Stability analysis of fractional difference equations with delay.

Long-term memory is a feature observed in systems ranging from neural networks to epidemiological models. The memory in such systems is usually modeled by the time delay. Furthermore, the nonlocal operators, such as the "fractional order difference," can also have a long-time memory. Therefore, the fractional difference equations with delay are an appropriate model in a range of systems. Even so, there are not many detailed studies available related to the stability analysis of fractional order systems with delay. In this work, we derive the stability conditions for linear fractional difference equations with an arbitrary delay τ and even for systems with distributed delay. We carry out a detailed stability analysis for the cases of single delay with τ=1 and τ=2. The results are extended to nonlinear maps. The formalism can be easily extended to multiple time delays.

Transition to period-3 synchronized state in coupled gauss maps.

We study coupled Gauss maps in one dimension with nearest-neighbor interactions. We observe transitions from spatiotemporal chaos to period-3 states in a coarse-grained sense and synchronized period-3 states. Synchronized fixed points are frequently observed in one dimension. However, synchronized periodic states are rare. The obvious reason is that it is very easy to create defects in one dimension. We characterize all transitions using the following order parameter. Let x∗ be the fixed point of the map. The values above (below) x∗ are classified as +1 (-1) spins. We expect all sites to return to the same band after three time steps for a coarse-grained periodic or three-period state. We define the flip rate F(t) as the fraction of sites i such that si(3t-3)≠si(t). It is zero in the coarse-grained periodic state. This state may or may not be synchronized. We observe three different transitions. (a) If different sites reach different bands, the transition is in the directed-percolation universality class. (b) If all sites reach the same band, we find an Ising-type transition. (c) A synchronized period-3 state where a new exponent is observed. We also study the finite-size scaling at critical points. The exponents obtained indicate that the synchronized period-3 transition is in a new universality class.

Study of low-dimensional nonlinear fractional difference equations of complex order.

We study the fractional maps of complex order, α e, for 0 < α < 1 and 0 ≤ r < 1 in one and two dimensions. In two dimensions, we study Hénon, Duffing, and Lozi maps, and in 1 d, we study logistic, tent, Gauss, circle, and Bernoulli maps. The generalization in 2 d can be done in two different ways, which are not equivalent for fractional order and lead to different bifurcation diagrams. We observed that the smooth maps, such as logistic, Gauss, Duffing, and Hénon maps, do not show chaos, while discontinuous maps, such as Bernoulli and circle maps,show chaos. The tent and Lozi map are continuous but not differentiable, and they show chaos as well. In 2 d, we find that the complex fractional-order maps that show chaos also show multistability. Thus, it can be inferred that the smooth maps of complex fractional order tend to show more regular behavior than the discontinuous or non-differentiable maps.

Absorbing phase transition in a unidirectionally coupled layered network.

We study the contact process on layered networks in which each layer is unidirectionally coupled to the next layer. Each layer has elements sitting on (i) an Erdös-Réyni network, and (ii) a d-dimensional lattice. The top layer is not connected to any layer and undergoes an absorbing transition in the directed percolation class for the corresponding topology. The critical infection probability p_{c} for the transition is the same for all layers. For an Erdös-Réyni network the order parameter decays as t^{-δ_{l}} at p_{c} for the lth layer with δ_{l}∼2^{1-l}. This can be explained with a hierarchy of differential equations in the mean-field approximation. The dynamic exponent z=0.5 for all layers and ν_{∥}→2 for larger l. For a d-dimensional lattice, we observe a stretched exponential decay of the order parameter for all but the top layer at p_{c}.

Investigation of structural evolution in the Cu-Zr metallic glass at cryogenic temperatures by using molecular dynamics simulations.

