Dynamic maps in phase-conjugated optical resonators

In this article the dynamical behavior of a beam within a ring-phase-conjugated resonator is modeled using two-dimensional iterative maps. In particular and as an example it is explicitly shown how the difference equations of the Duffing map can be used to describe the dynamic behavior of what we call Duffing beams i.e. beams that behave according to the Duffing map. The matrix of a Duffing map generating device is found in terms of the Duffing parameters, the state variables and the resonator parameters. To our knowledge this is the first time that the mathematical characteristics of an optical device in an optical cavity are stated so that a Duffing map is obtained as the dynamics for the ray beams.     

Bifurcation phenomena in two-dimensional piecewise smooth discontinuous maps

In recent years the theory of border collision bifurcations has been developed for piecewise smooth maps that are continuous across the border and has been successfully applied to explain nonsmooth bifurcation phenomena in physical systems. However, there exist a large number of switching dynamical systems that have been found to yield two-dimensional piecewise smooth maps that are discontinuous across the border. In this paper we present a systematic approach to the problem of analyzing the bifurcation phenomena in two-dimensional discontinuous maps, based on a piecewise linear approximation in the neighborhood of the border. We first motivate the analysis by considering the bifurcations occurring in a familiar physical system-the static VAR compensator used in electrical power systems-and then proceed to formulate the theory needed to explain the bifurcation behavior of such systems. We then integrate the observed bifurcation phenomenology of the compensator with the theory developed in this paper. This theory may be applied similarly to other systems that yield two-dimensional discontinuous maps.     

A novel bit-level image encryption algorithm based on chaotic maps

Recently, a number of chaos-based image encryption algorithms have been proposed at the pixel level, but little research at the bit level has been conducted. This paper presents a novel bit-level image encryption algorithm that is based on piecewise linear chaotic maps (PWLCM). First, the plain image is transformed into two binary sequences of the same size. Second, a new diffusion strategy is introduced to diffuse the two sequences mutually. Then, we swap the binary elements in the two sequences by the control of a chaotic map, which can permute the bits in one bitplane into any other bitplane. The proposed algorithm has excellent encryption performance with only one round. The simulation results and performance analysis show that the proposed algorithm is both secure and reliable for image encryption.     

Synthesizing chaotic maps with prescribed invariant densities

The Inverse Frobenius–Perron Problem (IFPP) concerns the creation of discrete chaotic mappings with arbitrary invariant densities. In this Letter, we present a new and elegant solution to the IFPP, based on positive matrix theory. Our method allows chaotic maps with arbitrary piecewise-constant invariant densities, and with arbitrary mixing properties, to be synthesized.     

Riddling and chaotic synchronization of coupled piecewise-linear Lorenz maps

We investigate the parametric evolution of riddled basins related to synchronization of chaos in two coupled piecewise-linear Lorenz maps. Riddling means that the basin of the synchronized attractor is shown to be riddled with holes belonging to another basin in an arbitrarily fine scale, which has serious consequences on the predictability of the final state for such a coupled system. We found that there are wide parameter intervals for which two piecewise-linear Lorenz maps exhibit riddled basins (globally or locally), which indicates that there are riddled basins in coupled Lorenz equations, as previously suggested by numerical experiments. The use of piecewise-linear maps makes it possible to prove rigorously the mathematical requirements for the existence of riddled basins.     

Stochastic resonance and chaotic resonance in bimodal maps: A case study

We present the results of an extensive numerical study on the phenomenon of stochastic resonance in a bimodal cubic map. Both Gaussian random noise as well as deterministic chaos are used as input to drive the system between the basins. Our main result is that when two identical systems capable of stochastic resonance are coupled, the SNR of either system is enhanced at an optimum coupling strength. Our results may be relevant for the study of stochastic resonance in biological systems.     

The structure of mode-locked regions in quasi-periodically forced circle maps

Using a mixture of analytic and numerical techniques we show that the mode-locked regions of quasi-periodically forced Arnold circle maps form complicated sets in parameter space. These sets are characterized by ‘pinched-off’ regions, where the width of the mode-locked region becomes very small. By considering general quasi-periodically forced circle maps we show that this pinching occurs in a broad class of such maps having a simple symmetry.     

