Multifractal scaling of crack images from pyroligneous acid dried sludge

Tannery effluent (sludge, wastewater) is treated by natural way. The waste sludge has been taken for two treatment process. The alkali chemicals are neutralized by pyroligneous acid which is obtained by pyrolysis process of wood. This sludge is sent out for drying. The dried sludge contains some crack pattern formation. Photographs were used to record two sludge cracking surfaces. Experiment has been performed to study the fracture pattern formation in thin film sludge. We studied changes of crack surface of a sludge by image analysis and also assessed applicability of fractal scaling to crack surfaces. The calculated crack surface dimension shows that the fracture surface exhibit fractal structure. Image size was 256 × 256 pixels. MFA (multifractal analysis) was carried out to the method of moments, i.e., the probability distribution was estimated for moments ranging from ?150 < q < 150 and the generalized dimension were calculated from the log/log slope of the probability distribution for the respective moments over box sizes. Generalized dimension D(q) were attained for this box size range, which are capable of characterizing heterogeneous spatial crack structure. Multifractal spectra analyzed two fracture surface of the image and results were indicated that the width of spectra increases due to pyroligneous acid. Multifractal method is sensitive enough to measure the fracture distribution and can be seen as a different approach for changing research of crack images of manure sludge drying.     

Multifractality in relativistic charged particles produced at SPS energies

The multifractal analysis of relativistic shower particles produced in ³²S–emulsion interactions at 200 AGeV has been investigated using the method of modified multifractal moments, G_q, in pseudo-rapidity space. The anomalous fractal dimension, d_q, and generalized fractal dimensions, D_q, are determined for the present data for different order of moment. The experimental data reflects multifractal geometry in a multipion production process. The downward concave shape of the multifractal spectral function, f(α_q), gives an evidence for self-similar cascade mechanism. The multifractal specific heat has also been evaluated for the present data using the generalized fractal dimensions, D_q. We compared our experimental results with those obtained from simulated events of the Lund Monte Carlo Code FRITIOF and uncorrelated Monte Carlo events, (MC-RAND) generated randomly in pseudorapidity space based on the assumption of independent emission of particles. The experimental data on multifractality has been found to exhibit a remarkable proximity to the analogous data obtained from the FRITIOF code and the uncorrelated Monte Carlo events.     

Strange attractors and synchronization dynamics of coupled Van der Pol–Duffing oscillators

We consider in this paper the synchronization dynamics of coupled chaotic Van der Pol–Duffing systems. We first find that with the judicious choose of the set of initial conditions, the model exhibits two strange chaotic attractors. The problem of synchronizing chaos both on the same and different chaotic orbits of two coupled Van der Pol–Duffing systems is investigated. The stability boundaries of the synchronization process between two coupled driven Van der Pol model are derived and the effects of the amplitude of the periodic perturbation of the coupling parameter on these boundaries are analyzed. The results are provided on the stability map in the (q, K) plane.     

Chaotic vibrations of flexible infinite length cylindrical panels using the Kirchhoff–Love model

Treated as continuous deformable systems with an infinite number of degrees of freedom, flexible infinite length cylindrical panels subject to harmonic load are studied. Using the finite difference method with respect to spatial coordinates, the continuous system is reduced to lumped one governed by ordinary differential equations. These equations are transformed to a normal form and then solved numerically using the fourth order Runge–Kutta method. In order to trace and explain vibrational behaviour, dependencies w_max(q₀) and Lyapunov exponents are calculated for panels with parameter value k_x = 48. The corresponding charts of the control parameters {q₀, ω_q} are also reported. Novel scenarios yielding chaotic dynamics exhibited by cylindrical panels are illustrated and discussed.     

Pattern Frequency Sequences and Internal Zeros

Let q be a pattern and let S_n, q(c) be the number of n-permutations having exactly c copies of q. We investigate when the sequence (S_n, q(c))_c ≥ 0 has internal zeros. If q is a monotone pattern it turns out that, except for q = 12 or 21, the nontrivial sequences (those where n is at least the length of q) always have internal zeros. For the pattern q = 1(l + 1)l…2 there are infinitely many sequences which contain internal zeros and when l = 2 there are also infinitely many which do not. In the latter case, the only possible places for internal zeros are the next-to-last or the second-to-last positions. Note that by symmetry this completely determines the existence of internal zeros for all patterns of length at most 3.     

Characterizations of PG(n − 1, q)\PG(k − 1, q) by Numerical and Polynomial Invariants

We show that the simple matroid PG(n − 1, q)\PG(k − 1, q), for n ≥ 4 and 1 ≤ k ≤ n − 2, is characterized by a variety of numerical and polynomial invariants. In particular, any matroid that has the same Tutte polynomial as PG(n − 1, q)\PG(k − 1, q) is isomorphic to PG(n − 1, q)\PG(k − 1, q).     