In the present work, investigation of structural evolution of Cu33Zr67 specimen during the cooling process from 2500 down to the 300 K, 200 K, 150 K, 100 K, 50 K, and 10 K has been performed at cooling rate of 5 K/ps using molecular dynamics simulation. The pair distribution function (PDF) reveals that Zr‒Zr pair causes the splitting of the first peak of the Cu33Zr67 glass at a lower temperature with an increase in height. Splitting of the first and second peaks supports the presence of the inhomogeneous structure with a statistical average of crystal-like and disordered structural regions in the Cu33Zr67 glass. Voronoi cluster analysis indicated that quasi icosahedral clusters such as < 284 > , < 0285 > , and < 0282 > ; mixed-type cluster such as < 0364 > ; and crystal-like clusters such as < 0446 > are responsible for stabilization of glassy phase at 300 K, 200 K, 150 K, 100 K, 50 K, and 10 K. Similarly, the maximum population of the Cu-centered and Zr-centered < 0286 > quasi icosahedral clusters support the stability of the glassy phase over the studied temperature range. Besides, the maximum population of Cu-centered < 0367 > and Zr-centered < 0364 > , < 0367 > , < 0363 > , and < 0365 > mixed-type clusters and Cu-centered < 0448 > and Zr-centered < 0448 > , < 0445 > , < 0446 > , and < 0444 > crystal-like clusters support the possibility of the presence of intermediate phase of CuZr2 at lower temperatures as observed from PDFs. Mean square displacement (MSD) for the Cu33Zr67 glass shows that the diffusion coefficient of Cu and Zr atoms reduces with decreasing temperature from 300 to 10 K. Diversity parameter (d) was found to decrease with decreasing temperature.

Emergence of logarithmic-periodic oscillations in contact process with topological disorder.

We present a model of contact process on Domany-Kinzel cellular automata with a geometrical disorder. In the 1D model, each site is connected to two nearest neighbors which are either on the left or the right. The system is always attracted to an absorbing state with algebraic decay of average density with a continuously varying complex exponent. The log-periodic oscillations are imposed over and above the usual power law and are clearly evident as p→1. This effect is purely due to an underlying topology because all sites have the same infection probability p and there is no disorder in the infection rate. An extension of this model to two and three dimensions leads to similar results. This may be a common feature in systems where quenched disorder leads to effective fragmentation of the lattice.

Dynamic phase transition in the contact process with spatial disorder: Griffiths phase and complex persistence exponents.

We present a model which displays the Griffiths phase, i.e., algebraic decay of density with continuously varying exponents in the absorbing phase. In the active phase, the memory of initial conditions is lost with continuously varying complex exponents in this model. This is a one-dimensional model where a fraction r of sites obey rules leading to the directed percolation class and the rest evolve according to rules leading to the compact directed percolation class. For infection probability p≤p_{c}, the fraction of active sites ρ(t)=0 asymptotically. For p>p_{c}, ρ(∞)>0. At p=p_{c}, ρ(t), the survival probability from a single seed and the average number of active sites starting from single seed decay logarithmically. The local persistence P_{l}(∞)>0 for p≤p_{c} and P_{l}(∞)=0 for p>p_{c}. For p≥p_{s}, local persistence P_{l}(t) decays as a power law with continuously varying exponents. The persistence exponent is clearly complex as p→1. The complex exponent implies logarithmic periodic oscillations in persistence. The wavelength and the amplitude of the logarithmic periodic oscillations increase with p. We note that the underlying lattice or disorder does not have a self-similar structure.

Universality of the local persistence exponent for models in the directed Ising class in one dimension.

We investigate local persistence in five different models and their variants in the directed Ising (DI) universality class in one dimension. These models have right-left symmetry. We study Grassberger's models A and B. We also study branching and annihilating random walks with two offspring: the nonequilibrium kinetic Ising model and the interacting monomer-dimer model. Grassberger's models are updated in parallel. This is not the case in other models. We find that the local persistence exponent in all these models is unity or very close to it. A change in the mode of the update does not change the exponent unless the universality class changes. In general, persistence exponents are not universal. Thus it is of interest that the persistence exponent in a range of models in the DI class is the same. Excellent scaling behavior of finite-size scaling is obtained using exponents in the DI class in all models. We also study off-critical scaling in some models and DI exponents yield excellent scaling behavior. We further study graded persistence, which shows similar behavior. However, for a logistic map with delay, which also has the transition in the DI class, there is no transition from nonzero to zero persistence at the critical point. Thus the accompanying transition in persistence and universality of the persistence exponent hold when the underlying model has right-left symmetry.

Lateral Semi-sitting Position: A Novel Method of Patient's Head Positioning in Suboccipital Retrosigmoid Approaches.

