Discrete Nonlinear Map Research Articles

Modulated structures in a Lebwohl–Lasher model with chiral interactions

We consider a Lebwohl–Lasher lattice model with nematic directors restricted to point along p planar directions. This XY Lebwohl–Lasher system is the nematic analogue of the standard p-state clock model. We then include chiral interactions, and introduce a chiral p-state nematic clock model. The statistical problem is formulated as a discrete non-linear map on a Cayley tree. The attractors of this map correspond to the physical solutions deep in the interior of the tree. It is possible to observe uniform and periodic structures, depending on temperature and a parameter of chirality. We find many different chiral nematic phases, and point out the effects of temperature and chirality on the modulation associated with these structures.

Fuzzy relaxed-finite step size method to enhance the instability of the fuzzy first-order reliability method using conjugate discrete map

Fuzzy reliability analysis can be implemented using two discrete optimization maps in the processes of reliability and fuzzy analysis. Actually, the efficiency and robustness of the iterative reliability methods are two main factors in the fuzzy-based reliability analysis due to the huge computational burdens and unstable results. In the structural fuzzy reliability analysis, the first-order reliability method (FORM) using discrete nonlinear map can provide a C membership function. In this paper, a discrete nonlinear conjugate map is proposed using a relaxed-finite step size method for fuzzy structural reliability analysis, namely Fuzzy conjugate relaxed-finite step size method fuzzy CRS. A discrete conjugate map is stabilized using two adaptive factors to compute the relaxed factor and step size in FORM. The framework of the proposed fuzzy structural reliability method is established using two linked iterative discrete maps as an outer loop, which constructs the membership function of the response using alpha level set optimization based on genetic operator, and the inner loop, implemented for reliability analysis using proposed conjugate relaxed-finite step size method. The fuzzy CRS and fuzzy HL-RF methods are compared to evaluate the membership functions of five structural problems with highly nonlinear limit state functions. Results demonstrated that the fuzzy CRS method is computationally more efficient and is strongly more robust than the HL-RF for fuzzy-based reliability analysis of the nonlinear structural reliability problems.

Heterogeneous fundamentalists and market maker inventories

In this paper, we develop a heterogeneous agents model of asset price and inventory with a market maker who considers the excess demand of two groups of agents that employ the same trading rule (i.e. fundamentalists) with different beliefs on the fundamental value. The dynamics of our model is driven by a bi-dimensional discrete non-linear map. We show that the market maker has a destabilizing role when she actively manages the inventory. Moreover, inventory share and the distance between agents’ beliefs strongly influence the results: market instability and periodic, or even, chaotic price fluctuations can be generated. Finally, we show through simulations that endogenous fluctuations of the fractions of agents may trigger instability for a larger set of parameters.

Text Encry ption Scheme Realized with a Chaotic Pseudo - Random Bit Generator

In this work a new encryption scheme, which is realized with a Chaotic Pseudo-Random Bit Generator (CPRBG) based on a Logistic map, is presented. The proposed system is used for encrypting text files for the purpose of creating secure data bases. The Logistic map is the most studied discrete nonlinear map because it has been used in many scientific fields. Also, the fact, that this discrete map has a known algebraic distribution, made the Logistic map a good candidate for use in the design of random bit generators. The proposed CPRBG, which is very easily implemented, uses the X-OR function, in the bit sequences, that are produced by two Logistic maps with different initial conditions and system’s parameters, to achieve better results concerning the “randomness” of the produced bits sequence. The detailed results of the statistical testing on generated bit sequences, done by the most well known tests of randomness: the FIPS-140-2 suite tests, confirmed the specific characteristics expected of random bit sequences.

Influence of dynamical noise on time series generated by nonlinear maps

We consider periodic and chaotic dynamics of discrete nonlinear maps in the presence of dynamical noise. We show that dynamical noise corrupting dynamics of a nonlinear map may be considered as a measurement “pseudonoise” with the distribution determined by the Jacobian of the map. The formula for the distribution of the measurement “pseudonoise” for one-dimensional quadratic maps has also been obtained in an explicit form. We expect that our results apply to an arbitrary distribution of low-level dynamical noise and hope that these results could help to find a universal method of discriminating dynamical from measurement noise.

