Superexponential stability of KAM tori

We study the dynamics in the neighborhood of an invariant torus of a nearly integrable system. We provide an upper bound to the diffusion speed, which turns out to be of superexponentially small size exp[-exp(1/)], being the distance from the invariant torus. We also discuss the connection of this result with the existence of many invariant tori close to the considered one.     

一种基于Arnold置乱和DCT编码的无载体信息隐藏方法北大核心CSCD

近年来十分火热的搜索映射式无载体信息隐藏虽具有一定的鲁棒性,但其隐藏容量较低、传输负载大且算法复杂度高。针对以上问题,该文章提出一种基于Arnold置乱和离散余弦变换(discrete cosine transform,DCT)编码的无载体信息隐藏方法。该算法先对图片进行Arnold置乱,再对DCT后的低频系数进行编码,接着更换置乱参数来构建索引表。选择索引表中与秘密信息相同的编码值所对应的参数构建候选队列,最后筛选出鲁棒性强的参数作为密钥发送给接收方。实验结果表明,该方法与现有方法相比大大提高了嵌入容量,拥有更强的抗JPEG压缩性能。并且减少了传输负载,算法简捷,具有较强的应用价值。     

Quantum qualitative dynamics

We describe our work on qualitative methods for visualizing the quantum eigenstates of systems with nonlinear classical dynamics. For two-degree-of-freedom systems, our approach is based on the use of generalized coherent states, and allows systems with nonoscillator kinematics to be investigated. The general approach is illustrated with two examples involving vibration-rotation interaction in polyatomic molecules. We apply the coherent states of the Lie groupH ₄SU(2) to define quantum surfaces of section for a model involving centrifugal coupling of a harmonic bend with molecular rotation, andSU(2)SU(2) coherent states to study two harmonic normal modes coupled to overall molecular rotation through coriolis interaction. In both systems, quantum states are visualized on the rotational surface of section and compared with the corresponding classical phase space structure. Striking classical-quantum correspondence is observed. We then describe recent results on the quantum states of (N 3)-dimensional systems of coupled nonlinear oscillators, which reveal a quantum delocalization that is reminiscent of classical Arnold diffusion.     

一种基于Arnold变换和RSA的图像加密算法

提出一种Arnold变换和RSA相结合的图像加密算法。利用Arnold变换对图像的像素位置进行链式迭代置乱。在对置乱后的图像像素值进行替代和扩散的过程中,采用基于大素数分解的RSA算法,构造一个位置置乱和灰度变换相结合的加密算法。实验仿真结果表明该算法支持变长密钥,具有敏感性强、去相关能力强及抵御统计和已知明文攻击等优点,能够取得很好的加密效果。     

Non-integrability vs. integrability in pentagram maps

We revisit recent results on integrable cases for higher-dimensional generalizations of the 2D pentagram map: short-diagonal, dented, deep-dented, and corrugated versions, and define a universal class of pentagram maps, which are proved to possess projective duality. We show that in many cases the pentagram map cannot be included into integrable flows as a time-one map, and discuss how the corresponding notion of discrete integrability can be extended to include jumps between invariant tori. We also present a numerical evidence that certain generalizations of the integrable 2D pentagram map are non-integrable and present a conjecture for a necessary condition of their discrete integrability.     

Autonomous coupled oscillators with hyperbolic strange attractors

We propose several examples of smooth low-order autonomous dynamical systems which have apparently uniformly hyperbolic attractors. The general idea is based on the use of coupled self-sustained oscillators where, due to certain amplitude nonlinearities, successive epochs of damped and excited oscillations alternate. Because of additional, phase sensitive coupling terms in the equations, the transfer of excitation from one oscillator to another is accompanied by a phase transformation corresponding to some chaotic map (in particular, an expanding circle map or Anosov map of a torus). The first example we construct is a minimal model possessing an attractor of the Smale-Williams type. It is a four-dimensional system composed of two oscillators. The underlying amplitude equations are similar to those of the predator-pray model. The other three examples are systems of three coupled oscillators with a heteroclinic cycle. This scheme presents more variability for the phase manipulations: in the six-dimensional system not only the Smale-Williams attractor, but also an attractor with Arnold cat map dynamics near a two-dimensional toral surface, and a hyperchaotic attractor with two positive Lyapunov exponents, are realized.     

On matrix analogs of Fermat’s little theorem

The theorem proved in this paper gives a congruence for the traces of powers of an algebraic integer for the case in which the exponent of the power is a prime power. The theorem implies a congruence in Gauss’ form for the traces of the sums of powers of algebraic integers, generalizing many familiar versions of Fermat’s little theorem. Applied to the traces of integer matrices, this gives a proof of Arnold’s conjecture about the congruence of the traces of powers of such matrices for the case in which the exponent of the power is a prime power.     

小波域分存和软判决恢复的隐藏技术

研究了一种把小波变换和可视密码技术相结合的信息隐藏方案,实现对灰度图像的信息隐藏。该方案把经过Arnold变换处理后的秘密信息,按照k阶线性方程求解理论产生n幅分享图片,再把n幅分享图片应用离散小波变换信息隐藏理论,分存到n幅公开图片中。提取秘密图片时,必须要大于k幅公开图片才能获得秘密图片。实验结果表明,该方案具有很强的鲁棒性,而且秘密信息不容易丢失。     

Image watermarking by using phase retrieval algorithm in gyrator transform domain

We propose an image watermarking scheme based on the phase retrieval algorithm in gyrator domain. The watermark is converted into a noise-like image by Arnold transform. The scrambled image is regarded as the amplitude of gyrator spectrum. The Gerchberg-Saxton algorithm is employed to obtain the unknown phase function in gyrator pair, in which the host image is the amplitude of input function. The phase information and the parameters of the two transforms serve as the key of watermarking algorithm. The numerical simulation has demonstrated the performance of the proposed algorithm.     

Arnold变换在LSB算法中的应用研究

为了更好地进行版权保护,设计了一种基于图像置乱技术的数字图像水印算法。将经过Arnold变换的水印信号嵌入在宿主图像的LSB位平面上,提高了LSB算法的鲁棒性,保留了盲提取特点。算法通过水印信号扩展使其与宿主图像相匹配,保证了水印嵌入位置的安全性。仿真实验比较了有无Arnold变换的水印算法,结果表明使用Arnold置乱的水印算法抗噪声、剪切能力较好,说明在数字图像水印中应用图像置乱技术可以有效提高数字图像水印的性能。