in the Gaussian map and its q-deformed version

In a recent study Jaganathan and Sinha [Jaganathan R, Sinha S. A q-deformed nonlinear map. Phys Lett A 2005;338:277–87] have introduced a scheme for the q-deformation of nonlinear maps using the logistic map as an example and shown that the q-logistic map exhibits a wide spectrum of dynamical behaviours including the co-existence of attractors (which is a rare phenomenon in one-dimensional maps). In this paper, we aim to analyze another famous one-dimensional map – the Gaussian map (a known one-dimensional map exhibiting co-existing attractors) subject to the same q-deformation scheme. We compare the dynamical behaviour of the Gaussian map and q-deformed Gaussian map with a special attention on the regions of the parameter space, where these maps exhibit co-existing attractors. An important conclusion of the present study is that the appearance of co-existing attractors for a particular choice of system parameters can be understood as a consequence of the presence of multiple fixed points in one-dimensional nonlinear maps; however the converse is not always true.     

Biharmonic Maps from Tori into a 2-Sphere

Biharmonic maps are generalizations of harmonic maps. A well-known result on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere(whatever the metrics chosen) in the homotopy class of maps of Brower degree±1. It would be interesting to know if there exists any biharmonic map in that homotopy class of maps. The authors obtain some classifications on biharmonic maps from a torus into a sphere, where the torus is provided with a flat or a class of non-flat metrics whilst the sphere is provided with the standard metric. The results in this paper show that there exists no proper biharmonic maps of degree±1 in a large family of maps from a torus into a sphere.     

Random approximations and random fixed point theorems for random 1-setn-contractive non-self-maps in abstract cones

In this paper, we will prove that the random version of Fan's Theorem [6, Theorem 2] is true for a random hemicompact 1-set-contractive map defined on a closed ball, a sphere and an annulus in cones. This class of random 1-set-contractive map includes random condensing maps, random continuous semicontractive maps, random LANE maps, random nonexpansive maps and others. As applications of our theorems, some random fixed point theorems of non-self-maps are proved under various well-known boundary conditions. Our results are generalizations, improvements or stochastic versions of the recent results obtained by many authors     

Banach空间中多类1-集-压缩映象的逼近定理和不动点定理

本文证明了 Ｋｙ Ｆａｎ定理［１］对定义在 Ｂａｎａｃｈ空间中含有内点的有界闭凸子集上的１－集－压缩映象正确．１－集－压缩映象类包含凝聚映象、非扩张映象、半压缩映象、局部几乎非扩张映象和其它映象．作为定理的应用,得到了一些非自映象在许多众知边界条件下的不动点定理,改进和推广了许多作者的最近结果．     

Functional equations with extrema

Exact solutions of sine Gordon and multiple sine Gordon equations are constructed in terms of solutions of a linear base equation, the Klein Gordon equation and also in terms of nonlinear base equations where the nonlinearity is polynomial in the dependent variable. Further, exact solutions of nonlinear generalizations of the Schrodinger equation and of additional nonlinear generalizations of the Klein Gordon equation are constructed in terms of solutions of linear base equations. Finally, solutions with spherical symmetry, of nonlinear Klein Gordon equations are given.     

Short-term memories with a stochastic perturbation

We investigate short-term memories in linear and weakly nonlinear coupled map lattices with a periodic external input. We use locally coupled maps to present numerical results about short-term memory formation adding a stochastic perturbation in the maps and in the external input.     

Periodic orbits in two classes of piecewise smooth maps with positive nonlinear parts

In this paper we consider two classes of one dimensional piecewise smooth continuous maps that have been derived as normal forms for grazing bifurcations of piecewise smooth dynamical systems. These maps are linear on one side of the phase space and nonlinear on the other side. The case of nonlinear parts with negative coefficients has been studied previously and it is proved that period-adding scenarios are generic in this case. In contrast to this result, in our analytical and numerical results, the period-adding scenarios are not observed when the nonlinear parts have positive coefficients. Furthermore, our results suggest that the typical bifurcation scenario is period doubling cascade leading to chaos in this case, which is similar to that of the smooth logistic map.