Simple One-dimensional Map Research Articles

Time crystals transforming frequency combs in tunable photonic oscillators.

The response of a tunable photonic oscillator, consisting of an optically injected semiconductor laser, under an injected frequency comb is considered with the utilization of the concept of the time crystal that has been widely used for the study of driven nonlinear oscillators in the context of mathematical biology. The dynamics of the original system reduce to a radically simple one-dimensional circle map with properties and bifurcations determined by the specific features of the time crystal fully describing the phase response of the limit cycle oscillation. The circle map is shown to accurately model the dynamics of the original nonlinear system of ordinary differential equations and capable for providing conditions for resonant synchronization resulting in output frequency combs with tunable shape characteristics. Such theoretical developments can have potential for significant photonic signal-processing applications.

A fractional-order discrete memristor neuron model: Nodal and network dynamics

<abstract><p>We discuss the dynamics of a fractional order discrete neuron model with electromagnetic flux coupling. The discussed neuron model is a simple one-dimensional map which is modified by considering flux coupling. We consider a discrete fractional order memristor to mimic the effects of electromagnetic flux on the neuron model. The bifurcation dynamics of the fractional order neuron map show an inverse period-doubling route to chaos as a function of control parameters, namely the fractional order of the map and the flux coupling coefficient. The bifurcation dynamics of the systems are derived both in the time and frequency domains. We present a two-parameter phase diagram using the Lyapunov exponent to categorize the various dynamics present in the system. In addition to the Lyapunov exponent, we use the entropy of the model to distinguish the various dynamics of the systems. To investigate the network behavior of the fractional order neuron map, a lattice array of $ N\times N $ nodes is constructed and external periodic stimuli are applied to the network. The formation of spiral waves in the network and the impact of various parameters, like the fractional order, flux coupling coefficient and the coupling strength on the wave propagation are also considered in our analysis.</p></abstract>

A new fractional one dimensional chaotic map and its application in high-speed image encryption

Chaos theory has been widely used in the design of image encryption schemes. Some low-dimensional chaotic maps have been proved to be easily predictable because of their small chaotic space. On the other hand, high-dimensional chaotic maps have a larger chaotic space. However, their structures are too complicated, and consequently, they are not suitable for real-time image encryption. Motivated by this, we propose a new fractional one-dimensional chaotic map with a large chaotic space. The proposed map has a simple structure and a high chaotic behavior in an extensive range of its control parameters values. Several chaos theoretical tools and tests have been carried out to analyze and prove the proposed map’s high chaotic behavior. Moreover, we use the proposed map in the design of a novel real-time image encryption scheme. In this new scheme, we combine the substitution and permutation stages to simultaneously modify both of the pixels’ positions and values. The merge of these two stages and the use of the new simple one-dimensional chaotic map significantly increase the proposed scheme’s security and speed. Besides, the simulation and experimental analysis prove that the proposed scheme has high performances.

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.

A new simple one-dimensional chaotic map and its application for image encryption

In this paper, we propose a new simple one-dimensional chaotic map. The chaotic characteristics have been declared by using bifurcation analysis and Lyapunov exponent analysis. Furthermore, we propose a new image encryption algorithm based on this new chaotic map. Both shuffling algorithm and substitution algorithm are related to this map. Many statistical tests and security analysis indicate that this algorithm has an excellent security performance, and can be competitive with some other recently proposed image encryption algorithms.

Deceleration of one-dimensional mixing by discontinuous mappings.

We present a computational study of a simple one-dimensional map with dynamics composed of stretching, permutations of equally sized cells, and diffusion. We observe that the combination of the aforementioned dynamics results in eigenmodes with long-time exponential decay rates. The decay rate of the eigenmodes is shown to be dependent on the choice of permutation and changes nonmonotonically with the diffusion coefficient for many of the permutations. The global mixing rate of the map M in the limit of vanishing diffusivity approximates well the decay rates of the eigenmodes for small diffusivity, however this global mixing rate does not bound the rates for all values of the diffusion coefficient. This counterintuitively predicts a deceleration in the asymptotic mixing rate with an increasing diffusivity rate. The implications of the results on finite time mixing are discussed.

Inverse period-doubling bifurcations determine complex structure of bursting in a one-dimensional non-autonomous map.

We propose a simple one-dimensional non-autonomous map, in which some novel bursting patterns (e.g., "fold/double inverse flip" bursting, "fold/multiple inverse flip" bursting, and "fold/a cascade of inverse flip" bursting) can be observed. Typically, these bursting patterns exhibit complex structures containing a chain of inverse period-doubling bifurcations. The active states related to these bursting can be period-2(n) (n = 1, 2, 3,…) attractors or chaotic attractors, which may evolve to quiescence by a chain of inverse period-doubling bifurcations when the slow excitation decreases through period-doubling bifurcation points of the map. This accounts for the complex inverse period-doubling bifurcation structures observed in bursting patterns. Our findings enrich the possible routes to bursting as well as the underlying mechanisms of bursting.

