Tent Map Research Articles

On the dynamics of the Tent function - Phase diagrams

This paper focuses on the study of a one-dimensional topological dynamical system, the tent function. We give a good background to the theory of dynamical systems while establishing the important asymptotic properties of topological dynamical systems that distinguishes it from other fields by way of an example - the tent function. A precise definition of the tent function is given and iterates are clearly shown using the phase diagrams. By this same method, chaos in the tent map is shown as iterates become higher. We also show that the tent map has infinitely many chaotic orbits and also express some important features of chaos such as topological transitivity, boundedness and sensitivity to change in initial conditions from a topological viewpoint.

Chaos in discrete fractional difference equations

Recently, the discrete fractional calculus (DFC) is receiving attention due to its potential applications in the mathematical modelling of real-world phenomena with memory effects. In the present paper, the chaotic behaviour of fractional difference equations for the tent map, Gauss map and 2x(mod 1) map are studied numerically. We analyse the chaotic behaviour of these fractional difference equations and compare them with their integer counterparts. It is observed that fractional difference equations for the Gauss and tent maps are more stable compared to their integer-order version.

An image encryption scheme based on chaotic tent map

Image encryption has been an attractive research field in recent years. The chaos-based cryptographic algorithms have suggested some new and efficient ways to develop secure image encryption techniques. This paper proposes a novel image encryption scheme, which is based on the chaotic tent map. Image encryption systems based on such map show some better performances. Firstly, the chaotic tent map is modified to generate chaotic key stream that is more suitable for image encryption. Secondly, the chaos-based key stream is generated by a 1-D chaotic tent map, which has a better performance in terms of randomness properties and security level. The performance and security analysis of the proposed image encryption scheme is performed using well-known ways. The results of the fail-safe analysis are inspiring, and it can be concluded that the proposed scheme is efficient and secure.

An Improved Chaotic Binary Bat Algorithm for QoS Multicast Routing

The quality of service (QoS) multicast routing problem is one of the main issues for transmission in communication networks. It is known to be an NP-hard problem, so many heuristic algorithms have been employed to solve the multicast routing problem and find the optimal multicast tree which satisfies the requirements of multiple QoS constraints such as delay, delay jitter, bandwidth and packet loss rate. In this paper, we propose an improved chaotic binary bat algorithm to solve the QoS multicast routing problem. We introduce two modification methods into the binary bat algorithm. First, we use the two most representative chaotic maps, namely the logistic map and the tent map, to determine the parameter [Formula: see text] of the pulse frequency [Formula: see text]. Second, we use a dynamic formulation to update the parameter α of the loudness [Formula: see text]. The aim of these modifications is to enhance the performance and the robustness of the binary bat algorithm and ensure the diversity of the solutions. The simulation results reveal the superiority, effectiveness and efficiency of our proposed algorithms compared with some well-known algorithms such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Jumping Particle Swarm Optimization (JPSO), and Binary Bat Algorithm (BBA).

Hybrid Artificial Bee Colony Algorithm with Differential Evolution and Free Search for Numerical Function Optimization

Artificial bee colony (ABC) algorithm invented by Karaboga has been proved to be an efficient technique compared with other biological-inspired algorithms for solving numerical optimization problems. Unfortunately, convergence speed of ABC is slow when working with certain optimization problems and some complex multimodal problems. Aiming at the shortcomings, a hybrid artificial bee colony algorithm is proposed in this paper. In the hybrid ABC, an improved search operator learned from Differential Evolution (DE) is applied to enhance search process, and a not-so-good solutions selection strategy inspired by free search algorithm (FS) is introduced to avoid local optimum. Especially, a reverse selection strategy is also employed to do improvement in onlooker bee phase. In addition, chaotic systems based on the tent map are executed in population initialization and scout bee's phase. The proposed algorithm is conducted on a set of 40 optimization test functions with different mathematical characteristics. The numerical results of the data analysis, statistical analysis, robustness analysis and the comparisons with other state-of-the-art-algorithms demonstrate that the proposed hybrid ABC algorithm provides excellent convergence and global search ability.

