One-dimensional Nonlinear Maps Research Articles

A novel multi-stable sinusoidal chaotic map with spectacular behaviors

Chaotic behavior can be observed in continuous and discrete-time systems. This behavior can appear in one-dimensional nonlinear maps; however, having at least three state variables in flows is necessary. Due to the lower mathematical complexity and computational cost of maps, lots of research has been conducted based on them. This paper aims to present a novel one-dimensional trigonometric chaotic map that is multi-stable and can act attractively. The proposed chaotic map is first analyzed using a single sinusoidal function; then, its abilities are expanded to a map with a combination of two sinusoidal functions. The stability conditions of both maps are investigated, and their different behaviors are validated through time series, state space, and cobweb diagrams. Eventually, the influence of parameter variations on the maps’ outputs is examined by one-dimensional and two-dimensional bifurcation diagrams and Lyapunov exponent spectra. Besides, the diversity of outputs with varying initial conditions reveals this map’s multi-stability. The newly designed chaotic map can be employed in encryption applications.

Various Auto-Correlation Functions of m-Bit Random Numbers Generated from Chaotic Binary Sequences.

This paper discusses the auto-correlation functions of m-bit random numbers obtained from m chaotic binary sequences generated by one-dimensional nonlinear maps. First, we provide the theoretical auto-correlation function of an m-bit sequence obtained by m binary sequences that are assumed to be uncorrelated to each other. The auto-correlation function is expressed by a simple form using the auto-correlation functions of the binary sequences. This implies that the auto-correlation properties of the m-bit sequences can be easily controlled by the auto-correlation functions of the original binary sequences. In numerical experiments using a computer, we generated m-bit random sequences using some chaotic binary sequences with prescribed auto-correlations generated by one-dimensional chaotic maps. The numerical experiments show that the numerical auto-correlation values are almost equal to the corresponding theoretical ones, and we can generate m-bit sequences with a variety of auto-correlation properties. Furthermore, we also show that the distributions of the generated m-bit sequences are uniform if all of the original binary sequences are balanced (i.e., the probability of 1 (or 0) is equal to ) and independent of one another.

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.

Relating high dimensional stochastic complex systems to low-dimensional intermittency

We evaluate the implication and outlook of an unanticipated simplification in the macroscopic behavior of two high-dimensional sto-chastic models: the Replicator Model with Mutations and the Tangled Nature Model (TaNa) of evolutionary ecology. This simplification consists of the apparent display of low-dimensional dynamics in the non-stationary intermittent time evolution of the model on a coarse-grained scale. Evolution on this time scale spans generations of individuals, rather than single reproduction, death or mutation events. While a local one-dimensional map close to a tangent bifurcation can be derived from a mean-field version of the TaNa model, a nonlinear dynamical model consisting of successive tangent bifurcations generates time evolution patterns resembling those of the full TaNa model. To advance the interpretation of this finding, here we consider parallel results on a game-theoretic version of the TaNa model that in discrete time yields a coupled map lattice. This in turn is represented, a la Langevin, by a one-dimensional nonlinear map. Among various kinds of behaviours we obtain intermittent evolution associated with tangent bifurcations. We discuss our results.

Flutter analysis of a nonlinear airfoil using stochastic approach

In this paper, the dynamic instability of a nonlinear system has been studied using the stochastic vibration analysis and employing statistical properties of the system response. In this method neither the time domain analysis nor limit cycle oscillations were used. A two degrees-of-freedom airfoil subjected to an aerodynamic quasi-steady flow with a nonlinear torsional spring was considered as the case study. The spring nonlinearity was examined in hardening and softening states. A random force in the form of the white noise with Gaussian function was added to the aerodynamic lift force. The statistical linearization and random vibration analysis were applied to the nonlinear system to obtain the equation of one-dimensional nonlinear map in terms of response variance and flow speed. Furthermore, the nonlinear map was solved to analyze the response variance of the system against the flow speed. The flow velocity of the maximum variance of the system response was regarded as the flutter speed. The bifurcation point and the approaching path to the chaos were determined by investigating the nonlinear map through the iteration process. In addition, a good vision of jump phenomenon in velocity-variance diagram was given through the current stochastic analysis using the chaos intermittency cascade and the tangent bifurcation point. The aforementioned result is new in the nonlinear aeroelastic dynamic instability field.

