Narrow Results

Coupled map lattices are a popular and computationally simpler model of pattern formation in nonlinear systems. In this work, we investigate three-site interactions with linear multiplicative coupling in one-dimensional coupled logistic maps that cannot be decomposed into pairwise interactions. We observe the transition to synchronization and the transition to long-range order in space. We coarse-grain the phase space in regions and denote them by spin values. We use two quantifiers the flip rate $F(t)$ that quantify departure from expected band-periodicity as an order parameter. We also study a non-Markovian quantity, known as persistence $P(t)$ to study dynamic phase transitions. Following transitions are observed. (a) Transition to two band attractor state: At this transition $F(t)$ as well as $P(t)$ shows a power-law decay in the range of coupling parameters. Here all sites reach one of the bands. The $F(t)$ as well as $P(t)$ decays as power-law with the decay exponent ${\delta }_1=0.46$ and ${\eta }_1=0.28$, respectively. (b) The transition from a fluctuating chaotic state to a homogeneous synchronized fixed point: Here both the quantifiers $F(t)$ and $P(t)$ show power-law decay with decay exponent ${\delta }_2=1$ and ${\eta }_2=0.11$, respectively. We compare the transitions with the case, where pairwise interactions are also present. The spatiotemporal evolution is analyzed as the coupling parameter is varied.

Reservoir Computing (RC) is a type of machine learning inspired by neural processes, which excels at handling complex and time-dependent data while maintaining low training costs. RC systems generate diverse reservoir states by extracting features from raw input and projecting them into a high-dimensional space. One key advantage of RC networks is that only the readout layer needs training, reducing overall training expenses. Memristors have gained popularity due to their similarities to biological synapses and compatibility with hardware implementation using various devices and systems. Chaotic events, which are highly sensitive to initial conditions, undergo drastic changes with minor adjustments. Cascade chaotic maps, in particular, possess greater chaotic properties, making them difficult to predict with memoryless devices. This study aims to predict 1D and 2D cascade chaotic time series using a memristor-based hierarchical RC system.

This short paper is focused on the bifurcation theory found in map functions called evolution functions that are used in dynamical systems. The most well-known example of discrete iterative function is the logistic map that puts into evidence bifurcation and chaotic behavior of the topology of the logistic function. We propose a new iterative function based on Lorentizan function and its generalized versions, based on numerical study, it is found that the bifurcation of the Lorentzian function is of second-order where it is characterized by the absence of chaotic region.

This paper explores the deep-zoom properties of the chaotic $k$-logistic map, in order to propose an improved chaos-based cryptosystem. This map was shown to enhance the random features of the Logistic map, while at the same time reducing the predictability about its orbits. We incorporate its strengths to security into a previously published cryptosystem to provide an optimal pseudo-random number generator (PRNG) as its core operation. The result is a reliable method that does not have the weaknesses previously reported about the original cryptosystem.

The permutation entropy (PE) is a statistical measure which can describe complexity of time series. In recent years, the research on PE is increasing gradually. As part of its application, the complexity–entropy causality plane (CECP) and weighted CECP (WCECP) have been recently used to distinguish the stage of stock market development. In this paper, we focus on weighted Rényi entropy causality plane (WRECP), and then extend WCECP and WRECP into multiscale WCECP (MWCECP) and multiscale WRECP (MWRECP) by introducing a new parameter scale. By data simulating and analyzing, we show the power of WRECP. Besides, we discuss the MWCECP and the MWRECP of adjacent scales. It reveals a gradual relationship between adjacent weighted scale entropies.

This paper analyzes the security of a recent cryptosystem based on the ergodicity property of chaotic maps. It is shown how to obtain the secret key using a chosen-ciphertext attack. Some other design weaknesses are also shown.

