One-dimensional Maps Research Articles

Dynamics of a new generalized fractional one-dimensional map: quasiperiodic to chaotic

Discovering new chaotic maps is always essential for secure communication, cryptography, image encryption and decryption when pseudo-number generation is mandatory; however, it is still very fascinating to come across new complex dynamics of very simple maps exhibiting chaotic behavior. Despite the various forms already presented in the literature, we deal with the fractional forms of one-dimensional chaotic map with one system parameter; yet while generalization, two parameters were inserted to the map as the multiplier and the power. Therefore, in this paper, we present a novel and generalized version of a map exhibiting a strange behavior in discrete time and real number space, while detailed analyses regarding the new map with intervals of various parameters are also included. We mainly focus on a simple one-dimensional chaotic map and propose various instances with linear stability, bifurcation and Lyapunov analyses for each instance, to enhance the understanding of unstable fractional chaotic maps. It is found that the fractional map exhibits quasiperiodicity as well as periodic behavior for the smallest power parameter; while the chaotic states emerge for larger values.

Pynamical: Model and visualize discrete nonlinear dynamical systems, chaos, and fractals

Author(s): Boeing, Geoff | Abstract: Pynamical is an educational Python package for introducing the modeling, simulation, and visualization of discrete nonlinear dynamical systems and chaos, focusing on one-dimensional maps (such as the logistic map and the cubic map). Pynamical facilitates defining discrete one-dimensional nonlinear models as Python functions with just-in-time compilation for fast simulation. It comes packaged with the logistic map, the Singer map, and the cubic map predefined. The models may be run with a range of parameter values over a set of time steps, and the resulting numerical output is returned as a pandas DataFrame. Pynamical can then visualize this output in various ways, including with bifurcation diagrams, two-dimensional phase diagrams, three-dimensional phase diagrams, and cobweb plots. These visualizations enable simple qualitative assessments of system behavior including phase transitions, bifurcation points, attractors and limit cycles, basins of attraction, and fractals.

Noise and multistability in the square root map

In this paper we describe the complex structure of the basins of attraction of stable periodic orbits of the one-dimensional square root map and how this produces sensitivity to the addition of small amplitude noise. In particular we focus on how noise of varying amplitudes affects the system in parameter regions of attractor coexistence and also how trajectories jump between different periodic behaviours. We show that there is a non-monotonic relationship between the noise amplitude and the proportion of time spent in each periodic behaviour. These relationships are explained by examining approximations of steady-state distributions of trajectory deviations due to noise and the complicated deterministic structures of the map. We also show how the effect of noise scales on consecutive intervals of multistability.

Coupled Map Lattices as Musical Instruments

Abstract We use one-dimensional coupled map lattices (CMLs) to generate sounds that reflect their spatial organization and temporal evolution from a random initial configuration corresponding to uncorrelated noise. In many instances, the process approaches an equilibrium, which generates a sustained tone. The pitch of this tone is proportional to the lattice size, so the CML behaves like an instrument that could be tuned. Among exceptional cases, we provide an example with a metastable strange attractor, which produces an evolving sound reminiscent of drone music.

An Efficient PV Power Optimizer With Reduced EMI Effects: Map-Based Analysis and Design Technique

The aim of this paper is to develop a one-dimensional (1-D) discrete-time model or map that will ensure reliable and safe chaotic operation of a modular photovoltaic (PV) power optimizer (PO) system. Based on this 1-D map, we perform the detailed bifurcation analysis and discuss a preliminary design guideline to operate the system into a desired chaotic regime, i.e., just after the golden mean. We show that operating POs in chaotic regime cannot only yield broader power spectrum with reduced spectral peak at the switching frequency harmonics of the system, but also exhibits faster tracking responses with overall conversion efficiency $\eta \geq$ 98%. Moreover, this optimized frequency-domain as well as time-domain performance are numerically analyzed and then verified experimentally using a prototype modular boost-type PV system.

