Tent Map Research Articles

Digital image watermarking using gyrator transform and chaotic maps

We propose a new method for digital image watermarking using gyrator transform and chaotic maps. Four chaotic maps have been used in the proposed technique. The four chaotic maps that have been used are the logistic map, the tent map, the Kaplan–Yorke map and the Ikeda map. These chaotic maps are used to generate the random phase masks and these random phase masks are known as chaotic random phase masks. A new technique has been proposed to generate the single chaotic random phase mask by using two chaotic maps together with different seed values. The watermark encoding method in the proposed technique is based on the double random phase encoding method. The gyrator transform and two chaotic random phase masks are used to encode the input image. The mean square error, the peak signal-to-noise ratio and the bit error rate have been calculated. Robustness of the proposed technique has been evaluated in terms of the chaotic maps, the number of the chaotic maps used to generate the CRPM, the rotation angle of the gyrator transform and the seed values of the chaotic random phase masks. Optical implementation of the technique has been proposed. The computer simulations are presented to verify the validity of the proposed technique.

Homeomorphisms of unimodal inverse limit spaces with a non-recurrent critical point

Let T be a tent map with the slope strictly between 2 and 2. Suppose that the critical point of T is not recurrent. Let K denote the inverse limit space obtained by using T repeatedly as the bonding map. We prove that every homeomorphism of K to itself is isotopic to some power of the natural shift homeomorphism.

A Novel Method for the Initial-Condition Estimation of a Tent Map

Based on the connection between the tent map and the saw tooth map or Bernoulli map, a novel method for the initial-condition estimation of the tent map is presented. In the method, firstly the symbolic sequence generated from the tent map is converted to the forms obtained from the saw tooth map and Bernoulli map, and then the relationship between the symbolic sequence and the initial condition of the tent map can be obtained from the initial-condition estimation equations, which can be easily obtained, hence the estimation of the tent map can be achieved finally. The method is computationally simple and the error of the estimator is less than 1/2N. The method is verified by software simulation.

Spectral properties of chaotic signals generated by the skew tent map

Chaotic signals have been considered potentially attractive in many signal processing applications ranging from wideband communication systems to cryptography and watermarking. Besides, some devices as nonlinear adaptive filters and phase-locked loops can present chaotic behavior. In this paper, we derive analytical expressions for the autocorrelation sequence, power spectral density and essential bandwidth of chaotic signals generated by the skew tent map. From these results, we suggest possible applications in communication systems.

Spectral properties of chaotic signals with applications in communications

This paper investigates the characteristics of the Power Spectral Density (PSD) of chaotic signals generated by skew tent maps. The influence of the Lyapunov exponent on the autocorrelation sequence and on the PSD is evaluated via computational simulations. We conclude that the essential bandwidth of these signals is strongly related to this exponent and they can be low-pass or high-pass depending on the family’s parameter. This way, the PSD of a chaotic signal is a function of the generating map although this is not a one-to-one relationship.

A classification of inverse limit spaces of tent maps with a nonrecurrent critical point

In this paper we prove the non-recurrent case of the Ingram conjecture by showing that if $T_s$ and $T_t$ are two tent maps with non-recurrent critical points then $\lim_{; ; ; ; \leftarrow}; ; ; ; \{; ; ; ; [0, s/2], T_s \}; ; ; ; $ is homeomorphic to $\lim_{; ; ; ; \leftarrow}; ; ; ; \{; ; ; ; [0, t/2], T_t \}; ; ; ; $ if, and only if $s=t$.

Robust chaos with variable Lyapunov exponent in smooth one-dimensional maps

We present several new easy ways of generating smooth one-dimensional maps displaying robust chaos, i.e., chaos for whole intervals of the parameter. Unlike what happens with previous methods, the Lyapunov exponent of the maps constructed here varies widely with the parameter. We show that the condition of negative Schwarzian derivative, which was used in previous works, is not a necessary condition for robust chaos. Finally we show that the maps constructed in previous works have always the Lyapunov exponent ln 2 because they are conjugated to each other and to the tent map by means of smooth homeomorphisms. In the methods presented here, the maps have variable Lyapunov coefficients because they are conjugated through non-smooth homeomorphisms similar to Minkowski’s question mark function.