With the occasional feedback method, the chaotic logistic map is stabilized at an unstable low-periodic orbit. In our method, the qualified feedback coefficient can be obtained through calculation instead of through simulation. Besides, the bifurcation control of the logistic map is studied, and a new scheme is proposed to change the parameter value of any one bifurcation point of this dynamic system optionally. Simulation results show the effectiveness of the methods.

This paper corrects some mistakes about three theorems, two figures and some expressions on “finite field” in a literature, and then it analyzes the digraphs and period properties of the logistic map on residue class rings $Z(3^n)$ and $Z(p^n)$. Some theorems and conjectures are proved or given. To validate these properties, we make some numeral experiments and draw the relative figures and tables. So the theoretical analysis and numeral experiments show that there exists periods in the logistic map on $Z(3^n)$ and on $Z(p^n)$ and it can be applied in the pseudo-random generators, cryptography, spread spectrum communications and so on.

Given the increasing complexity of cryptographic devices, testing their security level against existing attacks requires a fast simulation environment. SystemC is a standard language that is widely used for the modeling and the verification of complex systems. It is a promising candidate in Electronic System Level (ESL) which allows models to reach higher simulation speed. Accordingly, the Advanced Encryption Standard (AES) is one of the most known block ciphers. It is widely used in various applications in order to secure the sensitive data. It is considered to be secure. Still, some issues lie in the used key and the S-Box. This paper presents a SystemC implementation of a chaos-based crypto-processor for the AES algorithm. The design of the proposed architecture is studied using the SystemC tools. The proposed correction approach exploits the chaos theory properties to cope with the defaulting parameters of the AES algorithm. Detailed experimental results are given in order to evaluate the security level and the performance criteria. In fact, the proposed crypto-system presents numerous interesting features, including a high security level, a pixel distributing uniformity, a sufficiently large key-space with improved key sensitivity, and an acceptable speed.

We study the one-dimensional logistic map with control parameter perturbed by a small periodic function. In the pure constant case, scaling arguments are used to obtain the exponents related to the relaxation of the trajectories at the exchange of stability, period-doubling and tangent bifurcations. In particular, we evaluate the exponent z which describes the divergence of the relaxation time τ near a bifurcation by the relation τ ~ | R - R_c |^-z. Here, R is the control parameter and R_c is its value at the bifurcation. In the time-dependent case new attractors may appear leading to a different bifurcation diagram. Beside these new attractors, complex attractors also arise and are responsible for transients in many trajectories. We obtain, numerically, the exponents that characterize these transients and the relaxation of the trajectories.

We compute explicitly the irreducible polynomial corresponding to the bifurcation point B₄ of the logistic map. This settles in the positive recent two conjectures by Bailey and Broadhurst.

It is said that the numerical generation of exact chaotic time series by iterating, for example, the logistic map, will be impossible, because chaos has a high dependency on initial values. In this letter, an algorithm to generate them without the accumulation of inevitable round-off errors caused by the iteration is proposed, where rational numbers are introduced. Also, it is shown that the period of the chaotic time series depends on the rational numbers including large prime numbers, which are fundamentally related to the Mersenne and the Fermat prime ones. Since the time series are numerically regenerated by the proposed algorithm in an usual computer environment, it could be applied to cryptosystems which do not need the synchronization, and have a large key-space by using large prime numbers.

We present a computer-assisted analysis of the phase space features and bifurcations of a noninvertible, discrete-time system. Our focus is on the role played by noninvertibility in generating disconnected basins of attraction and the breakup of invariant circle solutions. Transitions between basin of attraction structures are identified and organized according to "levels of complexity," a term we define in this paper. In particular, we present an algorithm that provides a computational approximation to the boundary (in phase space) separating points with different preimage behavior. The interplay between this boundary and other phase space features is shown to be crucial in understanding global bifurcations and transitions in the structure of the basin of attraction.

The invariant Cantor sets of the logistic map g_μ(x) = μx(1 - x) for μ > 4 are hyperbolic and form a continuous family. We show that this family can be obtained explicitly through solutions of initial value problems for a system of infinitely coupled differential equations due to the hyperbolicity. The same result also applies to the tent map T_a(x) = a(1/2 - |1/2 - x|) for a > 2.