RGB image encryption using microcontroller ATMEGA 32

The encryption of RGB colour image using microcontroller ATMEGA 32 is studied in the present paper via synchronization with one-dimensional chaotic map. Two microcontrollers are used to execute encryption and decryption algorithms under synchronized condition to demonstrate the cryptosystem. Both the microcontrollers run the dimensional logistic map and they become synchronized through open plus closed loop (OPCL) coupling. Under synchronized condition, the stored data of the input image in one microcontroller is encrypted with its generated chaotic sequences and transmitted to the remaining microcontroller, known as the receiver. The receiver microcontroller decrypts the encrypted data and stores in its memory. After decryption of the full image data it is forwarded to LCD display to visualize the decrypted image. The strength of security of the algorithm is analyzed by a number of cryptanalysis techniques like, correlation coefficient, histogram analysis, information entropy, NPCR, UACI, MAE, etc.

Plasticity in inhomogeneously strained Au nanowires studied by Laue microdiffraction

ABSTRACTPlasticity in as-grown gold nanowires deformed in three-point bending configuration was studied by Laue microdiffraction. One-dimensional orientation maps of the Au crystal along the nanowire were generated from which the deformation profile was inferred. The crystal lattice was found to rotate continuously around the Au $[\bar{2}11]$ direction, which is transverse to the wire axis evidencing the storage of geometrically necessary dislocations (GNDs). The analysis of the diffraction peak shape points to the activation of multiple slip systems in contrast to the formation of wedge shaped twins predicted by MD simulations.

Discrete dynamical laser equation for the critical onset of bistability, entanglement and disappearance

We transform the semi-classical laser equation for single mode homogeneously broadened lasers to a one-dimensional nonlinear map by using the discrete dynamical approach. The obtained mapping, referred to as laser logistic mapping (LLM), characteristically exhibits convergent, cyclic and chaotic behavior depending on the control parameter. Thus, the so obtained LLM explains stable, bistable, multi-stable, and chaotic solutions for output field intensity. The onset of bistability takes place at a critical value of the effective gain coefficient. The obtained analytical results are confirmed through numerical calculations.

Circle maps with gaps: Understanding the dynamics of the two-process model for sleep–wake regulation

For more than 30 years the ‘two-process model’ has played a central role in the understanding of sleep/wake regulation. This ostensibly simple model is an interesting example of a non-smooth dynamical system, whose rich dynamical structure has been relatively unexplored. The two-process model can be framed as a one-dimensional map of the circle, which, for some parameter regimes, has gaps. We show how border collision bifurcations that arise naturally in maps with gaps extend and supplement the Arnold tongue saddle-node bifurcation set that is a feature of continuous circle maps. The novel picture that results shows how the periodic solutions that are created by saddle-node bifurcations in continuous maps transition to periodic solutions created by period-adding bifurcations as seen in maps with gaps.

A piecewise smooth model of evolutionary game for residential mobility and segregation.

The paper proposes an evolutionary version of a Schelling-type dynamic system to model the patterns of residential segregation when two groups of people are involved. The payoff functions of agents are the individual preferences for integration which are empirically grounded. Differently from Schelling's model, where the limited levels of tolerance are the driving force of segregation, in the current setup agents benefit from integration. Despite the differences, the evolutionary model shows a dynamics of segregation that is qualitatively similar to the one of the classical Schelling's model: segregation is always a stable equilibrium, while equilibria of integration exist only for peculiar configurations of the payoff functions and their asymptotic stability is highly sensitive to parameter variations. Moreover, a rich variety of integrated dynamic behaviors can be observed. In particular, the dynamics of the evolutionary game is regulated by a one-dimensional piecewise smooth map with two kink points that is rigorously analyzed using techniques recently developed for piecewise smooth dynamical systems. The investigation reveals that when a stable internal equilibrium exists, the bimodal shape of the map leads to several different kinds of bifurcations, smooth, and border collision, in a complicated interplay. Our global analysis can give intuitions to be used by a social planner to maximize integration through social policies that manipulate people's preferences for integration.

A financial market model with two discontinuities: Bifurcation structures in the chaotic domain.

We continue the investigation of a one-dimensional piecewise linear map with two discontinuity points. Such a map may arise from a simple asset-pricing model with heterogeneous speculators, which can help us to explain the intricate bull and bear behavior of financial markets. Our focus is on bifurcation structures observed in the chaotic domain of the map's parameter space, which is associated with robust multiband chaotic attractors. Such structures, related to the map with two discontinuities, have been not studied before. We show that besides the standard bandcount adding and bandcount incrementing bifurcation structures, associated with two partitions, there exist peculiar bandcount adding and bandcount incrementing structures involving all three partitions. Moreover, the map's three partitions may generate intriguing bistability phenomena.