Fractional dynamics of systems with long-range interaction

We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction defined by a term proportional to 1/∣n − m∣^α+1. Continuous medium equation for this system can be obtained in the so-called infrared limit when the wave number tends to zero. We construct a transform operator that maps the system of large number of ordinary differential equations of motion of the particles into a partial differential equation with the Riesz fractional derivative of order α, when 0 < α < 2. Few models of coupled oscillators are considered and their synchronized states and localized structures are discussed in details. Particularly, we discuss some solutions of time-dependent fractional Ginzburg–Landau (or nonlinear Schrodinger) equation.     

Noncommutative topological dynamics

We study noncommutative dynamical systems associated to unimodal and bimodal maps of the interval. To these maps we associate subshifts and the correspondent AF-algebras and Cuntz–Krieger algebras. As an example we consider systems having equal topological entropy log(1 + ϕ), where ϕ is the golden number, but distinct chaotic behavior and we show how a new numerical invariant allows to distinguish that complexity. Finally, we give a statistical interpretation to the topological numerical invariants associated to bimodal maps.     

Complexity and asymptotic stability in the process of biochemical substance exchange in a coupled ring of cells

We have considered the complexity and asymptotic stability in the process of biochemical substance exchange in a coupled ring of cells. We have used coupled maps to model this process. It includes the coupling parameter, cell affinity and environmental factor as master parameters of the model. We have introduced: (i) the Lempel–Ziv complexity spectrum and (ii) the Lempel–Ziv complexity spectrum highest value to analyze the dynamics of two cell model. The asymptotic stability of this dynamical system using an eigenvalue-based method has been considered. Using these complexity measures we have noticed an “island” of low complexity in the space of the master parameters for the weak coupling. We have explored how stability of the equilibrium of the biochemical substance exchange in a multi-cell system (N = 100) is influenced by the changes in the master parameters of the model for the weak and strong coupling. We have found that in highly chaotic conditions there exists space of master parameters for which the process of biochemical substance exchange in a coupled ring of cells is stable.     

Wave propagation properties in oscillatory chains with cubic nonlinearities via nonlinear map approach

Free wave propagation properties in one-dimensional chains of nonlinear oscillators are investigated by means of nonlinear maps. In this realm, the governing difference equations are regarded as symplectic nonlinear transformations relating the amplitudes in adjacent chain sites (n, n + 1) thereby considering a dynamical system where the location index n plays the role of the discrete time. Thus, wave propagation becomes synonymous of stability: finding regions of propagating wave solutions is equivalent to finding regions of linearly stable map solutions. Mechanical models of chains of linearly coupled nonlinear oscillators are investigated. Pass- and stop-band regions of the mono-coupled periodic system are analytically determined for period-q orbits as they are governed by the eigenvalues of the linearized 2D map arising from linear stability analysis of periodic orbits. Then, equivalent chains of nonlinear oscillators in complex domain are tackled. Also in this case, where a 4D real map governs the wave transmission, the nonlinear pass- and stop-bands for periodic orbits are analytically determined by extending the 2D map analysis. The analytical findings concerning the propagation properties are then compared with numerical results obtained through nonlinear map iteration.     

Equivalent Blocks of Finite General Linear Groups in Non-describing Characteristic

J. Chuang, R. Kessar, and J. Rickard have proved Broué's Abelian defect group conjecture for many symmetric groups. We adapt the ideas of Kessar and Chuang towards finite general linear groups (represented over non-describing characteristic). We then describe Morita equivalences between certain p-blocks of GL_n(q) with defect group C_p^α × C_p^α, as q varies (see Theorem 2). Here p and q are coprime. This generalizes work of S. Koshitani and M. Hyoue, who proved the same result for principal blocks of GL_n(q) when p = 3, α = 1, in a different way.     

The second law of thermodynamics and multifractal distribution functions: Bin counting,pair correlations,and the Kaplan–Yorke conjecture

We explore and compare numerical methods for the determination of multifractal dimensions for a doubly-thermostatted harmonic oscillator. The equations of motion are continuous and time-reversible. At equilibrium the distribution is a four-dimensional Gaussian, so that all the dimension calculations can be carried out analytically. Away from equilibrium the distribution is a surprisingly isotropic multifractal strange attractor, with the various fractal dimensionalities in the range 1 < D < 4. The attractor is relatively homogeneous, with projected two-dimensional information and correlation dimensions which are nearly independent of direction. Our data indicate that the Kaplan–Yorke conjecture (for the information dimension) fails in the full four-dimensional phase space. We also find no plausible extension of this conjecture to the projected fractal dimensions of the oscillator. The projected growth rate associated with the largest Lyapunov exponent is negative in the one-dimensional coordinate space.     

Numerical methods for a class of jump–diffusion systems with random magnitudes

Stochastic differential delay equations with Poisson driven jumps of random magnitude are popular as models in mathematical finance. In this paper, we shall deal with convergence of the semi-implicit Euler method for nonlinear stochastic differential delay equations with random jump magnitudes and show that the approximate solutions strongly converge to the exact solutions with the order 1 − 1/q (q > 1). This result is more general than what they deal with the jump of deterministic magnitude.     