Heart rate variability as determinism with jump stochastic parameters

We use measured heart rate information (RR intervals) to develop a one-dimensional nonlinear map that describes short term deterministic behavior in the data. Our study suggests that there is a stochastic parameter with persistence which causes the heart rate and rhythm system to wander about a bifurcation point. We propose a modified circle map with a jump process noise term as a model which can qualitatively capture such this behavior of low dimensional transient determinism with occasional (stochastically defined) jumps from one deterministic system to another within a one parameter family of deterministic systems.

Measuring complexity with multifractals in texts. Translation effects

Should quality be almost a synonymous of complexity? To measure quality appears to be audacious, even very subjective. It is hereby proposed to use a multifractal approach in order to quantify quality, thus through complexity measures. A one-dimensional system is examined. It is known that (all) written texts can be one-dimensional nonlinear maps. Thus, several written texts by the same author are considered, together with their translation, into an unusual language, Esperanto, and asa baseline their corresponding shuffled versions.Different one-dimensional time series can be used: e.g. (i) one based on word lengths, (ii) the other based on word frequencies; both are used for studying, comparing and discussing the map structure. It is shown that a variety in style can be measured through the D(q) and f(α) curves characterizing multifractal objects. This allows to observe on the one hand whether natural and artificial languages significantly influence the writing and the translation, and whether one author’s texts differ technically from each other. In fact, the f(α) curves of the original texts are similar to each other, but the translated text shows marked differences. However in each case, the f(α) curves are far from being parabolic, – in contrast to the shuffled texts. Moreover, the Esperanto text has more extreme values. Criteria are thereby suggested for estimating a text quality, as if it is a time series only. A model is introduced in order to substantiate the findings: it consists in considering a text as a random Cantor set resulting from a binomial cascade of long and short words with appropriate weights. In an appendix, a connection is given with an analysis of turbulence by statistics based on Tsallis generalized entropy. In a second appendix, another view of text (language) complexity is outlined within the copying mistake map concept.

A comparative study on the co-existing attractors in the Gaussian map and its q-deformed version

In a recent study Jaganathan and Sinha [Jaganathan R, Sinha S. A q-deformed nonlinear map. Phys Lett A 2005;338:277–87] have introduced a scheme for the q-deformation of nonlinear maps using the logistic map as an example and shown that the q-logistic map exhibits a wide spectrum of dynamical behaviours including the co-existence of attractors (which is a rare phenomenon in one-dimensional maps). In this paper, we aim to analyze another famous one-dimensional map – the Gaussian map (a known one-dimensional map exhibiting co-existing attractors) subject to the same q-deformation scheme. We compare the dynamical behaviour of the Gaussian map and q-deformed Gaussian map with a special attention on the regions of the parameter space, where these maps exhibit co-existing attractors. An important conclusion of the present study is that the appearance of co-existing attractors for a particular choice of system parameters can be understood as a consequence of the presence of multiple fixed points in one-dimensional nonlinear maps; however the converse is not always true.

The dynamics of a bouncing superball with spin

When a superball is thrown forwards but with backspin, it is observed to reverse both direction and spin for a few bounces before settling to bouncing motion in one direction. The bouncing ball is modelled by a two-dimensional iterated map in terms of the horizontal velocity and spin immediately after each bounce. The asymptotic motion of this system is easily determined. However, of more interest is the transient behaviour. The two-dimensional linear map is reduced to a one-dimensional non-linear map and this map is used to determine the number of direction and spin reversals that can occur for given initial conditions. The model is then generalised to describe a ball bouncing up and down a single step or successively down a staircase.

Analysis and comparison of multiple-delay schemes for controlling unstable fixed points of discrete maps

We investigate theoretically the stabilization of a fixed point of a discrete one-dimensional nonlinear map by applying small perturbations to an accessible system parameter or variable. The size of the perturbations is determined in real time using feedback schemes incorporating only the dynamical state of the system and its state at previous iterates without making a comparison to a reference state. In particular, we compare and contrast two algorithms: extended time-delay autosynchronization, which uses an infinite series of past iterates with weights that decay by a factor of R with each time step, and N-time-delay autosynchronization, which uses an average of N past iterates with equal weights. The range of feedback parameters that successfully stabilize the fixed point and the robustness of the schemes to noise are determined. It is found that the domain of control for the two schemes is similar for appropriately matched values of R and N , and that N-time-delay autosynchronization tends to be less sensitive to noise. @S1063-651X~98!02706-8#