Chaotic Itinerancy Observed in Mutually Coupled Gaussian Maps

A one-dimensional Gaussian map defined by a Gaussian function describes a discrete-time dynamical system. Chaotic behavior can be observed in both Gaussian and logistic maps. This study analyzes the bifurcation structure corresponding to the fixed and periodic points of a coupled system comprising two Gaussian maps. The bifurcation structure of a mutually coupled Gaussian map is more complex than that of a mutually coupled logistic map. In a coupled Gaussian map, it was confirmed that after a stable fixed point or stable periodic points became unstable through the bifurcation, the points were able to recover their stability while the system parameters were changing. Moreover, we investigated a parameter region in which symmetric and asymmetric stable fixed points coexisted. Asymmetric unstable fixed point was generated by the [Formula: see text]-type branching of a symmetric stable fixed point. The stability of the unstable fixed point could be recovered through period-doubling and tangent bifurcations. Furthermore, a homoclinic structure related to the occurrence of chaotic behavior and invariant closed curves caused by two-periodic points was observed. The mutually coupled Gaussian map was merely a two-dimensional dynamical system; however, chaotic itinerancy, known to be a characteristic property associated with high-dimensional dynamical systems, was observed. The bifurcation structure of the mutually coupled Gaussian map clearly elucidates the mechanism of chaotic itinerancy generation in the two-dimensional coupled map. We discussed this mechanism by comparing the bifurcation structures of the Gaussian and logistic maps.

A hybrid intelligent technique for model order reduction in the delta domain: a unified approach

This paper presents a unified method for reduced-order modelling of higher-order discrete-time systems in the complex delta domain. The approach is unified in the sense that it is fundamentally discrete time, but at a high sampling frequency converges to its continuous-time counterpart. New hybrid algorithms with a blend of grey wolf optimizer (GWO) and chaotic firefly algorithm (CFA) using iterative chaotic map have been proposed for the order reduction of large-scale systems in the delta domain. GWO explores the entire search domain while CFA carries out local search in the proposed hybrid technique with different tuning parameters of FA improvised by one-dimensional iterative map, thus improving the convergence speed and the quality of the solutions. Two examples of single-input single-output systems are taken up to establish the usefulness of the proposed method. The hybrid approach not only yields good matching in the delta domain, but also delivers quality solutions (minimum fitness value) and provides faster convergence speed than the existing techniques. The performances of the reduced systems are also compared with their respective original systems as well as reduced systems reported in the literature in terms of time-domain and frequency-domain parameters. The proposed topology also proved its superiority in terms of some benchmark error indices well established in the literature of systems approach and control.

Constructions of conformal measures for generalized interval exchange transformations

For onedimensional piecewise monotone discontionuous maps without periodic points, the 1-conformal measures are constructed and, as a corollary, semiconjugacy to piecewise linear models of constant (in absolute value) slope is obtained. It turns out that for generalized interval exchange transformations, the constructed semiconjugacy is, in fact, conjugacy.

Sine-Transform-Based Chaotic System With FPGA Implementation

As chaotic dynamics is widely used in nonlinear control, synchronization communication, and many other applications, designing chaotic maps with complex chaotic behaviors is attractive. This paper proposes a sine-transform-based chaotic system (STBCS) of generating one-dimensional (1-D) chaotic maps. It performs a sine transform to the combination of the outputs of two existing chaotic maps (seed maps). Users have the flexibility to choose any existing 1-D chaotic maps as seed maps in STBCS to generate a large number of new chaotic maps. The complex chaotic behavior of STBCS is verified using the principle of Lypunov exponent. To show the usability and effectiveness of STBCS, we provide three new chaotic maps as examples. Theoretical analysis shows that these chaotic maps have complex dynamics properties and robust chaos. Performance evaluations demonstrate that they have much larger chaotic ranges, better complexity, and unpredictability, compared with chaotic maps generated by other methods and the corresponding seed maps. Moreover, to show the simplicity of STBCS in hardware implementation, we simulate the three new chaotic maps using the field-programmable gate array (FPGA).