Multifractal rainfall extremes: Theoretical analysis and practical estimation

We study the extremes generated by a multifractal model of temporal rainfall and propose a practical method to estimate the Intensity–Duration–Frequency (IDF) curves. The model assumes that rainfall is a sequence of independent and identically distributed multiplicative cascades of the beta-lognormal type, with common duration D. When properly fitted to data, this simple model was found to produce accurate IDF results [Langousis A, Veneziano D. Intensity–duration–frequency curves from scaling representations of rainfall. Water Resour Res 2007;43. doi:10.1029/2006WR005245]. Previous studies also showed that the IDF values from multifractal representations of rainfall scale with duration d and return period T under either d → 0 or T → ∞, with different scaling exponents in the two cases. We determine the regions of the (d, T)-plane in which each asymptotic scaling behavior applies in good approximation, find expressions for the IDF values in the scaling and non-scaling regimes, and quantify the bias when estimating the asymptotic power-law tail of rainfall intensity from finite-duration records, as was often done in the past. Numerically calculated exact IDF curves are compared to several analytic approximations. The approximations are found to be accurate and are used to propose a practical IDF estimation procedure.     

Global chaos synchronization of new chaotic systems via nonlinear control

Nonlinear control is an effective method for making two identical chaotic systems or two different chaotic systems be synchronized. However, this method assumes that the Lyapunov function of error dynamic (e) of synchronization is always formed as V (e) = 1/2e^Te. In this paper, modification based on Lyapunov stability theory to design a controller is proposed in order to overcome this limitation. The method has been applied successfully to make two identical new systems and two different chaotic systems (new system and Lorenz system) globally asymptotically synchronized. Since the Lyapunov exponents are not required for the calculation, this method is effective and convenient to synchronize two identical systems and two different chaotic systems. Numerical simulations are also given to validate the proposed synchronization approach.     

A cartographic study of the phase space of the elliptic restricted three body problem. Application to the Sun–Jupiter–Asteroid system

This work deals with numerical investigations of the phase space of the planar elliptic restricted three body model. The Sun–Jupiter–Asteroid system is considered and the fast Lyapunov indicator (FLI) is used as a tool to examine various types of orbits on which the infinitesimal mass can undergo. The FLI is computed on given grids of initial conditions regularly spaced in the domain 1.5 AU ? a ? 6 AU and 0 ? e ? 0.5 and for various choices of initial angles: the argument of perihelion ω and mean anomaly M. On the obtained charts the stability regions, the chaotic zones and the geography of resonances are clearly displayed. Moreover, the ‘V’ shaped layers associated with the mean motion resonances of low order with its chaotic zones due to separatrix splitting and libration regions are clearly distinguished. Their size is discussed as a function of the resonance order and the parameters entering into the perturbing function. The results are discussed and compared with analytical studies concerning the subject.     

Complex dynamics in Duffing–Van der Pol equation

Duffing–Van der Pol equation with fifth nonlinear-restoring force and two external forcing terms is investigated. The threshold values of existence of chaotic motion are obtained under the periodic perturbation. By second-order averaging method and Melnikov method, we prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation for ω₂ = nω₁ + εσ, n = 1, 3, 5, and cannot prove the criterion of existence of chaos in second-order averaged system under quasi-periodic perturbation for ω₂ = nω₁ + εσ, n = 2, 4, 6, 7, 8, 9, 10, where σ is not rational to ω₁, but can show the occurrence of chaos in original system by numerical simulation. Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent, phase portraits and Poincaré map, not only show the consistence with the theoretical analysis but also exhibit the more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations from period-2 to -4 and -6 orbits, interleaving occurrence of chaotic behaviors and quasi-periodic orbits, transient chaos with a great abundance of period windows, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos which occurs more than one, chaos suddenly disappearing to period orbits, interior crisis, strange non-chaotic attractor, non-attracting chaotic set and nice chaotic attractors. Our results show many dynamical behaviors and some of them are strictly departure from the behaviors of Duffing–Van der Pol equation with a cubic nonlinear-restoring force and one external forcing.     

Construction of analytic solution to chaotic dynamical systems using the Homotopy analysis method

Based on a new kind of analytic method, namely the Homotopy analysis method, an analytic approach to solve non-linear, chaotic system of ordinary differential equations is presented. The method is applied to Lorenz system; this system depends on the three parameters: σ, b and the so-called bifurcation parameter R are real constants. Two cases are considered. The first case is when R = 20.5 which corresponds to the transition region and the second case corresponds to R = 23.5 which corresponds to the chaotic region.The validity of the method is verified by comparing the approximation series solution with the results obtained using the standard numerical techniques such as Runge-Kutta method.     

Analytic integrability of the Bianchi class A cosmological models with 0 ⩽ k < 1

Many works study the integrability of the Bianchi class A cosmologies with k = 1, where k is the ratio between the pressure and the energy density of the matter. Here we characterize the analytic integrability of the Bianchi class A cosmological models when 0 ⩽ k < 1. We conclude that Bianchi types VI₀, VII₀, VIII and IX can exhibit chaos whereas Bianchi type I is not chaotic and Bianchi type II is at most partially chaotic.