arXiv subjects

Prashant M. Gade

Publications and source records attributed to Prashant M. Gade.

At least 19 recordsLinked to original sources

arXiv2026

Gap distributions between successive personal bests in cricket: Data and Models

Successive personal best performances provide a natural measure of progression in an athlete's career. Classical record theory predicts a universal gap distribution, $P(g)\sim 1/g$, for independent and identically distributed (i.i.d.) sequences. However, sporting careers are shaped by learning, aging, changes in ability, and external influences that violate these assumptions. We investigate the statistics of inter-record gaps, defined as the number of innings between successive personal best scores, in cricket. Using career records of leading Test, ODI, and T20 players obtained from ESPN Cricinfo. We find that the empirical distributions are well described by truncated power law $P(g) \propto g^{-\alpha} e^{-\lambda g}$ with exponents in the range (0.799 $\leq \alpha \leq$ 0.843). Much of this deviation disappears when the temporal ordering of innings is destroyed, indicating that career evolution plays a key role in shaping record occurrence. Bootstrap-shuffled careers, which preserve individual score distributions and career lengths while removing temporal ordering, yield significantly larger exponents ($\alpha \approx 0.939\text{--}0.979$). These findings show that the progression of personal best performances retains information about the temporal organization of a player's career and cannot be fully explained by simple stochastic record processes. More generally, they illustrate how record statistics are altered in nonstationary and path-dependent systems.

cond-mat.stat-mech↗

arXiv2026

Deterministic cascade coarsening in a Bistable Gene Toggle model

We investigate deterministic coarsening dynamics in a spatially extended bistable gene toggle model with diffusive coupling. Unlike classical curvature-driven coarsening, where domain walls move continuously and annihilate gradually, the present system exhibits a qualitatively different mechanism. The domain walls remain pinned for long intervals and disappear abruptly through collective cascade events. The density of domain walls decays approximately as $\rho(t)\sim t^{-\delta}$, but the coarsening exhibits clear log-periodic oscillations superimposed on the power-law behavior. For all values of the promoter strength $\alpha$ considered, the measured exponent satisfies $\delta<0.5$, indicating a systematic deviation from the classical Allen--Cahn prediction $\delta=1/2$ for curvature-driven coarsening. We show that log-periodic oscillations are not controlled by the density of domain walls, but by the \emph{domains that disappear} in each cascade. The average size of disappearing domains grows roughly linearly with cascade index, producing a constant geometric spacing of cascade times, consistent with discrete scale invariance.

cond-mat.stat-mech↗

arXiv2026

Maps of q-deformed fractional order: From circle to cardioid via crescent

We introduce a class of $q$-deformed fractional order maps by replacing the classical binomial memory kernel in discrete fractional dynamics with Gaussian ($q$-) binomial coefficients. The proposed framework interpolates between memoryless discrete maps and classical fractional order maps, unifying circle- and cardioid-shaped stability regions through intermediate crescent geometries. Using the $Z$-transform and the $q$-binomial theorem, we derive characteristic equations and determine the associated stability regions in the complex plane. We further analyze the asymptotic behavior of the memory kernels, showing that the classical fractional kernel exhibits power-law decay, whereas the $q$- and $(p,q)$-deformed kernels exhibit exponential-type localization. The theory is extended to nonlinear logistic-type maps and to a broader $(p,q)$-deformed framework, where the regime $p>q$ yields decaying memory kernels and stable dynamics. Numerical simulations illustrate the theoretical results and the interplay between the deformation parameters, memory effects, and stability geometry.

math.DS↗

arXiv2026

Zigzag ordering, defects, and anomalous relaxation in antiferromagnetic Kuramoto lattices

We investigate the nonequilibrium ordering dynamics of coupled Kuramoto oscillators with negative nearest-neighbor coupling, which induces a zigzag antiferromagnetic ordering. In one dimension, the defect density exhibits anomalously slow coarsening, decaying as $(D(t)\sim t^{-1/4})$ before saturating at a system-size-dependent time $(t_c(N)\sim N^z)$ with (z=2). The local persistence probability follows a stretched-exponential form, $(P(t)\sim \exp(-c t^\alpha))$, with $(\alpha=1/4)$. These exponents are observed are independent of the magnitude of the coupling, which merely rescales the characteristic time scale. The equality $(\alpha=\delta=1/4)$ together with (z=2) is consistent with a distinct universality class. These results demonstrate that deterministic nonlinear dynamics and geometric frustration alone are sufficient to generate slow relaxation and anomalous scaling, without quenched disorder or stochastic noise. A continuum approximation and the corresponding coarse-grained partial differential equation provide a theoretical explanation for the observed anomalous exponents, while linear stability analysis accounts for the emergence of the zigzag ordered state. In two dimensions, geometric frustration inhibits complete ordering and gives rise to long-lived metastable domain-wall structures. An initial transient defect decay is observed before crossover and saturation. These results demonstrate how frustration and continuous phase variables can fundamentally modify coarsening dynamics and generate anomalously slow relaxation in deterministic many-body systems.

cond-mat.stat-mech↗

arXiv2026

Pattern Formation in Excitable Neuronal Maps

Coupled excitable systems can generate a variety of patterns. In this work, we investigate coupled Chialvo maps in two dimensions under two types of nearest-neighbor couplings. One coupling produces ringlike patterns, while the other produces spirals. The rings expand with increasing coupling, whereas spirals evolve into turbulence and dissipate at stronger coupling. To quantify these patterns, we introduce an analogue of the discriminant of the velocity gradient tensor and examine the persistence of its sign. For ring-type patterns, the persistence decays more slowly than exponentially, often following a power law or stretched exponential. When spiral structures remain intact, persistence saturates asymptotically and can exhibit superposed periodic oscillations, suggesting complex exponents at early times. These behaviors highlight deep connections with the underlying dynamics.

