Ordering of the Mandelbrot-like set of the exponential map

We graphically study the Mandelbrot-like set of the complex exponential family of maps E(lambda)(z)=lambdae(z), which we call the Baker-Rippon-Devaney (BRD) set. We observe that the period of every hyperbolic component can be deduced with the naked eye by using two simple rules. (c) 1998 American Institute of Physics.     

Decay of Correlations,Quantitative Recurrence and Logarithm Law for Contracting Lorenz Attractors

In this paper we prove that a class of skew products maps with non uniformly hyperbolic base has exponential decay of correlations. We apply this to obtain a logarithm law for the hitting time associated to a contracting Lorenz attractor at all the points having a well defined local dimension, and a quantitative recurrence estimation.     

Central Limit Theorems and Invariance Principles¶for Time-One Maps of Hyperbolic Flows

We give a general method for deducing statistical limit laws in situations where rapid decay of correlations has been established. As an application of this method, we obtain new results for time-one maps of hyperbolic flows. In particular, using recent results of Dolgopyat, we prove that many classical limit theorems of probability theory, such as the central limit theorem, the law of the iterated logarithm, and approximation by Brownian motion (almost sure invariance principle), are typically valid for such time-one maps. The central limit theorem for hyperbolic flows goes back to Ratner 1973 and is always valid, irrespective of mixing hypotheses. We give examples which demonstrate that the situation for time-one maps is more delicate than that for hyperbolic flows, illustrating the need for rapid mixing hypotheses. Received: 4 January 2002 / Accepted: 16 February 2002?Published online: 24 July 2002     

Exact Solutions to a Combined sinh-cosh-Gordon Equation

Based on a transformed Painlevé property and the variable separated ODE method, a function transformation method is proposed to search for exact solutions of some partial differential equations (PDEs) with hyperbolic or exponential functions. This approach provides a more systematical and convenient handling of the solution process of this kind of nonlinear equations. Its key point is to eradicate the hyperbolic or exponential terms by a transformed Painlevé property and reduce the given PDEs to a variable-coefficient ordinary differential equations, then we seek for solutions to the resulting equations by some methods. As an application, exact solutions for the combined sinh-cosh-Gordon equation are formally derived.     

Hyperbolic reduction

The notions of weak Darboux integrability and hyperbolic reduction are introduced, and their potential is gauged as a means of extending the range of application of geometric methods for solving hyperbolic partial differential equations. For directness, our work is expressed in local coordinates and formulated for semilinear hyperbolic systems in two independent variables. The theory is applied to the study of 1+1-wave maps into surfaces of revolution. It is shown that the differential system for any such wave map may be viewed as an integrable extension of a certain scalar, semilinear, hyperbolic partial differential equation which is explicitly constructed. Using this we discover a new integrable wave map system for which hyperbolic reduction leads to a large family of explicit wave maps.     

Strange Attractors with One Direction of Instability

We give simple conditions that guarantee, for strongly dissipative maps, the existence of strange attractors with a single direction of instability and certain controlled behaviors. Only the d= 2 case is treated in this paper, although our approach is by no means limited to two phase-dimensions. We develop a dynamical picture for the attractors in this class, proving they have many of the statistical properties associated with chaos: positive Lyapunov exponents, existence of SRB measures, and exponential decay of correlations. Other results include the geometry of fractal critical sets, nonuniform hyperbolic behavior, symbolic coding of orbits, and formulas for topological entropy. Received: 25 April 2000 / Accepted: 17 October 2000     

A remark on papers by pixton and oliveira: Genericity of symplectic diffeomorphisms ofS 2 with positive topological entropy

We prove the existence of an open and dense subset of mapsfDiff (S²) which have positive topological entropy. It follows that these maps have infinitely many hyperbolic periodic points and an exponential growth rate of hyperbolic periodic points. The proof is an application of Pixton's theorem     

Fan sub-equation method for Wick-type stochastic partial differential equations

An improved algorithm is devised for using Fan sub-equation method to solve Wick-type stochastic partial differential equations. Applying the improved algorithm to the Wick-type generalized stochastic KdV equation, we obtain more general Jacobi and Weierstrass elliptic function solutions, hyperbolic and trigonometric function solutions, exponential function solutions and rational solutions.     

Exponential mixing for the geodesic flow on hyperbolic three-manifolds

We give a short and direct proof of exponential mixing of geodesic flows on compact hyperbolic three-manifolds with respect to the Liouville measure. This complements earlier results of Collet-Epstein-Gallovotti, Moore, and Ratner for hyperbolic surfaces. Furthermore, since the analysis is even easier in three dimensions than in two dimensions (because of the absence of discrete series and the simplicity of the zonal spherical functions in this case), this apparently gives the simplest example of a flow with exponential mixing.     

变阻抗传输线能量传输效率分析

采用数值方法分析了阻抗各向同性、波速不变的变阻抗传输线（指数型、双曲型和线性型）的电压反射系数,得到了能量传输效率与归一化频率、阻抗变比之间的关系。结果表明,在低频条件下(归一化频率小于2),指数传输线的能量传输效率最高。计算了Z300装置变阻抗传输线的能量传输线效率,指数线为91.6%、双曲线为90.4%,圆锥线为91.2%。     

The fractional coupled KdV equations： Exact solutions and white noise functional approach

Variable coefficients and Wick-type stochastic fractional coupled KdV equations are investigated. By using the mod- ified fractional sub-equation method, Hermite transform, and white noise theory the exact travelling wave solutions and white noise functional solutions are obtained, including the generalized exponential, hyperbolic, and trigonometric types.     