Modeling the dynamics of natural rotifer populations: Phase-parametric analysis

A model of the dynamics of natural rotifer populations is described as a discrete non-linear map depending on three parameters, which reflect characteristics of the population and environment. Model dynamics and their change by variation of these parameters were investigated by methods of bifurcation theory. A phase-parametric portrait of the model was constructed and domains of population persistence (stable equilibrium, periodic and a-periodic oscillations of population size) as well as population extinction were identified and investigated. The criteria for population persistence and approaches to determining critical parameter values are described. The results identify parameter values that lead to population extinction under various environmental conditions. They further illustrate that the likelihood of extinction can be substantially increased by small changes in environmental quality, which shifts populations into new dynamical regimes.

NONLINEAR DYNAMICS OF A BOUNCING BALL DRIVEN BY ELECTRIC FORCES

This paper is devoted to a theoretical and experimental study of the dynamics of a bouncing ball driven by an electric force. The experimental model consists of a metallic ball immersed in a poorly conducting liquid between two horizontal electrodes. The ball bounces upon the lower electrode as a high voltage is applied between the two plates. The measurement of the time between successive impacts produces a time series, which depends on two control parameters, the amplitude and the frequency of the applied voltage. A theoretical model is proposed, which provides a discrete nonlinear map, and discussed in comparison with the experimental results. It is shown that the system exhibits a period doubling route to chaos and a non-Feigenbaum universal scaling at the onset of chaos. Chaotic motion is investigated using the usual tools: Lyapunov exponents, correlation dimensions and entropies. Fractal structure of the chaotic attractor is also brought to evidence in experimental time series as well as in numerical simulations.

Mixed spin- [formula omitted] and spin-1 Blume-Capel Ising ferrimagnetic system on the Bethe lattice

The mixed spin- 1 2 and spin-1 Blume-Capel Ising ferrimagnetic system is studied on the Bethe lattice by using the exact recursion equations. Exact expressions for the magnetization, the quadrupolar moment, the Curie temperature and the free energy are found and the phase diagrams are constructed on the Bethe lattice with the coordination numbers q=3, 4, 5 and 6. The existence of a tricritical point is investigated for different values of q. The results are compared with those of other approximate methods and with the exact result on the Bethe lattice by using a discrete nonlinear map and also the exact results that are available for the case of the honeycomb lattice.

Existence of localized periodic motions in systems of coupled integrable symplectic maps

We study a one-dimensional chain of discrete non-linear maps with a weak coupling. We consider solutions of the integrable anticontinuous limit, where one or more “central” oscillators move in resonant non-isolated periodic orbits while the other oscillators are at rest. We prove the continuation of these motions for weak non-zero coupling and determine their initial conditions and stability. We apply the above results and perform numerical investigation in the case where the uncoupled motion of each oscillator is described by the integrable Suris map.

Equilibrium states of adaptive algorithms for delay differential equations

This work examines the performance of explicit, adaptive, Runge-Kutta based algorithms for solving delay differential equations. The results of Hall (1985) for ordinary differential equation (ODE) solvers are extended by adding a constant-delay term to the test equation. It is shown that by regarding an algorithm as a discrete nonlinear map, fixed points or equilibrium states can be identified and their stability can be determined numerically. Specific results are derived for a low order Runge-Kutta pair coupled with either a linear or cubic interpolant. The qualitative performance is shown to depend upon the interpolation process, in addition to the ODE formula and the error control mechanism. Furthermore, and in contrast to the case for standard ODEs, it is found that the parameters in the test equation also influence the behaviour. This phenomenon has important implications for the design of robust algorithms. The choice of error tolerance, however, is shown not to affect the stability of the equilibrium states. Numerical tests are used to illustrate the analysis. Finally, a general result is given which guarantees the existence of equilibrium states for a large class of algorithms.

Period matching in modulated maps

We study discrete nonlinear maps in which the control parameter is itself “modulated” by another discrete nonlinear map. We show that for a certain class of such maps, which includes for example the logistic map, the periodicity of the modulated signal is either one, independent of the periodicity of the modulating signal, or its periodicity is an integral multiple of the periodicity of the modulating signal or it is chaotic.

Mixed-spin Ising model on the Bethe lattice.

We obtain the phase diagram of a ferromagnetic mixed Ising system, consisting of spin-1/2 and spin-S variables, on a Bethe lattice of coordination number z, with nearest-neighbor exchange interactions and single-ion terms. The problem is formulated as a discrete nonlinear map. There is a tricritical point for S integer and z\ensuremath{\ge}5. In the infinite-coordination-number limit, we regain the results of an exact calculation for a Curie-Weiss version of the model.