Robust chaos in a credit cycle model defined by a one-dimensional piecewise smooth map

We consider a family of one-dimensional continuous piecewise smooth maps with monotone increasing and monotone decreasing branches. It is associated with a credit cycle model introduced by Matsuyama, under the assumption of the Cobb-Douglas production function. We offer a detailed analysis of the dynamics of this family. In particular, using the skew tent map as a border collision normal form we obtain the conditions of abrupt transition from an attracting fixed point to an attracting cycle or a chaotic attractor (cyclic chaotic intervals). These conditions allow us to describe the bifurcation structure of the parameter space of the map in a neighborhood of the boundary related to the border collision bifurcation of the fixed point. Particular attention is devoted to codimension-two bifurcation points. Moreover, the described bifurcation structure confirms that the chaotic attractors of the considered map are robust, that is, persistent under parameter perturbations.

RFID Bi-Directional Authentication Protocol Based on Dynamic Coupled Integer Tent Map

To meet the special needs of RFID, it is proposed to use a coupled dynamic integer tent map lattices model (DCML) to implement a Hash function and a random number generation scheme. The scheme is featured with fast operation. Based on this, this paper proposes RFID bi-directional authentication protocol. This has strictly proved by BAN logic. This protocol can resolve eavesdropping, illegal access, location tracking, impersonation and replay attack.

Chaos based crossover and mutation for securing DICOM image

This paper proposes a novel encryption scheme based on combining multiple chaotic maps to ensure the safe transmission of medical images. The proposed scheme uses three chaotic maps namely logistic, tent and sine maps. To achieve an efficient encryption, the proposed chao-cryptic system employs a bio-inspired crossover and mutation units to confuse and diffuse the Digital Imaging and Communications in Medicine (DICOM) image pixels. The crossover unit extensively permutes the image pixels row-wise and column-wise based on the chaotic key streams generated from the Combined Logistic–Tent (CLT) system. Prior to mutation, the pixels of the crossed over image are decomposed into two images with reduced bit depth. The decomposed images are then mutated by XOR operation with quantized chaotic sequences from Combined Logistic–Sine (CLS) system. In order to validate the sternness of the proposed algorithm, the developed chao-cryptic scheme is subjected to various security analyses such as statistical, differential, key space, key sensitivity, intentional cropping attack and chosen plaintext attack analyses. The experimental results prove the proposed DICOM cryptosystem has achieved a desirable amount of protection for real time medical image security applications.

Chaos synchronization by resonance of multiple delay times

Chaos synchronization may arise in networks of nonlinear units with delayed couplings. We study complete and sublattice synchronization generated by resonance of two large time delays with a specific ratio. As it is known for single-delay networks, the number of synchronized sublattices is determined by the greatest common divisor (GCD) of the network loop lengths. We demonstrate analytically the GCD condition in networks of iterated Bernoulli maps with multiple delay times and complement our analytic results by numerical phase diagrams, providing parameter regions showing complete and sublattice synchronization by resonance for Tent and Bernoulli maps. We compare networks with the same GCD with single and multiple delays, and we investigate the sensitivity of the correlation to a detuning between the delays in a network of coupled Stuart-Landau oscillators. Moreover, the GCD condition also allows detection of time-delay resonances, leading to high correlations in nonsynchronizable networks. Specifically, GCD-induced resonances are observed both in a chaotic asymmetric network and in doubly connected rings of delay-coupled noisy linear oscillators.

Collision analysis and improvement of a hash function based on chaotic tent map

We analyze the computational collision problem on a hash algorithm based on chaotic tent map, and then present an improvement of the original algorithm in this paper. More specifically, we utilize message extension to enhance the correlation of plaintexts in the message and aggregation operation to improve the correlation of sequences of message blocks, which significantly increase the sensitivity between message and hash values, thereby greatly resisting the collision. Finally, we evaluate the performance of the improved algorithm by computer simulation, and the results show that it can resist the computational collision and can satisfy the requirements of a more secure hash algorithm.