Stability of minimal periodic orbits

Symplectic twist maps are obtained from a Lagrangian variational principle. It is well known that nondegenerate minima of the action correspond to hyperbolic orbits of the map when the twist is negative definite and the map is two-dimensional. We show that for more than two dimensions, periodic orbits with minimal action in symplectic twist maps with negative definite twist are not necessarily hyperbolic. In the proof we show that in the neighborhood of a minimal periodic orbit of period n, the nth iterate of the map is again a twist map. This is true even though in general the composition of twist maps is not a twist map.     

On dynamical behavior for optical solitons sustained by the perturbed Chen–Lee–Liu model

This study investigates the perturbed Chen–Lee–Liu model that represents the propagation of an optical pulse in plasma and optical fiber. The generalized exponential rational function method is used for this purpose. As a result, we obtain some non-trivial solutions such as the optical singular, periodic, hyperbolic, exponential, trigonometric soliton solutions. We aim to express the pulse propagation of the generated solutions, by taking specific values for the free parameters existed in the obtained solutions. The obtained results show that the generalized exponential rational function technique is applicable, simple and effective to get the solutions of nonlinear engineering and physical problems. Moreover, the acquired solutions display rich dynamical evolutions that are important in practical applications.     

Exponential wave maps

We generalize wave maps to exponential wave maps. We compute the first and second variations of the exponential energy, and obtain results concerning the stability of exponential wave maps. We prove a theorem which relates wave maps, exponential wave maps, and the conservation law of second-order symmetric tensors. We show that if f is an exponential wave and a pseudo-weakly conformal map, then f is homothetic. We finally discuss the applications of exponential wave maps in relativity.     

Ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the 2-torus

We study generic piecewise linear hyperbolic automorphisms of the 2-torus. We explain why the resulting dynamical system is ergodic and mixing and prove the exponential decay of correlations.     

New type of heteroclinic tangency in two-dimensional maps

A new mechanism of heteroclinic tangency is investigated by using two-dimensional maps. First, it is numerically shown that the unstable manifold from a hyperbolic fixed point accumulates to the stable manifold of a nearby period-2 hyperbolic point in a piecewise linear map and that the unstable manifold from a hyperbolic fixed point accumulates to the accumulation of the stable manifold of a nearby period-2 hyperbolic point in a cubic map. Second, a theorem on the impossibility of heteroclinic tangency (in the usual sense) is given for a particular type of map. The notions ofdirect andasymptotic heteroclinic tangencies are introduced and heteroclinic tangency is classified into four types.     

Exponential sine chaotification model for enhancing chaos and its hardware implementation

Chaotic systems have been intensively studied for their roles in many applications, such as cryptography, secure communications, nonlinear controls, etc. However, the limited complexity of existing chaotic systems weakens chaos-based practical applications. Designing chaotic maps with high complexity is attractive. This paper proposes the exponential sine chaotification model (ESCM), a method of using the exponential sine function as a nonlinear transform model, to enhance the complexity of chaotic maps. To verify the performance of the ESCM, we firstly demonstrated it through theoretical analysis. Then, to exhibit the high efficiency and usability of ESCM, we applied ESCM to one-dimensional (1D) and multi-dimensional (MD) chaotic systems. The effects were examined by the Lyapunov exponent and it was found that enhanced chaotic maps have much more complicated dynamic behaviors compared to their originals. To validate the simplicity of ESCM in hardware implementation, we simulated three enhanced chaotic maps using a digital signal processor (DSP). To explore the ESCM in practical application, we applied ESCM to image encryption. The results verified that the ESCM can make previous chaos maps competitive for usage in image encryption.     

(2+1) 维Broer-Kau-Kupershmidt方程一系列新的精确解

借助于符号计算软件Maple，通过一种构造非线性偏微分方程(组)更一般形式精确解的直接方法即改进的代数方法，求解(2+1) 维 Broer-Kau-Kupershmidt方程，得到该方程的一系列新的精确解，包括多项式解、指数解、有理解、三角函数解、双曲函数解、Jacobi 和 Weierstrass 椭圆函数双周期解. 关键词： 代数方法 (2+1) 维 Broer-Kau-Kupershmidt 方程 精确解 行波解     

Computational modeling of MR flow imaging by the lattice Boltzmann method and Bloch equation

In this work, a computational model of magnetic resonance (MR) flow imaging is proposed. The first model component provides fluid dynamics maps by applying the lattice Boltzmann method. The second one uses the flow maps and couples MR imaging (MRI) modeling with a new magnetization transport algorithm based on the Eulerian coordinate approach. MRI modeling is based on the discrete time solution of the Bloch equation by analytical local magnetization transformations (exponential scaling and rotations).     

Entropy Production for a Class of Inverse SRB Measures

We study the entropy production for inverse SRB measures for a class of hyperbolic folded repellers presenting both expanding and contracting directions. We prove that for most such maps we obtain strictly negative entropy production of the respective inverse SRB measures. Moreover we provide concrete examples of hyperbolic folded repellers where this happens.