In this paper, we propose a robust hyper-chaotic system that is suitable for digital secure-communication. The system consists of many coupled robust logistic maps that form a hyper-chaotic system. It has a higher degree of complexity than traditional discrete-time secure-communication systems that use only a single map. Moreover, the system has a very large parameter space which grows along with system precision. Hence, attacking the system by the method of map reconstruction in current computation technology is not feasible. Statistical analysis shows that the system achieves very high security level. Finally, two hardware architectures (multiple-cycle and pipelined) are proposed for area and performance optimization, respectively.

This paper studies the construction of one-dimensional real chaotic polynomial maps. Given an arbitrary nonzero polynomial of degree m (≥ 0), two methods are derived for constructing chaotic polynomial maps of degree m + 2 by simply multiplying the given polynomial with suitably designed quadratic polynomials. Moreover, for m + 2 arbitrarily given different positive constants, a method is given to construct a chaotic polynomial map of degree 2m based on the coupled-expansion theory. Furthermore, by multiplying a real parameter to a special kind of polynomial, which has at least two different non-negative or nonpositive zeros, the chaotic parameter region of the polynomial is analyzed based on the snap-back repeller theory. As a consequence, for any given integer n ≥ 2, at least one polynomial of degree n can be constructed so that it is chaotic in the sense of both Li–Yorke and Devaney. In addition, two natural ways of generalizing the logistic map to higher-degree chaotic logistic-like maps are given. Finally, an illustrative example is provided with computer simulations for illustration.

This paper proposes a new technique for generating random-looking binary digits based on an irregularly decimated chaotic map. We present a class of irregularly decimated chaos-based keystream generators, related to the shrinking generator, for the generation of binary sequences. Each generator consists of two subsystems: a control subsystem and a generating subsystem, where each subsystem is based on a single chaotic map. This chaotic map is presented as a 1-D piecewise chaotic map related to the chaotic logistic map. We conduct an analysis of the dynamical behavior of the proposed map to integrate it as a component in the proposed generators subsystems. The output bits of these keystream generators are produced by applying a threshold function to convert the floating-point iterates of the irregularly decimated map into a binary form. The generated keystream bits are demonstrated to exhibit high level of security, long period length, high linear complexity measure and random-like properties at given certain parameter values. Standard statistical tests on the proposed generators, as well as other keystream generators, are performed and compared.

In this paper, we present a class of chaotic clock-controlled cipher systems based on two exact chaotic maps, where each map is capable of generating exact chaotic time series of the logistic map. This class is designed in such a way that one map controls the iterations of the second map. The suggested technique results in generating orbits possessing long cycle length and high level of security from the two periodic exact maps. In the first part of this paper, two keystream generators based on two exact chaotic logistic maps are suggested for use in cryptographic applications. The necessary conditions to generate orbits with guaranteed long enough cycle length are established. Furthermore, the generated keystreams are demonstrated to possess excellent randomness properties. In the second part, we suggest a clock-controlled encryption scheme related to Baptista's scheme and based on two exact chaotic logistic maps. This technique results in increasing the size of the key space, and hence may increase the security level against some existing cryptanalytic attacks. Furthermore, it leads to reducing the size of the ciphertext file and propably increasing the encryption speed.

When memory is endowed in the dynamics of the logistic map, its parameter may reach, without diverging, values than are greater than its upper limit in the ahistoric, conventional formulation. A numerical study is made in this study on the dynamics of the logistic map with memory m_T: x_T+1 = λm_T(1-m_T), when λ > 4.0 or λ < -2.0.

We provide a local saddle-node bifurcation result for quasiperiodically forced interval maps. As an application, we give a rigorous description of saddle-node bifurcations of 3-periodic graphs in the quasiperiodically forced logistic map with small forcing amplitude.