Difference in the dynamics of logistic maps

2019 ◽  
Vol 33 (20) ◽  
pp. 1950226
Author(s):  
Tianshu Song ◽  
Hui Xia
Keyword(s):  
Logistic Map ◽  
Jigsaw Puzzle ◽  
Initial Values ◽  
The Difference

We investigate the evolution of the difference in the conventional logistic map and observed complicated but interesting dynamics. Our results show that the bifurcation, the attractor and the asymptotic measure depend on initial values. The reverse bifurcation, the multiple bifurcation and other bifurcation properties obtained here are different from those of the original logistic map. Furthermore, we find that attractors form a structure which looks like a jigsaw puzzle, and the corresponding distributions related to the Feigenbaum constant are also analyzed.

scholarly journals Global Behavior of the Difference Equationxn+1=xn-1g(xn)

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Hongjian Xi ◽  
Taixiang Sun ◽  
Bin Qin ◽  
Hui Wu
Keyword(s):  
Continuous Function ◽  
Difference Equation ◽  
Positive Solution ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Global Behavior ◽  
Initial Values ◽  
The Difference ◽  
Surjective Function

We consider the following difference equationxn+1=xn-1g(xn),n=0,1,…,where initial valuesx-1,x0∈[0,+∞)andg:[0,+∞)→(0,1]is a strictly decreasing continuous surjective function. We show the following. (1) Every positive solution of this equation converges toa,0,a,0,…,or0,a,0,a,…for somea∈[0,+∞). (2) Assumea∈(0,+∞). Then the set of initial conditions(x-1,x0)∈(0,+∞)×(0,+∞)such that the positive solutions of this equation converge toa,0,a,0,…,or0,a,0,a,…is a unique strictly increasing continuous function or an empty set.

scholarly journals On the Difference Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Candace M. Kent ◽  
Witold Kosmala ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Real Numbers ◽  
Initial Values ◽  
The Difference ◽  
Long Term Behavior

The long-term behavior of solutions of the following difference equation: , , where the initial values , , are real numbers, is investigated in the paper.

scholarly journals On the Difference Equationxn=anxn-k/(bn+cnxn-1⋯xn-k)

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Stevo Stević ◽  
Josef Diblík ◽  
Bratislav Iričanin ◽  
Zdeněk Šmarda
Keyword(s):  
Difference Equation ◽  
Real Numbers ◽  
Initial Values ◽  
The Difference

The behavior of well-defined solutions of the difference equationxn=anxn-k/(bn+cnxn-1⋯xn-k), n∈ℕ0, wherek∈ℕis fixed, the sequencesan,bnandcnare real,(bn,cn)≠(0,0),n∈ℕ0, and the initial valuesx-k,…,x-1are real numbers, is described.

scholarly journals Chaotic Image Encryption Design Using Tompkins-Paige Algorithm

2009 ◽  
Vol 2009 ◽  
pp. 1-22 ◽  
Author(s):  
Shahram Etemadi Borujeni ◽  
Mohammad Eshghi
Keyword(s):  
Image Encryption ◽  
Logistic Map ◽  
Correlation Coefficients ◽  
Chi Square ◽  
Encrypted Image ◽  
Key Space ◽  
Chi Square Test ◽  
Plain Image ◽  
The Difference ◽  
Pixel Substitution

In this paper, we have presented a new permutation-substitution image encryption architecture using chaotic maps and Tompkins-Paige algorithm. The proposed encryption system includes two major parts, chaotic pixels permutation and chaotic pixels substitution. A logistic map is used to generate a bit sequence, which is used to generate pseudorandom numbers in Tompkins-Paige algorithm, in 2D permutation phase. Pixel substitution phase includes two process, the tent pseudorandom image (TPRI) generator and modulo addition operation. All parts of the proposed chaotic encryption system are simulated. Uniformity of the histogram of the proposed encrypted image is justified using the chi-square test, which is less than (255, 0.05). The vertical, horizontal, and diagonal correlation coefficients, as well as their average and RMS values for the proposed encrypted image are calculated that is about 13% less than previous researches. To quantify the difference between the encrypted image and the corresponding plain-image, three measures are used. These are MAE, NPCR, and UACI, which are improved in our proposed system considerably. NPCR of our proposed system is exactly the ideal value of this criterion. The key space of our proposed method is large enough to protect the system against any Brute-force and statistical attacks.

