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In this work, we propose a generalization to the classical logistic map. The generalized map preserves most properties of the classical map and has richer dynamics as it contains the fractional order and one more parameter. We propose the stability bounds for each equilibrium point. The detailed bifurcation analysis with respect to both parameters is presented using the bifurcation diagrams in one and two dimensions. The chaos in this system is controlled using delayed feedback. We provide some non-linear feedback controllers to synchronize the system. The multistability in the proposed system is also discussed., Comment: 25 pages, 46 figures

There are few known universality classes of absorbing phase transitions in one dimension and most models fall in the well-known directed percolation (DP) class. Synchronization is a transition to an absorbing state and this transition is often DP class. With local coupling, the transition is often to a fixed point state. Transitions to a periodic synchronized state are seldom observed. Recently a transition to a synchronized period-3 state that is not in DP class is observed. We model it using a cellular automata model with states 1 to $n$. The rules are a) Each site in state $i$ changes to state $i+1$ for $i2$. For $n=2$, a different exponent is observed.

Synchronization of two replicas of coupled map lattices for continuous maps is known to be in the multiplicative noise universality class. We study this transition in the presence of quenched disorder in coupling. The disorder is identical in both replicas. We study one-dimensional, two-dimensional, and globally coupled logistic and tent maps. We observe a clear second-order transition with new exponents. The order parameter decays as $t^{-\delta}$ and $\delta$ depends on the map and its parameters. The asymptotic order parameter for $\Delta$ distance from a critical point grows as $\Delta^{\beta}$ with $\beta=\delta$. The quenched disorder in coupling is a relevant perturbation for the replica synchronization of coupled map lattices. The critical exponents are different from those of the multiplicative noise universality class. However, it does not depend on dimensionality if the transition is continuous for the cases studied.

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 a delay term $\tau$. We have given detailed stability analysis for the cases $\tau=1$ and $\tau=2$. The results are extended to nonlinear maps., Comment: 21 pages, 48 figures

We consider the stability of periodic map with period-$2$ in linear fractional difference equations where the function is $f(x)=ax$ at even times and $f(x)=bx$ at odd times. The stability of such a map for an integer order map depends on product $ab$. The conditions are much complex for fractional maps and depend on $ab$ as well as $a+b$. There are no superstable period-2 orbits. These conditions are useful in obtaining stability conditions of asymptotically periodic orbits with period-$2$ in the nonlinear case. The stability conditions are demonstrated numerically. The formalism can be generalized to higher periods., Comment: 22 pages, 10 figures

One of the most popular methods of controlling dynamical systems is feedback. It can be used without acquiring detailed knowledge of the underlying system. In this work, we study the stability of fractional-order linear difference equations under feedback. The stability results are derived for an arbitrary feedback time $\tau$. We study the cases of $\tau=1$ and $\tau=2$ in further detail. The extension to the stability of fixed points under feedback for nonlinear fractional order difference equations with fixed points $ x_{*}=0$ is also carried out., Comment: 27 pages, 44 figures

Dynamical behaviour of discrete dynamical systems has been investigated extensively in the past few decades. However, in several applications, long term memory plays an important role in the evolution of dynamical variables. The definition of discrete maps has recently been extended to fractional maps to model such situations. We extend this definition to a spatiotemporal system. We define a coupled map lattice on different topologies, namely, one-dimensional coupled map lattice, globally coupled system and small-world network. The spatiotemporal patterns in the fractional system are more ordered. In particular, synchronization is observed over a large parameter region. For integer order coupled map lattice in one dimension, synchronized periodic states with a period greater than one are not obtained. However, we observe synchronized periodic states with period-3 or period-6 in one dimensional coupled fractional maps even for a large lattice. With nonlocal coupling, the synchronization is reached over a larger parameter regime. In all these cases, the standard deviation decays as power-law in time with the power same as fractional-order. The physical significance of such studies is also discussed.

We study coupled Gauss maps in one dimension and observe a transition to band periodic state with 2 bands. This is a periodic state with period-2 in a coarse-grained sense. This state does not show any long-range order in space. We compute two different order parameters to quantify the transition a) Flipping rate $F(t)$ which measures departures from period-2 and b) Persistence $P(t)$ which quantifies the loss of memory of initial conditions. At the critical point, $F(t)$ shows a power-law decay with exponent 0.158 which is close to 1-D directed percolation (DP) transition. The persistence exponent at the critical point is found to be 1.51 which matches with several models in 1-D DP class. We also study the finite-size scaling and off-critical scaling to estimate other exponents $z$ and $\nu_{\parallel}$. We observe excellent scaling for both $F(t)$ as well as $P(t)$ and the exponents obtained are clearly in DP class. We believe that DP transition could be observed in systems where activity goes to zero even if the spatial profile could be inhomogeneous and lacking any long-range order.