BACKGROUND: The most common methods of positioning patients for suboccipital approaches are the lateral, lateral oblique, sitting, semisitting, supine with the head turn, and park bench. The literature on the positioning of patients for these approaches does not mention the use of lateral semisitting position. This position allows utilization of the benefits of both semisitting and lateral position without causing any additional morbidity to the patient. AIMS: The aim of the present study is to highlight the advantages of the lateral semisitting position while operating various cerebellopontine angle (CPA) and posterior fossa lesions. MATERIALS AND METHODS: The position involved placing the patient in a lateral position with torso flexed to 45° and head tilted toward opposite shoulder by 20°. The most common approach taken was retrosigmoid suboccipital craniotomy. RESULTS: The advantages of lateral semisitting position were early decompression of cisterna magna, and the surgical field remained relatively clear, due to gravity-assisted drainage of blood and irrigating fluid. We could perform all the surgeries without the use of any retractors. The position allowed better delineation of surrounding structures resulting in achieving correct dissection plane and also permitted early caudal to cranial dissection of tumor capsule, thereby increasing chances of facial nerve preservation. Importantly, there is less engorgement of the cerebellum as the venous outflow is promoted. We have not experienced any increased rate of complications, such as venous air embolism, tension pneumocephalus with this lateral semisitting position. CONCLUSIONS: Lateral semisitting position is a relatively safe modification, which combines the benefits of semisitting and lateral position, and avoids the disadvantages of sitting position in operating CPA tumors. This position can provide quick and better exposure of the CPA without any significant complications.

Frustration induced oscillator death on networks.

An array of identical maps with Ising symmetry, with both positive and negative couplings, is studied. We divide the maps into two groups, with positive intra-group couplings and negative inter-group couplings. This leads to antisynchronization between the two groups which have the same stability properties as the synchronized state. Introducing a certain degree of randomness in signs of these couplings destabilizes the anti-synchronized state. Further increasing the randomness in signs of these couplings leads to oscillator death. This is essentially a frustration induced phenomenon. We explain the observed results using the theory of random matrices with nonzero mean. We briefly discuss applications to coupled differential equations.

Transition to almost periodic patterns in circle map with delay: persistence as order parameter.

We study delayed circle map. A previously proposed analogy between delayed map and spatiotemporal system [F. T. Arecchi et al., Phys. Rev. A 45, R4225 (1992)] is employed to study this system. In the phase diagram, we observe laminar phase, travelling defect phase, and standing defect phase. We push this analogy further; and in this pseudo-spatiotemporal system, we investigate "phases" and define "order parameter" to describe transition between phases. We find that persistence (defined as the probability that a given site has not deviated even once from its coarse grained initial state upto time t) works as an "order parameter" for the transition from a travelling wave phase to standing wave phase. We observe an interesting finite "size" scaling and off-critical scaling above the critical point.

Universal persistence exponent in transition to antiferromagnetic order in coupled logistic maps.

We study coupled logistic maps on a one-dimensional lattice. We coarse grain the system by labeling the local state + if the value of the variable is above the fixed point and - if the value is below the (nonzero) fixed point, x(*)=1-1/µ. We find that the system exhibits a transition from a global state, in which moving defects occur, to a global state, which is frozen in time (modulo-2). All such states display nonzero value of persistence. Besides, we also observe a state which displays long-range antiferromagnetic order. Onset of such a state is accompanied with a power-law behavior of persistence P(t) as a function of time at the critical line. Usually, persistence transitions show nonuniversal exponent. However, here persistence exponent for synchronous dynamics is 3/8 over the entire lower critical curve. This exponent is the same as one observed for asynchronous Glauber dynamics simulation of zero-temperature Ising model in one dimension. The expected value for synchronous dynamics is 3/4. We present a plausible argument for this scaling.

A mathematical model for dynamics of CD40 clustering.