Capturing correlations in chaotic diffusion by approximation methods

We investigate three different methods for systematically approximating the diffusion coefficient of a deterministic random walk on the line that contains dynamical correlations that change irregularly under parameter variation. Capturing these correlations by incorporating higher-order terms, all schemes converge to the analytically exact result. Two of these methods are based on expanding the Taylor-Green-Kubo formula for diffusion, while the third method approximates Markov partitions and transition matrices by using a slight variation of the escape rate theory of chaotic diffusion. We check the practicability of the different methods by working them out analytically and numerically for a simple one-dimensional map, study their convergence, and critically discuss their usefulness in identifying a possible fractal instability of parameter-dependent diffusion, in the case of dynamics where exact results for the diffusion coefficient are not available.

MODELING AND BIFURCATION BEHAVIOR OF MULTI-PHASE SIMIMO DC–DC SWITCHING REGULATORS

In this paper, different discrete-time models in the form of maps are proposed and analyzed in order to describe the dynamics of single inductor multiple-input multiple-output (SIMIMO) switching DC–DC converters. These systems can be used to regulate generally multiple (positive and/or negative) outputs by means of individual switches associated to each of the outputs. These switches are current mode controlled through corresponding channels. The discrete-time approach allows the dynamical behavior of these systems to be accurately predicted as well as to detect possible subharmonic oscillations and chaotic behavior. Under certain operating conditions, for which the system can be modeled by a one-dimensional piecewise constant vector field, a simple one-dimensional and piecewise-linear (PWL) map can be obtained. Some closed form expressions for ensuring stability are derived from this map in terms of a stability index λ, which is, in turn, expressed in terms of system parameters. However, some discrepancies have been found between the switched model and this simpler map, and therefore a full order model is derived to obtain more accurate information about the actual dynamical behavior of these converters. The theoretical results are confirmed by one-dimensional bifurcation diagrams and codimension 1 two-parameter bifurcation curves obtained by standard continuation methods applied to the derived discrete-time models as well as from computer simulations from the switched model.

Strange nonchaotic repellers

We show the existence of strange nonchaotic repellers--that is, systems with transient dynamics whose nonattracting invariant set is fractal, but whose maximum Lyapunov coefficient is zero. We introduce the concept using a simple one-dimensional map and argue that strange nonchaotic repellers are a general phenomenon, occurring in bifurcation points of transient chaotic systems. All strange nonchaotic systems studied to date have been attractors; here, it is revealed that strange nonchaotic sets are also present in transient systems.

Fractal diffusion coefficient from dynamical zeta functions

Dynamical zeta functions provide a powerful method to analyse low-dimensional dynamical systems when the underlying symbolic dynamics is under control. On the other hand, even simple one-dimensional maps can show an intricate structure of the grammar rules that may lead to a non-smooth dependence of global observables on parameters changes. A paradigmatic example is the fractal diffusion coefficient arising in a simple piecewise linear one-dimensional map of the real line. Using the Baladi–Ruelle generalization of the Milnor–Thurnston kneading determinant, we provide the exact dynamical zeta function for such a map and compute the diffusion coefficient from its smallest zero.

Nonlinear Analysis of the Iterative Decoding of Parallel Concatenated Convolutional Codes

In this correspondence, we introduce a simple one-dimensional (1-D) nonlinear map to describe the iterates of the bit-error rate (BER) of parallel-concatenated convolutional codes (PCCC) on the binary-input Gaussian channel. A lower bound on this map is derived based upon the weight enumerator of the constituent codes, thus enabling the characterization of the dynamics of the decoder in terms of fixed points, along with the associated stability analysis.

Observation of a continuous interior crisis in the Hindmarsh-Rose neuron model.

Interior crises are understood as discontinuous changes of the size of a chaotic attractor that occur when an unstable periodic orbit collides with the chaotic attractor. We present here numerical evidence and theoretical reasoning which prove the existence of a chaos-chaos transition in which the change of the attractor size is sudden but continuous. This occurs in the Hindmarsh-Rose model of a neuron, at the transition point between the bursting and spiking dynamics, which are two different dynamic behaviors that this system is able to present. Moreover, besides the change in attractor size, other significant properties of the system undergoing the transitions do change in a relevant qualitative way. The mechanism for such transition is understood in terms of a simple one-dimensional map whose dynamics undergoes a crossover between two different universal behaviors.