We study the fractional maps of complex order, $\alpha_0e^{i r \pi/2}$ for $0<\alpha_0<1$ and $0\le r<1$ in 1 and 2 dimensions. In two dimensions, we study H{\'e}non and Lozi map and in $1d$, we study logistic, tent, Gauss, circle, and Bernoulli maps. The generalization in $2d$ 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, and H{\'e}non maps do not show chaos while discontinuous maps such as Lozi, Bernoulli, and circle maps show chaos. The tent map is continuous but not differentiable and it shows chaos as well. In $2d$, 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., Comment: 16 pages, 35 figures

We study the stability of synchronized fixed-point state for linear fractional-order coupled map lattice(CML). We observe that the eigenvalues of the connectivity matrix determine the stability as for integer-order CML. These eigenvalues can be determined exactly in certain cases. We find exact bounds in one-dimensional lattice with translationally invariant coupling using the theory of circulant matrices. This can be extended to any finite dimension. Similar analysis can be carried out for the synchronized fixed point of nonlinear coupled fractional maps where eigenvalues of the Jacobian matrix play the same role. The analysis is generic and demonstrates that the eigenvalues of connectivity matrix play a pivotal role in stability analysis of synchronized fixed point even in coupled fractional maps., Comment: 26 pages. 19 figures

We extend the definition of $n$-dimensional difference equations to complex order $\alpha\in \mathbb{C} $. We investigate the stability of linear systems defined by an $n$-dimensional matrix $A$ and derive conditions for the stability of equilibrium points for linear systems. For the one-dimensional case where $A =\lambda \in \mathbb {C}$, we find that the stability region, if any is enclosed by a boundary curve and we obtain a parametric equation for the same. Furthermore, we find that there is no stable region if this parametric curve is self-intersecting. Even for $ \lambda \in \mathbb{R} $, the solutions can be complex and dynamics in one-dimension is richer than the case for $ \alpha\in \mathbb{R} $. These results can be extended to $n$-dimensions. For nonlinear systems, we observe that the stability of the linearized system determines the stability of the equilibrium point., Comment: 21 pages, 17 figures

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) Erd{\"o}s-R{\'e}yni network, ii) a $d$-dimensional lattice. The layer at the top which is not connected to any layer. The top layer undergoes absorbing transition in the directed percolation class for the corresponding topology. The critical point for absorbing transition is the same for all layers. For Erdos-Reyni network order parameter $\rho(t)$ decays as $t^{-\delta_l}$ at the critical point for $l'$th layer with $\delta_l \sim 2^{1-l}$. This can be explained with a hierarchy of differential equations in the mean-field approximation. The dynamic exponent $z$ is $0.5$ for all layers and the value of $\nu_{\parallel}$ tends to 2 for larger $l$. For a d-dimensional lattice, we observe stretched exponential decay of order parameter for all but top layer at the critical point., Comment: 5 pages, 9 figures

We study one-dimensional coupled logistic maps with delayed linear or nonlinear nearest-neighbor coupling. Taking the nonzero fixed point of the map x* as reference, we coarse-grain the system by identifying values above x* with the spin-up state and values below x* with the spin-down state. We define persistent sites at time T as the sites which did not change their spin state even once for all even times till time T. A clear transition from asymptotic zero persistence to non-zero persistence is seen in the parameter space. The transition is accompanied by the emergence of antiferromagnetic, or ferromagnetic order in space. We observe antiferromagnetic order for nonlinear coupling and even delay, or linear coupling and odd delay. We observe ferromagnetic order for linear coupling and even delay, or nonlinear coupling and odd delay. For symmetric coupling, we observe a power-law decay of persistence. The persistence exponent is close to 0.375 for the transition to antiferromagnetic order and close to 0.285 for ferromagnetic order. The number of domain walls decays with an exponent close to 0.5 in all cases as expected. The persistence decays as a stretched exponential and not a power-law at the critical point, in the presence of asymmetry., Comment: 18 pages, 7 figures

We study the stability of linear fractional order maps. We show that in the stable region, the evolution is described by Mittag-Leffler functions and a well defined effective Lyapunov exponent can be obtained in these cases. For one-dimensional systems, this exponent can be related to the corresponding fractional differential equation. A fractional equivalent of map $f(x)=ax$ is stable for $a_c(\alpha)

We present a model of contact process on Domany-Kinzel cellular automata with a geometrical disorder. In the 1-D 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. We also study a model with fixed immunization rate p0 in one and two dimensions. If p0 > ps, where ps is percolation threshold, the system always tends to an absorbing state. As infection rate p --> 1, we observe a power law decay of order parameter with complex exponent. This may be a common feature in systems where quenched disorder leads to effective fragmentation of the lattice.