Optical image encryption using improper Hartley transforms and chaos

We propose a new method for image encryption using improper Hartley transform and chaos theory. Improper Hartley transform is a Hartley transform in which the phase between the two Fourier transforms is a fractional multiple of π/2. This fractional order is called fractional parameter and serves as a key in the image encryption and decryption process. Four types of chaos functions have been used. These functions are the logistic map, the tent map, the Kaplan–Yorke map and the Ikeda map. Random intensity masks have been generated using these chaotic functions and are called chaotic random intensity masks. The image is encrypted by using improper Hartley transform and two chaotic random intensity masks. The mean square error has been calculated. The robustness of the proposed technique in terms of blind decryption has been tested. The computer simulations are presented to verify the validity of the proposed technique.

Reconstruction of driving forces through recurrence plots

We consider the problem of reconstructing one-dimensional driving forces only from the observations of driven systems. We extend the approach presented in a seminal paper [M.C. Casdagli, Physica D 108 (1997) 12] and propose a method that is robust and has wider applicability. By reinterpreting the work of Thiel et al. [M. Thiel, M.C. Romano, J. Kurths, Phys. Lett. A 330 (2004) 343], we formulate the reconstruction problem as a combinatorial optimization problem and relax conditions by assuming that a driving force is continuous. The method is demonstrated by using a tent map driven by an external force.

Statistics of the maximum for the tent map

In this work we derive the asymptotic distribution for the partial maximum of orbits generated by the tent map.

Performance analysis of chaotic and white watermarks in the presence of common watermark attacks

Digital watermarking is a technique that aims to embed a piece of information permanently into some digital media, which may be used at a later stage to prove owner authentication and attempt to provide protection to documents. The most common watermark types used to date are pseudorandom number sequences which possess a white spectrum. Chaotic watermark sequences have been receiving increasing interest recently and have been shown to be an alternative to the pseudorandom watermark types. In this paper the performance of pseudorandom watermarks and chaotic watermarks in the presence of common watermark attacks is performed. The chaotic watermarks are generated from the iteration of the skew tent map, the Bernoulli map and the logistic map. The analysis focuses on the watermarked images after they have been subjected to common image distortion attacks. The capacities of each of these images are also calculated. It is shown that signals generated from lowpass chaotic signals have superior performance over the other signal types analysed for the attacks studied.

A point is normal for almost all mapsβx+αmod 1 or generalizedβ-transformations

AbstractWe consider the mapTα,β(x):=βx+α mod 1, which admits a unique probability measureμα,βof maximal entropy. Forx∈[0,1], we show that the orbit ofxisμα,β-normal for almost all (α,β)∈[0,1)×(1,∞) (with respect to Lebesgue measure). Nevertheless, we construct analytic curves in [0,1)×(1,∞) along which the orbit ofx=0 isμα,β-normal at no more than one point. These curves are disjoint and fill the set [0,1)×(1,∞). We also study the generalizedβ-transformations (in particular, the tent map). We show that the critical orbitx=1 is normal with respect to the measure of maximal entropy for almost allβ.

Complexity measures from interaction structures

We evaluate information-theoretic quantities that quantify complexity in terms of kth-order statistical dependences that cannot be reduced to interactions among k-1 random variables. Using symbolic dynamics of coupled maps and cellular automata as model systems, we demonstrate that these measures are able to identify complex dynamical regimes.

Gyrator transform-based optical image encryption, using chaos

We propose a new method for image encryption, using gyrator transform and chaos theory. Random phase masks are generated using chaos functions and are called as chaotic random phase masks. In the proposed technique, the image is encrypted using gyrator transform and two chaotic random phase masks. Three types of chaos functions have been used to generate the chaotic random phase masks. These chaos functions are the logistic map, the tent map and the Kaplan-Yorke map. The computer simulations are presented to verify the validity of the proposed technique. The mean square errors have been calculated. The robustness of the proposed technique to blind decryption in terms of rotation angle and the seed values of the chaotic random phase mask have been evaluated. The optical implementation of the encryption and the decryption technique has been proposed.