Ligand bound-receptors in a signalosome complex trigger signals to determine cellular functions. Upon ligand binding, the ligand-receptor complexes form clusters on cell membrane. Guided by the previous experimental reports on the cluster formation of CD40, a trans membrane receptor for CD40-ligand, we built a minimal model of the receptor cluster formation. In this model, we studied co-operative and non-co-operative clustering of a maximum of four CD40 molecules assuming a positive mediator of clustering such as cholesterol to be present in both cases. We observed that co-operative interactions between CD40 molecules resulted in more of the largest CD40 clusters than that observed with the non-co-operatively interacting CD40 molecules. We performed global sensitivity analysis on the model parameters and the analyses suggested that cholesterol influenced only the initial stage of the co-operatively clustering CD40 molecules but it affects both the initial and the final stages in case of the non-co-operatively clustering CD40 molecules. Robustness analyses revealed that in both co-operative and non-co-operative interactions, the higher order clusters beyond a critical size are more robust with respect to alterations in the environmental parameters including the cholesterol. Thus, the role of co-operative and non-co-operative interactions in environment-influenced receptor clustering is reported for the first time.

Dynamic phase transition from localized to spatiotemporal chaos in coupled circle map with feedback.

We investigate coupled circle maps in the presence of feedback and explore various dynamical phases observed in this system of coupled high dimensional maps. We observe an interesting transition from localized chaos to spatiotemporal chaos. We study this transition as a dynamic phase transition. We observe that persistence acts as an excellent quantifier to describe this transition. Taking the location of the fixed point of circle map (which does not change with feedback) as a reference point, we compute a number of sites which have been greater than (less than) the fixed point until time t. Though local dynamics is high dimensional in this case, this definition of persistence which tracks a single variable is an excellent quantifier for this transition. In most cases, we also obtain a well defined persistence exponent at the critical point and observe conventional scaling as seen in second order phase transitions. This indicates that persistence could work as a good order parameter for transitions from fully or partially arrested phase. We also give an explanation of gaps in eigenvalue spectrum of the Jacobian of localized state.

Transition from clustered state to spatiotemporal chaos in a small-world networks.

We study the spatiotemporal patterns in coupled circle maps on a small-world network. This system shows a rich phase diagram with several interesting phases. In particular, we make a detailed study of transition from clustered phase to spatiotemporal chaos. In the clustered state, observed at smaller coupling values, some sites stay close to the fixed point forever while others explore a larger part of the phase space. For stronger coupling, there is a transition to spatiotemporal chaos where no site stays close to fixed point forever. We study this transition as a dynamic phase transition. Persistence acts as a good order parameter for this transition. We find that this transition is continuous. We also briefly discuss other phases observed in this system.

Power-law persistence characterizes traveling waves in coupled circle maps with repulsive coupling.

We study persistence in coupled circle maps with repulsive (inhibitory) coupling, and find that it offers an effective way to characterize the synchronous, traveling wave and spatiotemporally chaotic states of the system. In the traveling wave state, persistence decays as a power law and, in contrast to earlier observations in dynamical systems, this power-law scaling does not occur at the transition point alone, but over the entire dynamical phase (with the same exponent). We give a cellular automata model displaying the qualitative features of the traveling wave regime and provide an argument based on the theory of Motzkin numbers in combinatorics to explain the observed scaling.

Synchronization in a network of model neurons.

We study the spatiotemporal dynamics of a network of coupled chaotic maps modelling neuronal activity, under variation of coupling strength epsilon and degree of randomness in coupling p. We find that at high coupling strengths (epsilon>epsilonfixed) the unstable saddle point solution of the local chaotic maps gets stabilized. The range of coupling where this spatiotemporal fixed point gains stability is unchanged in the presence of randomness in the connections, namely epsilonfixed is invariant under changes in p. As coupling gets weaker (epsilon0) one obtains spatial synchronization in the network. We find that this range of synchronized chaos increases exponentially with the fraction of random links in the network. Further, in the space of fixed coupling strengths, the synchronization transition occurs at a finite value of p, a scenario quite distinct from the many examples of synchronization transitions at p-->0. Further we show that the synchronization here is robust in the presence of parametric noise, namely in a network of nonidentical neuronal maps. Finally we check the generality of our observations in networks of neurons displaying both spiking and bursting dynamics.

Scaling and universality in transition to synchronous chaos with local-global interactions.

We study the coupled-map lattice model with both local and global couplings. We find necessary conditions for observing synchronous chaos and investigate the transition to synchronization as a dynamic phase transition. We discover that this transition, if continuous, shows scaling and universal behavior with the dynamic exponent z = 2. We also define and illustrate an interesting quantity similar to persistence at critical point.