We present a model which displays Griffiths phase i.e. algebraic decay of density with continuously varying exponent in the absorbing phase. In active phase, the memory of initial conditions is lost with continuously varying complex exponent in this model. This is 1-D model where fraction r of sites obey rules leading to directed percolation (DP) class and the rest evolve according to rules leading to compact directed percolation (CDP) class. For infection probability $p < p_c$, the fraction of active sites $\rho(t) = 0$ asymptotically. For $p > p_c$, $\rho(infty) > 0$. At $p = p_c$, $\rho(t)$, the survival probability from single seed and the average number of active sites starting from single seed decay logarithmically. The local persistence $P_l(\infty) > 0$ for $p < p_c$ and $P_l(\infty) = 0$ 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 $p\rightarrow 1$. The complex exponent implies logarithmic periodic oscillations in persistence. The wavelength and the amplitude of the logarithmic periodic oscillations increases with p. We note that underlying lattice or disorder does not have self-similar structure.

We investigate the dynamic phase transition from partially or fully arrested state to spatiotemporal chaos in coupled logistic maps on a small-world network. Persistence of local variables in coarse grained sense acts as an excellent order parameter to study this transition. We investigate the phase diagram by varying coupling strength and small-world rewiring probability p of nonlocal connections. The persistent region is a compact region bounded by two critical lines where band-merging crisis occurs. On one critical line, the persistent sites shows a nonexponential (stretched exponential) decay for all p while for another one, it shows crossover from nonexponential to exponential behavior as p tends to 1. With an effectively antiferromagnetic coupling, coupling to two neighbors on either side leads to exchange frustration. Apart from exchange frustration, non-bipartite topology and nonlocal couplings in a small-world networks could be reason for anomalous relaxation. The distribution of trap times in asymptotic regime has a long tail as well. The dependence of temporal evolution of persistence on initial conditions is studied and a scaling form for persistence after waiting time is proposed. We present a simple possible model for this behavior., Comment: 12 pages, 23 figures

The question of universality class of pair contact process with diffusion (PCPD) is revisited with an alternative approach. We study persistence in Generalized Pair-Contact Process with diffusion (GPCPD) introduced by Noh and Park, (Phys. Rev. E 69,016122(2004)). This model allows us to interpolate between directed percolation (DP) and PCPD universality classes. We find that transition to nonzero persistence is at same parameter value as transition to zero number density. We obtain finite size scaling and off-critical scaling collapse for persistence and find critical exponents by fitting phenomenological scaling laws to persistence. While the dynamic scaling exponent $z$ varies continuously in GPCPD, the correlation-time exponent $\nu_\parallel$ matches with directed percolation universality class.

We investigate coupled circle maps in 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 number of sites which have been greater than (less than) the fixed point till 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 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.

We present a detailed study of prisoner's dilemma game with stochastic modifications on a two-dimensional lattice, in presence of evolutionary dynamics. By very nature of the rules, the cooperators have incentive to cheat and the fear to being cheated in prisoner's dilemma and may cheat even when not dictated by evolutionary dynamics. We consider two variants. In either case, the agents mimic the action (cooperation or defection) in the previous timestep of the most successful agent in the neighborhood. Over and above this, the fraction p of cooperators spontaneously change their strategy to pure defector at every time step in the first variant. In the second variant, there are no pure cooperators. All cooperators keep defecting with probability p at every time-step. In both cases, the system switches from coexistence state to an all-defector state for higher values of p. We show that the transition between these states unambiguously belongs to directed percolation universality class in 2 + 1 dimension. We also study the local persistence and the persistence exponents are higher than ones obtained in previous studies underlining their dependence on details of dynamics, Comment: 20 pages, 25 figures. to appear in JSTAT