The invariant densities for maps modeling intermittency

For a one-parameter family of maps modeling intermittency the explicit formula of the invariant density is presented.     

Collisional effects in the tokamap

Plasmas confined in tokamaks with non-symmetric perturbations are surrounded by a chaotic layer of magnetic field lines that guide charged particles to the tokamak wall. We use an analytical two-dimensional symplectic mapping to study the resulting fractal patterns of field line escape. However, particles may experience several collisions before escaping toward the tokamaks wall. We add a random collisional term to the field line mapping to investigate how the particle collisions modify their escape patterns.     

Quantization of nonintegrable maps

Using Heisenberg's matrix formulation of quantum mechanics, a method is given for quantizing volume-preserving polynomial mappings. The energy levels of the linear map are obtained exactly and those of the cubic, nonintegrable map are obtained approximately and numerically.     

Delay-coupled discrete maps: Synchronization, bistability, and quasiperiodicity

The synchronization transition is studied in delay-coupled logistic maps. For low coupling, in-phase and out-of-phase synchronous dynamics coexist, and with increasing coupling there is a regime of quasiperiodicity before eventual attraction to a fixed point at a critical value of coupling that depends on the nonlinearity. The presence of a region of asynchrony separating two synchronized regimes—termed anomalous behaviour—has been observed earlier in continuous systems and is shown here to occur in delay mappings as well. There are regions of in-phase, anti-phase, and out-of-phase dynamics of periodic as well as chaotic attractors.     

Spectral decomposition of tent maps using symmetry considerations

The spectral decomposition of the Frobenius-Perron operator of maps composed of many tents is determined from symmetry considerations. The eigenstates involve Euler as well as Bernoulli polynomials.     

Fattening complex manifolds: Curvature and Kodaira—Spencer maps

We present a calculus whereby the curvature of a geometry arising from any generalized twistor correspondence is related to an obstruction-theoretic classification of the infinitesimal neighborhoods of submanifolds of its twistor space. The crux of the argument involves a relation between Kodaira—Spencer maps and the Penrose transform.     

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.     

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.     

Effective suppression of period-doubling in two diffusively coupled logistic maps

We consider a system of two delay diffusively coupled logistic maps. We find that for moderate values of diffusion coupling, the period-doubling sequence is effectively suppressed. Our study supports the existence of certain generic features for systems consisting of two coupled maps.     

New type of heteroclinic tangency in two-dimensional maps

A new mechanism of heteroclinic tangency is investigated by using two-dimensional maps. First, it is numerically shown that the unstable manifold from a hyperbolic fixed point accumulates to the stable manifold of a nearby period-2 hyperbolic point in a piecewise linear map and that the unstable manifold from a hyperbolic fixed point accumulates to the accumulation of the stable manifold of a nearby period-2 hyperbolic point in a cubic map. Second, a theorem on the impossibility of heteroclinic tangency (in the usual sense) is given for a particular type of map. The notions ofdirect andasymptotic heteroclinic tangencies are introduced and heteroclinic tangency is classified into four types.     

Chaos in discrete maps,deterministic scattering,and nondifferentiable functions

Arguments in favor of the nondifferentiability with respect to initial data of some functions associated with deterministic discrete-time dynamical systems are presented. A correspondence between a discrete-time dynamical system and a deterministic scattering model is found and used to interpret nondifferentiability conditions. A connection with random walks is also found.     

Dense mode clustering in brain maps

A mode-based clustering method is developed for identifying spatially dense clusters in brain maps. This type of clustering focuses on identifying clusters in brain maps independent of their shape or overall variance. This can be useful for both localization in terms of interpretation and for subsequent graphical analysis that might require more coherent or dense regions of interest as starting points. The method automatically does signal/noise sharpening through density mode seeking. We also discuss the problem of parameter selection with this method and propose a new method involving 2-parameter control surface, in which we show that the same cluster solution results from tradeoff of these 2 parameters (the local density k and the radius r of the spherical kernel). We benchmark the new dense mode clustering by using several artificially created data sets and brain imaging data sets from an event perception task by perturbing the data set with noise and measuring three kinds of deviation from the original cluster solution. We present benchmark results that demonstrate that the mode clustering method consistently outperforms the commonly used single-linkage clustering, k means method (centroid method) and Ward's method (variance method).