Robust PRNG based on homogeneously distributed chaotic dynamics

This paper is devoted to the design of new chaotic Pseudo Random Number Generator (CPRNG). Exploring several topologies of network of 1-D coupled chaotic mapping, we focus first on two dimensional networks. Two topologically coupled maps are studied: TTLrc non-alternate, and TTLSC alternate. The primary idea of the novel maps has been based on an original coupling of the tent and logistic maps to achieve excellent random properties and homogeneous /uniform/ density in the phase plane, thus guaranteeing maximum security when used for chaos base cryptography. In this aim two new nonlinear CPRNG: MTTL2sc and NTTL2 are proposed. The maps successfully passed numerous statistical, graphical and numerical tests, due to proposed ring coupling and injection mechanisms.

Fast and Secure Chaos-Based Cryptosystem for Images

Nonlinear dynamic cryptosystems or chaos-based cryptosystems have been attracting a large amount of research since 1990. The critical aspect of cryptography is to face the growth of communication and to achieve the design of fast and secure cryptosystems. In this paper, we introduce three versions of a chaos-based cryptosystem based on a similar structure of the Zhang and Fridrich cryptosystems. Each version is composed of two layers: a confusion layer and a diffusion layer. The confusion layer is achieved by using a modified 2-D cat map to overcome the fixed-point problem and some other weaknesses, and also to increase the dynamic key space. The 32-bit logistic map is used as a diffusion layer for the first version, which is more robust than using it in 8-bit. In the other versions, the logistic map is replaced by a modified Finite Skew Tent Map (FSTM) for three reasons: to increase the nonlinearity properties of the diffusion layer, to overcome the fixed-point problem, and to increase the dynamic key space. Finally, all versions of the proposed cryptosystem are more resistant against known attacks and faster than Zhang cryptosystems. Moreover, the dynamic key space is much larger than the one used in Zhang cryptosystems. Performance and security analysis prove that the proposed cryptosystems are suitable for securing real-time applications.

A new data hiding method based on chaos embedded genetic algorithm for color image

Data hiding algorithms, which have many methods describing in the literature, are widely used in information security. In data hiding applications, optimization techniques are utilized in order to improve the success of algorithms. The genetic algorithm is one of the largely using heuristic optimization techniques in these applications. Long running time is a disadvantage of the genetic algorithm. In this paper, chaotic maps are used to improve the data hiding technique based on the genetic algorithm. Peak signal to noise ratio (PSNR) is chosen as the fitness function. Different sized secret data are embedded into the cover object using random function of MATLAB and chaotic maps. Randomness of genetic is performed by using different chaotic maps. The success of the proposed method is presented with comparative results. It is observed that gauss, logistic and tent maps are faster than random function for proposed data hiding method.

Establishment of Light Weight Cryptography for Resource Constraint Environment Using FPGA

Wireless Sensor Networks (WSN) is the accumulation of large number of tiny, low power autonomous devices. By the very nature, WSN are prone to attack and difficult to secure, manage, even for most savvy network administrators. Security algorithms are requisite for protecting data transmission in WSN, since unlicensed wireless bandwidth is used for data communication. In this work a lightweight encryption algorithm with modified key generation by fusing logistic map and tent map is proposed and the same is implemented in ALTERA DE1 cyclone II FPGA which occupies only 1550 logic element for 128 bit key size and a maximum throughput of 200 Kbps is achieved.

Null

In this paper, firstly we construct a quadratic chaotic system and prove that it is a topological conjugate system of Tent map. Secondly, having analyzed the probability density function of the system, we propose an anti-trigonometric function map. Additionally, the performances of the quadratic chaotic system such as information entropy, Kolmogorov entropy and discrete entropy are tested for both the original systems and the homogenized systems with different parameters. Numerical simulations show that the information entropy of the uniformly distributed sequence is close to the theoretical limit and the discrete entropy remains unchanged. This result shows that the homogenization method is effective. In conclusion, the chaotic sequence after homogenization not only inherits the diverse properties of the original sequence, but also exhibits better uniformity.