scholarly journals On boundedness of the solutions of the difference equationxn+1=xn−1/(p+xn)

2006 ◽  
Vol 2006 ◽  
pp. 1-7 ◽  
Author(s):  
Taixiang Sun ◽  
Hongjian Xi ◽  
Hui Wu
Keyword(s):  
Difference Equation ◽  
Positive Solution ◽  
Open Problem ◽  
Initial Values ◽  
The Difference

We study the difference equationxn+1=xn−1/(p+xn),n=0,1,…,where initial valuesx−1,x0∈(0,+∞)and0<p<1, and obtain the set of all initial values(x−1,x0)∈(0,+∞)×(0,+∞)such that the positive solution{xn}n=−1∞is bounded. This answers the Open Problem 2 proposed by Kulenović and Ladas.

scholarly journals Long-Term Behavior of Solutions of the Difference Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Candace M. Kent ◽  
Witold Kosmala ◽  
Stevo Stević
Keyword(s):  
Difference Equation ◽  
Real Numbers ◽  
Initial Values ◽  
The Difference ◽  
Long Term Behavior

We investigate the long-term behavior of solutions of the following difference equation: , , where the initial values , , and are real numbers. Numerous fascinating properties of the solutions of the equation are presented.

scholarly journals A Novel Encryption Scheme for Digital Image - Based on One Dimensional Logistic Map

2014 ◽  
Vol 7 (4) ◽  
pp. 65 ◽  
Author(s):  
Obaida M. Al-hazaimeh
Keyword(s):  
Image Encryption ◽  
Digital Image ◽  
Chaotic Map ◽  
Logistic Map ◽  
Encryption Scheme ◽  
Table Lookup ◽  
One Dimensional ◽  
The Difference ◽  
Digital Image Encryption ◽  
High Level

In this paper, an implementation of digital image encryption scheme based on one dimensional logistic map is proposed. The chaotic cryptography technique concentrates in general on the symmetric key cryptographic technique. In the proposed algorithm, a random key table lookup criterion was combined with a one-dimensional chaotic map were used for high degree 2-stage security image encryption while maintaining acceptable overhead delay time. The proposed algorithm is based on image row shuffling and pixel-wise XOR encryption. To increase the security of row shuffling variable rotation and inversion were applied to each shuffled row, based on the difference between old and new row location. The experimental results showed that the proposed algorithm is effective and applicable. The combination of logistic map and key table lookup shows advantages of large random key space and high-level of security. The resulting cipher image is suitable for practical use in secure image storing and transmission.

scholarly journals On the Difference Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
İbrahim Yalçinkaya
Keyword(s):  
Difference Equation ◽  
Higher Order ◽  
Real Numbers ◽  
Positive Real ◽  
Global Behaviour ◽  
Initial Values ◽  
The Difference

We investigate the global behaviour of the difference equation of higher order , where the parameters and the initial values and are arbitrary positive real numbers.

scholarly journals Quasi-equilibrium relaxation of two identical quantum oscillators with arbitrary coupling strength

2015 ◽  
Vol 93 (7) ◽  
pp. 750-759 ◽  
Author(s):  
Illarion Dorofeyev
Keyword(s):  
Open Systems ◽  
Quasi Equilibrium ◽  
Initial Values ◽  
Integral Methods ◽  
Fluctuation Dissipation ◽  
Fluctuation Dissipation Theorem ◽  
Different Temperatures ◽  
Long Time ◽  
The Difference ◽  
Identical Quantum

This paper deals with the problem of open systems out of equilibrium. An analytical expression for a time-dependent density matrix of two arbitrarily coupled identical quantum oscillators interacting with separate reservoirs is derived using path integral methods. The temporal behavior of spatial variances and of covariance from given initial values up to stationary values is investigated. It is shown that at comparatively low coupling strengths, the asymptotic variances in the long-time limit achieve steady states independently of initial values. Stationary variances differ from the case of total equilibrium due to their coupling with separate thermal reservoirs of different temperatures. The larger the difference in temperatures of thermal baths, the larger is the difference of the stationary values of variances of coupled oscillators compared with values given by the fluctuation dissipation theorem. At strong couplings the variances have divergent character in the framework of the accepted model. Otherwise, in the weak coupling limit the asymptotic stationary variances are always in accordance with the fluctuation dissipation theorem with some accuracy.

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