On a flexible class of continuous functions with uniform local structure

This paper considers a class of continuous functions constructed as a series of iterates of the “tent map” multiplied by variable signs. This class includes Takagi's nowhere-differentiable function, and contains the functions studied by Hata and Yamaguti [Japan J. Appl. Math., 1 (1984), 183-199] and Kono [Acta Math. Hungar., 49 (1987), 315-324] as a proper subclass. A complete description is given of the differentiability properties of the functions in this class, and several statements are proved concerning their uniform and local moduli of continuity. The results are applied to generation of random functions.

Research on the stability of complexity of chaos-based pseudorandom sequence

Intensive statistical complexity can reflect the random nature of chaos-based pseudorandom sequence. Based on this property, the definition of k-error intensive statistical complexity is presented and two basic properties of it are proved in this paper, which can be used to measure the stability of complexity of chaos-based pseudorandom sequences. Based on chaos-based pseudorandom sequences produced via Logistic, Henon, Cubic, Chebyshev and Tent maps, an example is given to demonstrate how it works. Simulation results indicate that the approach is effective, it can distinguish the stability of diverse chaos-based pseudorandom sequences, and is an effective way for evaluating the stability of chaos-based sequences.

전동기의 소음 저감을 위한 카오스 2중 텐트 사상 PWM기법

본 연구에서는 전동기 소음 저감을 위한 카오스 2중 텐트 사상(double tent mapping)PWM 기법에 대하여 고찰하였다. 카오스 수 발생은 2중 텐트 사상의 카오스 말생 영역인 λ=0.99에서의 분기 트리(bifurcation tree)를 사용하였다. 카오스 수 발생은 80C196 마이크로 콘트롤라가 담당하고 있으며, 80C196으로부터 발생된 카오스 수와 MAX038 주파수 변조기를 이용하여 카오스 특성을 갖는 삼각파 캐리어가 발생한다. 2.5[A] 부하조건에 대한 1.5[kw]급 유도 선동기 구동 시스템에 제안된 방법과 종전의 고정 주파수 방법을 적용하였으며 각각의 방법에 대하여 캐리어 및 전동기 전압 그리고 3차원 스위칭 소음의 고조파 스펙트럼을 비교 검토하였다. 그 결과 제안된 방법의 고조파 스펙트럼은 특정 주파수애 집중되지 않는 카오스 분포를 하였으며, 이로 인하여 전동기에서 발생하는 날카로운 스위칭 소음이 저감됨을 알 수 있었다.

A block cipher with dynamic S-boxes based on tent map

In this paper, a block encryption scheme based on dynamic substitution boxes (S-boxes) is proposed. Firstly, the difference trait of the tent map is analyzed. Then, a method for generating S-boxes based on iterating the tent map is presented. The plaintexts are divided into blocks and encrypted with different S-boxes. The cipher blocks are obtained by 32 rounds of substitution and left cyclic shift. To improve the security of the cryptosystem, a cipher feedback is used to change the state value of the tent map, which makes the S-boxes relate to the plaintext and enhances the confusion and diffusion properties of the cryptosystem. Since dynamic S-boxes are used in the encryption, the cryptosystem does not suffer from the problem of fixed structure block ciphers. Theoretical and experimental results indicate that the cryptosystem has high security and is suitable for secure communications.

A simple electronic circuit realization of the tent map

We present a very simple electronic implementation of the tent map, one of the best-known discrete dynamical systems. This is achieved by using integrated circuits and passive elements only. The experimental behavior of the tent map electronic circuit is compared with its numerical simulation counterpart. We find that the electronic circuit presents fixed points, periodicity, period doubling, chaos and intermittency that match with high accuracy the corresponding theoretical values.

One-way hash function construction based on chaotic map network

A novel chaotic hash algorithm based on a network structure formed by 16 chaotic maps is proposed. The original message is first padded with zeros to make the length a multiple of four. Then it is divided into a number of blocks each contains 4 bytes. In the hashing process, the blocks are mixed together by the chaotic map network since the initial value and the control parameter of each tent map are dynamically determined by the output of its neighbors. To enhance the confusion and diffusion effect, the cipher block chaining (CBC) mode is adopted in the algorithm. Theoretic analyses and numerical simulations both show that the proposed hash algorithm possesses good statistical properties, strong collision resistance and high flexibility, as required by practical keyed hash functions.