An Efficient Image Encryption Scheme Based on Quintuple Encryption Using Gumowski-Mira and Tent Maps

This paper proposes an efficient image encryption scheme based on quintuple encryption using two chaotic maps. The encryption process is realized with quintuple encryption by calling the encrypt(E) and decrypt(D) functions five times with five different keys in the form EDEEE. The decryption process is accomplished in the reverse direction by invoking the encrypt and decrypt functions in the form DDDED. The keys for the quintuple encryption/decryption processes are generated by using a Tent map. The chaotic values for the encrypt/decrypt operations are generated by using a Gumowski-Mira map. The encrypt function E is composed of three stages: permutation, pixel value rotation and diffusion. The permutation stage scrambles all the rows and columns to chaotically generated positions. This stage reduces the correlation radically among the neighboring pixels. The pixel value rotation stage circularly rotates all the pixels either left or right, and the amount of rotation is based on chaotic values. The last stage performs the diffusion four times by scanning the image in four different directions: Horizontally, Vertically, Principal diagonally and Secondary diagonally. Each of the four diffusion steps performs the diffusion in two directions (forward and backward) with two previously diffused pixels and two chaotic values. This stage ensures the resistance against the differential attacks. The security and performance of the proposed method is investigated thoroughly by using key space, statistical, differential, entropy and performance analysis. The experimental results confirm that the proposed scheme is computationally fast with security intact.

An application of a tent map initiated Chaotic Firefly algorithm for optimal overcurrent relay coordination

Abstract Over-current relays provide primary as well as backup protection to electrical distribution systems. These relays should be coordinated and set at the optimum values, to minimize the total operating time and hence ensure that least damage is caused when a fault occurs. While they provide backup protection, it is also imperative to ensure that their settings should not cause their inadvertent operation and subsequent sympathy trips. This paper describes a Chaotic Firefly algorithm (CFA) for optimal time coordination of these relays. The algorithm has been implemented in MATLAB and tested on several systems, out of which two have been illustrated in this paper. The results obtained by the Chaotic Firefly algorithm are compared with those obtained by the conventional Firefly algorithm (FA). The novel feature of this paper is the application of the Chaotic Firefly algorithm to the problem of over-current relay coordination.

On the Convergence to Equilibrium of Unbounded Observables Under a Family of Intermittent Interval Maps

We consider a family $\{ T_{r} \colon [0, 1] \circlearrowleft \}_{r \in [0, 1]}$ of Markov interval maps interpolating between the Tent map $T_{0}$ and the Farey map $T_{1}$. Letting $\mathcal{P}_{r}$ denote the Perron-Frobenius operator of $T_{r}$, we show, for $\beta \in [0, 1]$ and $\alpha \in (0, 1)$, that the asymptotic behaviour of the iterates of $\mathcal{P}_{r}$ applied to observables with a singularity at $\beta$ of order $\alpha$ is dependent on the structure of the $\omega$-limit set of $\beta$ with respect to $T_{r}$. Having a singularity it seems that such observables do not fall into any of the function classes on which convergence to equilibrium has been previously shown.

Fibonacci-like unimodal inverse limit spaces and the core Ingram conjecture

We study the structure of inverse limit space of so-called Fibonacci-like tent maps. The combinatorial constraints implied by the Fibonacci-like assumption allow us to introduce certain chains that enable a more detailed analysis of symmetric arcs within this space than is possible in the general case. We show that link-symmetric arcs are always symmetric or a well-understood concatenation of quasi-symmetric arcs. This leads to the proof of the Ingram Conjecture for cores of Fibonacci-like unimodal inverse limits.

Bifurcation structure in the skew tent map and its application as a border collision normal form

The goal of the present paper is to collect the results related to dynamics of a one-dimensional piecewise linear map widely known as the skew tent map. These results may be useful for the researchers working on theoretical and applied problems in the field of nonsmooth dynamical systems. In particular, we propose the complete description of the bifurcation structure of the parameter space of the skew tent map, providing the related proofs. It is also shown how these results can be used to classify border collision bifurcations in one-dimensional piecewise smooth maps.