Homotopy of exact coherent structures in plane shear flows

Three-dimensional steady states and traveling wave solutions of the Navier–Stokes equations are computed in plane Couette and Poiseuille flows with both free-slip and no-slip boundary conditions. They are calculated using Newton’s method by continuation of solutions that bifurcate from a two-dimensional streaky flow then by smooth transformation (homotopy) from Couette to Poiseuille flow and from free-slip to no-slip boundary conditions. The structural and statistical connections between these solutions and turbulent flows are illustrated. Parametric studies are performed and the parameters leading to the lowest onset Reynolds numbers are determined. In all cases, the lowest onset Reynolds number corresponds to spanwise periods of about 100 wall units. In particular, the rigid-free plane Poiseuille flow traveling wave arises at Reτ=44.2 for Lx+=273.7 and Lz+=105.5, in excellent agreement with observations of the streak spacing. A simple one-dimensional map is proposed to illustrate the possible nature of the “hard” transition to shear turbulence and connections with the unstable exact coherent structures.

A multiple-access technique for differential chaos-shift keying

A multiple-access technique for use with differential chaos shift keying is proposed and analyzed in this paper. A simple one-dimensional iterative map is used to generate the chaotic signals for all users and the average data rates for the users are identical. Bit-error rates are derived numerically for different numbers of users and computer simulations are performed to verify the results.

Protein analysis of an individual Caenorhabditis elegans single-cell embryo by capillary electrophoresis

We present a simple one-dimensional electrophoretic map of the expressed proteins in a Caenorhabditis elegans embryo. The embryo was taken from an adult nematode, injected into a 50-μm I.D. capillary, and lysed. The proteins were fluorescently labeled and then separated by capillary electrophoresis and detected by laser-induced fluorescence. Over 20 components were resolved in the 22-min separation. The dynamic range was outstanding for this separation, noise in the baseline was less than 0.01% the amplitude of the largest component.

Jordan blocks in a one-dimensional Markov map

A simple one-dimensional chaotic map, whose spectral decomposition of the Frobenius-Perron operator in the space of polynomials has Jordan block structure is given and analyzed. The Jordan block structure implies that some correlation functions have exponential decay modified by coefficients, which are polynomials in t. Such a correlation is given for this system and a numerical simulation, which agrees with the theoretical prediction is presented.

Low Voltage Bifurcations in a Diode Resonator

We have observed an interesting bifurcation sequence at very low driving voltages applied to a diode resonator. As the driving voltage increases, an ordinary period-doubling bifurcation appears. Further increase of the driving voltage results in a reverse period-doubling bifurcation and a sudden jump of an amplitude of the current oscillation of the diode. The experimental results were compared with the results of numerical simulations of the equivalent circuit of the diode resonator and they were in good agreement. This suggests that these low-voltage bifurcations are due to nonlinearity of the capacitance and current of the diode to the applied voltage. A simple one-dimensional map explaining these low-voltage bifurcations of the diode resonator is introduced and its properties are discussed.

Fine Structures in Stationary and Non-Stationary Chaos: Numerical Approach to the Law of Large Number

Large deviation properties in the stationary and non-stationary chaos regimes are numerically studied by using a simple one-dimensional chaotic map. Some ergodic-theoretical quantities and scaling indices are precisely determined by computer simulations, and they are compared with theoretical predictions which were derived in previous papers. The behaviors in stationary regime are almost completely consistent with theoretical estimations, but in the non-stationary region some small difference between numerical results and theoretical predictions are observed in the global characters of the entropy function, the Allan variance and the distribution function of the Lyapunov exponent. New results for several scaling indices in non-stationary chaos are derived

A low-order model of the shear instability of convection: chaos and the effect of noise

The authors consider a third-order system of ordinary differential equations aimed at modelling the shear instability of tall thin cells as found, for example, in thermohaline convection. The nonlinear interactions between the zeroth and first harmonics of the convective cells (A and B) fuel the growth of a horizontal shear (C) which, in turn, becomes of sufficient magnitude to destabilise the convective motions. The most interesting feature of the model is that in certain parameter regimes a small amount of random noise has a tremendous influence on the dynamics-in others the noise is unimportant and the system displays many typical features of low-order dynamical systems, such as sequences of period-doubling bifurcations. When the decay rates of B and C are comparable, analytic solutions can be found for both regimes; for the former by solving the Fokker-Planck equation describing the probability distribution of A, B and C, for the latter by reducing the equations to a simple one-dimensional map. These analytic results are a considerable aid in understanding the numerical solutions of the equations for general parameter values.