cond-mat.stat-mech↗

arXiv2025

Approach to zigzag and checkerboard patterns in spatially extended systems

Zigzag patterns in one dimension or checkerboard patterns in two dimensions occur in a variety of pattern-forming systems. We introduce an order parameter `phase defect' to identify this transition and help to recognize the associated universality class on a discrete lattice. In one dimension, if $x_{i}(t)$ is a variable value at site $i$ at time $t$. We assign spin $s_i(t)=1$ for $x_{i}(t)>x_{i-1}(t)$, $s_i(t)=-1$ if $x_{i}(t)<x_{i-1}(t)$, and $s_i(t)=0$ if $x_{i}(t)=x_{i-1}(t)$. The phase defect $D(t)$ is defined as $D(t)={\frac{\sum_{i=1}^N \vert s_i(t)+s_{i-1}(t)\vert} {2N}}$ for a lattice of $N$ sites with periodic boundary conditions. It is zero for a zigzag pattern. In two dimensions, $D(t)$ is the sum of row-wise as well as column-wise phase defects and is zero for the checkerboard pattern. The persistence $P(t)$ is the fraction of sites whose spin value did not change even once till time $t$. We find that $D(t)\sim t^{-\delta}$ and $P(t)\sim t^{-\theta}$ for the parameter range over which the zigzag or checkerboard pattern is realized. We observe that $\delta=0.5$ and $\theta=3/8$ for 1-d coupled logistic maps or Gauss maps, and $\theta=0.22$ and $\delta=0.45$ in 2-d logistic or Gauss maps. The exponent $\theta$ matches with the persistence exponent at zero temperature for the Ising model, and $\delta$ matches with the exponent for the Ising model at the critical temperature. This power-law decay is observed over a range of parameter values and not just critical point.

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arXiv2025

Is directed percolation class for synchronization transition robust with multi-site interactions?

Coupled map lattice with pairwise local interactions is a well-studied system. However, in several situations, such as neuronal or social networks, multi-site interactions are possible. In this work, we study the coupled Gauss map in one dimension with 2-site, 3-site, 4-site and 5-site interaction. This coupling cannot be decomposed in pairwise interactions. We coarse-grain the variable values by labeling the sites above $x^{\star}$ as up spin (+1) and the rest as down spin (-1) where $x^{\star}$ is the fixed point. We define flip rate $F(t)$ as the fraction of sites $i$ such that $s_{i}(t-1) \neq s_{i}(t)$ and persistence $P(t)$ as the fraction of sites $i$ such that $s_{i}(t')=s_{i}(0)$ for all $t' \le t$. The dynamic phase transitions to a synchronized state is studied above quantifiers. For 3 and 5 sites interaction, we find that at the critical point, $F(t) \sim t^{-δ}$ with $δ=0.159$ and $P(t) \sim t^{-θ}$ with $θ=1.5$. They match the directed percolation (DP) class. Finite-size and off-critical scaling is consistent with DP class. For 2 and 4 site interactions, the exponent $δ$ and behavior of $P(t)$ at critical point changes. Furthermore, we observe logarithmic oscillations over and above power-law decay at the critical point for 4-site coupling. Thus multi-site interactions can lead to new universality class(es).

cond-mat.stat-mech↗

arXiv2025

Analysis of the maps with variable fractional order

Fractional order differential and difference equations are used to model systems with memory. Variable order fractional equations are proposed to model systems where the memory changes in time. We investigate stability conditions for linear variable order difference equations where the order is periodic function with period $T$. We give a general procedure for arbitrary $T$ and for $T=2$ and $T=3$, we give exact results. For $T=2$, we find that the lower order determines the stability of the equations. For odd $T$, numerical simulations indicate that we can approximately determine the stability of equations from the mean value of the variables.

math.DS↗

arXiv2024

Cellular Automata model for period-$n$ synchronization: A new universality class

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 possible. We model those 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 $i 2$. For $n=2$, a different exponent is observed.

cond-mat.stat-mech↗

arXiv2024

Dynamical Analysis Of Fractional Order Generalized Logistic Map

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.

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arXiv2023

Synchronization transition in space-time chaos in the presence of quenched disorder

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.

cond-mat.stat-mech↗

arXiv2023

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 a delay term $τ$. We have given detailed stability analysis for the cases $τ=1$ and $τ=2$. The results are extended to nonlinear maps.

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arXiv2023

Fractional Order Periodic Maps: Stability Analysis and Application to the Periodic-2 Limit Cycles in the Nonlinear Systems

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.

math.DS↗

arXiv2023

Controlling Fractional Difference Equations Using Feedback

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 $τ$. We study the cases of $τ=1$ and $τ=2$ 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.

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arXiv2022

Emergence of Order in Dynamical Phases in Coupled Fractional Gauss Map

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.

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arXiv2022

Novel Transition to fully absorbing state without long-range spatial order in Directed Percolation class

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 $ν_{\parallel}$. 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.

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arXiv2022

Stability analysis of fixed point of fractional-order coupled map lattices

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.

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arXiv2021

Stability and Dynamics of Complex Order Fractional Difference Equations

We extend the definition of $n$-dimensional difference equations to complex order $α\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 $A =λ\in \mathbb {C}$, 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 $ λ\in \mathbb{R} $, the solutions can be complex and dynamics in one-dimension is richer than the case for $ α\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.

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