Lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms

1989 ◽  
Vol 81 (4) ◽  
pp. 521-554 ◽  
Author(s):  
Peter H. Baxendale
Keyword(s):  
Lyapunov Exponents ◽  
Relative Entropy ◽  
Stochastic Flow ◽  
Stochastic Flow Of Diffeomorphisms

LYAPUNOV SPECTRUM OF NONAUTONOMOUS LINEAR STOCHASTIC DIFFERENTIAL ALGEBRAIC EQUATIONS OF INDEX-1

2012 ◽  
Vol 12 (04) ◽  
pp. 1250002 ◽  
Author(s):  
NGUYEN DINH CONG ◽  
NGUYEN THI THE
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Lyapunov Exponents ◽  
Algebraic Equation ◽  
Differential Algebraic Equations ◽  
Lyapunov Spectrum ◽  
Stochastic Flow ◽  
Algebraic Equations ◽  
Differential Algebraic Equation ◽  
Two Parameter

We introduce a concept of Lyapunov exponents and Lyapunov spectrum of a stochastic differential algebraic equation (SDAE) of index-1. The Lyapunov exponents are defined samplewise via the induced two-parameter stochastic flow generated by inherent regular stochastic differential equations. We prove that Lyapunov exponents are nonrandom.

DYNAMICAL PROPERTIES OF LÉVY PROCESSES IN LIE GROUPS

2002 ◽  
Vol 02 (01) ◽  
pp. 1-23 ◽  
Author(s):  
MING LIAO
Keyword(s):  
Vector Field ◽  
Homogeneous Space ◽  
Lie Groups ◽  
Lyapunov Exponents ◽  
Lie Group ◽  
Stationary Points ◽  
Stochastic Flow ◽  
Semisimple Lie Group ◽  
Noncompact Type ◽  
Dynamical Properties

Let ϕt be a Lévy process in a semisimple Lie group G of noncompact type regarded as a stochastic flow on a homogeneous space of G, called a G-flow. We will determine the Lyapunov exponents and the stable manifolds of ϕt, and the stationary points of an associated vector field. As examples, SL (d,R)-flows and SO (1,d)-flows on SO (d) and Sd - 1 are discussed in details.

Necessary and sufficient conditions for stable synchronization in random dynamical systems

2017 ◽  
Vol 38 (5) ◽  
pp. 1857-1875 ◽  
Author(s):  
JULIAN NEWMAN
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Compact Space ◽  
Sufficient Conditions ◽  
Invariant Set ◽  
Stochastic Flow ◽  
Asymptotically Stable ◽  
Random Maps ◽  
Necessary And Sufficient ◽  
Being Moved

For a composition of independent and identically distributed random maps or a memoryless stochastic flow on a compact space$X$, we find conditions under which the presence of locally asymptotically stable trajectories (e.g. as given by negative Lyapunov exponents) implies almost-sure mutual convergence of any given pair of trajectories (‘synchronization’). Namely, we find that synchronization occurs and is ‘stable’ if and only if the system exhibits the following properties: (i) there is asmallestnon-empty invariant set$K\subset X$; (ii) any two points in$K$are capable of being moved closer together; and (iii) $K$admits asymptotically stable trajectories.

Lyapunov Exponents

2016 ◽  
Author(s):  
Arkady Pikovsky ◽  
Antonio Politi
Keyword(s):  
Lyapunov Exponents

Efficient Steganography Algorithm Based On DCT And Entropy Thresholding Technique

2018 ◽  
Vol 8 (1) ◽  
pp. 45
Author(s):  
Satvir Singh
Keyword(s):  
Relative Entropy ◽  
Experimental Work ◽  
Random Function ◽  
Signal To Noise Ratio ◽  
Binary Sequence ◽  
Threshold Value ◽  
Absolute Difference ◽  
Secret Message ◽  
Signal To Noise ◽  
Confidential Information

Steganography is the special art of hidding important and confidential information in appropriate multimedia carrier. It also restrict the detection of  hidden messages. In this paper we proposes steganographic method based on dct and entropy thresholding technique. The steganographic algorithm uses random function in order to select block of the image where the elements of the binary sequence of a secret message will be inserted. Insertion takes place at the lower frequency  AC coefficients of the  block. Before we insert the secret  message. Image under goes dc transformations after insertion of the secret message we apply inverse dc transformations. Secret message will only be inserted into a particular block if  entropy value of that particular block is greater then threshold value of the entropy and if block is selected by the random function. In  Experimental work we calculated the peak signal to noise ratio(PSNR), Absolute difference , Relative entropy. Proposed algorithm give high value of PSNR  and low value of Absolute difference which clearly indicate level of distortion in image due to insertion of secret message is reduced. Also value of  relative entropy is close to zero which clearly indicate proposed algorithm is sufficiently secure. 

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