Pareto law, which states that wealth distribution in societies have a power-law tail, has been a subject of intensive investigations in statistical physics community. Several models have been employed to explain this behavior. However, most of the agent based models assume the conservation of number of agents and wealth. Both these assumptions are unrealistic. In this paper, we study the limiting wealth distribution when one or both of these assumptions are not valid. Given the universality of the law, we have tried to study the wealth distribution from the asset exchange models point of view. We consider models in which a) new agents enter the market at constant rate b) richer agents fragment with higher probability introducing newer agents in the system c) both fragmentation and entry of new agents is taking place. While models a) and c) do not conserve total wealth or number of agents, model b) conserves total wealth. All these models lead to a power-law tail in the wealth distribution pointing to the possibility that more generalized asset exchange models could help us to explain emergence of power-law tail in wealth distribution., Comment: 15 pages and 6 figures

We investigate the problem of wealth distribution from the viewpoint of asset exchange. Robust nature of Pareto's law across economies, ideologies and nations suggests that this could be an outcome of trading strategies. However, the simple asset exchange models fail to reproduce this feature. A yardsale(YS) model in which amount put on the bet is a fraction of minimum of the two players leads to condensation of wealth in hands of some agent while theft and fraud(TF) model in which the amount to be exchanged is a fraction of loser's wealth leads to an exponential distribution of wealth. We show that if we allow few agents to follow a different model than others, {\it i.e.} there are some agents following TF model while rest follow YS model, it leads to distribution with power law tails. Similar effect is observed when one carries out transactions for a fraction of one's wealth using TF model and for the rest YS model is used. We also observe a power law tail in wealth distribution if we allow the agents to follow either of the models with some probability., Comment: 18 pages and 9 figures

We study the transition to phase synchronization in a model for the spread of infection defined on a small world network. It was shown (Phys. Rev. Lett. {\bf 86} (2001) 2909) that the transition occurs at a finite degree of disorder $p$, unlike equilibrium models where systems behave as random networks even at infinitesimal $p$ in the infinite size limit. We examine this system under variation of a parameter determining the driving rate, and show that the transition point decreases as we drive the system more slowly. Thus it appears that the transition moves to $p=0$ in the very slow driving limit, just as in the equilibrium case., Comment: 8 pages, 2 figures

We study the dynamical behaviour of the collective field of chaotic systems on small world lattices. Coupled neuronal systems as well as coupled logistic maps are investigated. We observe that significant changes in dynamical properties occur only at a reasonably high strength of nonlocal coupling. Further, spectral features, such as signal-to-noise ratio (SNR), change monotonically with respect to the fraction of random rewiring, i.e. there is no optimal value of the rewiring fraction for which spectral properties are most pronounced. We also observe that for small rewiring, results are similar to those obtained by adding small noise., Comment: 16 pages including 7 figures. To be published in Int. J. of Bif. and Chaos

We demonstrate the phenomenon of self organized criticality (SOC) in a simple random walk model described by a random walk of a myopic ant. The ant acts on the underlying lattice aiming at uniform digging of the surface but is unaffected by the underlying lattice. In 1-d, 2-d and 3-d we have explored this model and have obtained power laws in the time intervals between consecutive events of `digging'. Being a simple random walk, the power laws in space translate to power laws in time. We also study the finite size scaling of asymptotic scale invariant process as well as dynamic scaling in this system. This model differs qualitatively from the cascade models of SOC., Comment: Submitted to Phys. Rev. Lett., 7 .eps files, Revtex document

We demonstrate the phenomenon of stochastic resonance (SR) for discrete-time dynamical systems. We investigate various systems that are not necessarily bistable, but do have two well defined states, switching between which is aided by external noise which can be additive or multiplicative. Thus we find it to be a fairly generic phenomenon. In these systems, we investigate kinetic aspects like hysteresis which reflect the nonlinear and dissipative nature of the response of the system to the external field. We also explore spatially extended systems with additive or parametric noise and find that they differ qualitatively., Comment: 6 pages revtex document, 15 tex figures. (Submitted to Phys. Rev. E)

We study coupled maps on a Cayley tree, with local (nearest-neighbor) interactions, and with a variety of boundary conditions. The homogeneous state (where every lattice site has the same value) and the node-synchronized state (where sites of a given generation have the same value) are both shown to occur for particular values of the parameters and coupling constants. We study the stability of these states and their domains of attraction. As the number of sites that become synchronized is much higher compared to that on a regular lattice, control is easier to effect. A general procedure is given to deduce the eigenvalue spectrum for these states. Perturbations of the synchronized state lead to different spatio-temporal structures. We find that a mean-field like treatment is valid on this (effectively infinite dimensional) lattice., Comment: latex file (25 pages), 4 figures included. To be published in Phys. Rev. E