On the Transfer of a Number of Concepts of Statistical Radiophysics to the Theory of One-dimensional Point Mappings

In the article, the possibility of using a bispectrum under the investigation of regular and chaotic behaviour of one-dimensional point mappings is discussed. The effectiveness of the transfer of this concept to nonlinear dynamics was demonstrated by an example of the Feigenbaum mapping. Also in the work, the application of the Kullback-Leibler entropy in the theory of point mappings is considered. It has been shown that this information-like value is able to describe the behaviour of statistical ensembles of one-dimensional mappings. In the framework of this theory some general properties of its behaviour were found out. Constructivity of the Kullback-Leibler entropy in the theory of point mappings was shown by means of its direct calculation for the ”saw tooth” mapping with linear initial probability density. Moreover, for this mapping the denumerable set of initial probability densities hitting into its stationary probability density after a finite number of steps was pointed out.

Laminar Chaos.

We show that the output of systems with time-varying delay can exhibit a new kind of chaotic behavior characterized by laminar phases, which are periodically interrupted by irregular bursts. Within each laminar phase the output intensity remains almost constant, but its level varies chaotically from phase to phase. In scalar systems, the periodic dynamics of the lengths and the chaotic dynamics of the intensity levels can be understood and also tuned via two one-dimensional maps, which can be deduced from the nonlinearity of the delay equation and from the delay variation, respectively.

Robustification of a One-Dimensional Generic Sigmoidal Chaotic Map with Application of True Random Bit Generation.

The search for generation approaches to robust chaos has received considerable attention due to potential applications in cryptography or secure communications. This paper is of interest regarding a 1-D sigmoidal chaotic map, which has never been distinctly investigated. This paper introduces a generic form of the sigmoidal chaotic map with three terms, i.e., xn+1 = ∓AfNL(Bxn) ± Cxn ± D, where A, B, C, and D are real constants. The unification of modified sigmoid and hyperbolic tangent (tanh) functions reveals the existence of a “unified sigmoidal chaotic map” generically fulfilling the three terms, with robust chaos partially appearing in some parameter ranges. A simplified generic form, i.e., xn+1 = ∓fNL(Bxn) ± Cxn, through various S-shaped functions, has recently led to the possibility of linearization using (i) hardtanh and (ii) signum functions. This study finds a linearized sigmoidal chaotic map that potentially offers robust chaos over an entire range of parameters. Chaos dynamics are described in terms of chaotic waveforms, histogram, cobweb plots, fixed point, Jacobian, and a bifurcation structure diagram based on Lyapunov exponents. As a practical example, a true random bit generator using the linearized sigmoidal chaotic map is demonstrated. The resulting output is evaluated using the NIST SP800-22 test suite and TestU01.

S-box design method based on improved one-dimensional discrete chaotic map

ABSTRACTA new method for obtaining random bijective S-boxes based on improved one-dimensional discrete chaotic map is presented. The proposed method uses a new special case of discrete chaotic map based on the composition of permutations, in order to overcome the problem with potentially short length of the orbits. The proposed special case is based on the composition of permutations and sine function and has a larger minimum length of the orbits compared to the previous special case of the discrete-space chaotic map. The results of performance test show that the example of S-box generated by the proposed method has good cryptographic properties. The proposed method can achieve large key space, which makes it suitable for generation of larger S-boxes, and the process of generation of S-boxes is not affected by approximations of any kind. Also, proposed method has potential to operate at greater speed and with smaller memory requirements than previous S-box generation method based on discrete space chaotic map, which can be particularly useful for lightweight devices such as wireless sensor networks.

Parametrization of unstable manifolds and Fatou disks for parabolic skew-products

We study the existence of Fatou components on parabolic skew product maps. We focus on skew products in which each coordinate has a fixed point that is parabolic. As in the geometrically attracting case, we prove that there exists maps F that have one-dimensional disks that are mapped to a point in the Julia set of the restriction of F to an invariant one-dimensional fiber. We first prove a linearization theorem for a one-dimensional map, then for a parabolic skew product. Finally, we apply this result to construct the skew product map described above.