The prediction of the critical point of a phase transition is a topic of great current interest, and is of utility in many practical contexts. Therefore, the identification of precursors, or early warning signals of the critical point, has become the focus of current interest. Recent model studies have shown that a series of small transitions, which have been called microtransitions, act as precursors to the percolation transition. Here, we identify the existence of microtransitions in two distinct networks, for two distinct processes. The first case is the process of avalanche transmission on branching hierarchical networks. Here, typical realizations of the original lattice of this network exhibit a second order transition.We note that microtransitions in the variance of the order parameter are seen in this case. Additionally, the positions of the microtransitions follow ascaling relation. The scaling relation can be used to calculate the position of the critical point, which is seen to be in agreement with the observed result.We also introduce this method of identifying the microtransitions occurring before the tipping point to a complex real world system, the climate system. We analyse the discontinuous first order phase transition occurring in the climate networks. We apply the percolation framework to these networks to analyse the structural changes in the network and construct an order parameter and a susceptibility. Microtransitions can be found in the behaviour of the susceptibility. These can be used to predict the tipping point in the system. We discuss possible applicationsof this.

We employ Floquet theory to study the spectral properties of the Floquet Hamiltonian, also known as the effective static Hamiltonian of periodically driven kicked systems. In general, the Floquet Hamiltonian cannot be determined exactly, and therefore one has to employ some perturbation theory. Here we apply a recently proposed perturbation theory to obtain the Floquet Hamiltonian periodically kicked systems at very high-frequency limit. We studied the spectral properties of two well-known kicked systems: single and double-kicked top. Classical dynamics of these systems is chaotic, but their quantum mechanical spectrum is very different: the first one follows the Bohigas–Giannoni–Schmit conjecture of random matrix theory, but the latter one shows self-similarfractal behavior. Here we show that the fractal spectrum of the double-kicked top system shares some number of theoretical properties with the famous Hoftstadter butterfly.

We study the equal-mass classical three rotor problem, a variant of the three body problem of celestial mechanics. The quantum N-rotor problem has been used to model chains of coupled Josephson junctions and also arises via a partial continuum limit of the Wick-rotated XY model. In units of the coupling, the energy serves as a control parameter. We find periodic ‘pendulum’ and ‘breather’ orbits at all energies and choreographies at relatively low energies. They furnish analogs of the Euler–Lagrange and figure-8 solutions of the planar three body problem. Integrability at very low energies gives way to a rather marked transition to chaos at $E_{c} \approx 4$, followed by a gradual return to regularity as $E \rightarrow \infty$. We find four signatures of this transition: (a) the fraction of the areaof Poincare surfaces occupied by chaotic sections rises sharply at $E_{c}$, (b) discrete symmetries are spontaneously broken at $E_{c}$, (c) $E = 4$ is an accumulation point of stable to unstable transitions in pendulum solutions and (d) the Jacobi–Maupertuis curvature goes from being positive to having both signs above $E = 4$. Moreover, Poincare plots also reveal a regime of global chaos slightly above $E_{c}$.

We consider an ensemble of complex random matrices interpolating between Wishart–Laguerre and Wigner–Gaussian ensembles, and use the Dyson’s Brownian motion approach to obtain the corresponding eigenvaluestatistics. The crossover parameter ($\tau$) in this case serves as a positive-definiteness violation parameter. The joint probability density of eigenvalues of this random matrix model evolves from that of Laguerre unitary ensemble(LUE) to Gaussian unitary ensemble (GUE) as $\tau$ is varied from 0 to $\infty$. It exhibits a biorthogonal structure and hence eigenvalue correlation functions of all orders follow using a generalization of Andreief’s integration formula.

We investigate the effect of deterministic input signals on a quasiperiodically driven Murali–Lakshmanan–Chua (QPDMLC) circuit, which exhibits strange nonchaotic attractors (SNAs). We show that if one uses two-square waves in an aperiodic manner as an input to a QPDMLC circuit, the response of the circuit canproduce a logical output, controlled by such forcing. We also demonstrate that one of the variables of the circuit exhibits one logic element, while the other variable shows its complementary logic operation. It is further shown that these logical behaviors persist even for an experimental noise floor. Thus we confirm that SNA is an efficient tool for computation as in the case of the quasiperiodically driven Duffing oscillator.

In this short paper we explore whether nature can be described by a single law in the reductionist paradigm, or it is to be described by a set of laws based on multiscale perspectives.

Intermittency arising due to parametric noise has been observed across a variety of physical systems. A simple and generic approach for characterizing the onset of such intermittency in one-dimensional maps due to a periodic fluctuations of a system parameter is presented. Two mutually complementary conditions forthe onset of intermittency, with respect to the evolution during the two distinct phases constituting intermittent response are derived.We illustrate these ideas with numerical results from a noisy logistic map, with intermittencyabout a noise-free transcritical bifurcation and an aeroelastic system with intermittency about a noise-free Hopf bifurcation.

Zeros of a function encode significant information about the patterns they represent. Owing to nonseparable or/and non-integrable nature of the classical system, the nodal curves display a complex morphology. As they must avoid to cross, the domains of positive or negative sign they enclose look very complex and intriguing. Probability distribution functions of the normalized-domain number will be presented for simple billiards. For some of these systems, the amplitude distribution functions are found. Several open problems will be discussed.

Extreme events occur in the complex system without generic early warning signals. When the predictability of the extreme events in complex systems is not possible, we can design an environment, which suppresses the extreme event from the systems. In this paper, we report the annihilation of extreme events in thedynamical system under environmental coupling. We consider two systems such as CO$_2$ laser and FitzHugh–Nagumo neuron models to examine the suppression of extreme events. When the systems are coupled with anenvironment, depending on the coupling strength the probability for the occurrence of extreme events decreases and after a critical coupling strength, the extreme events are annihilated from the system.

Experimental and numerical observations of reviving oscillations from amplitude death (AD) in two coupled mismatched nonlinear electronic circuits are described. The inclusion of processing delay in the coupling node of the coupled Chua’s oscillators creates a memory of the past state for a finite time duration. Similarly, the fractional order dynamical system has a memory function and remembers the past state of the system. We show that these memory functions are responsible for the reviving and sustained oscillations in the coupled oscillators.

We present in this paper, the synchronization dynamics observed in a network of mutually coupled simple chaotic systems. The network consisting of chaotic systems arranged in a square matrix network is studied for its different types of synchronization behavior. The chaotic attractors of the simple 2 × 2 matrix networkexhibiting strange non-chaotic attractors (SNAs) in their synchronization dynamics for smaller values of the coupling strength are reported. Further, the existence of islands of unsynchronized and synchronized states of SNAs for smaller values of coupling strength is observed. The process of complete synchronization (CS) observed in the network with all the systems exhibiting strange non-chaotic behavior is reported. The variation of the slope of the singular continuous spectra as a function of the coupling strength confirming the strange non-chaotic state of each of the system in the network is presented. The stability of CS observed in the network is studied using the master stability function.

In this paper, we briefly review a few interesting dynamical aspects and applications of nonlinear $PT$-symmetric systems. Being at the boundary between dissipative and conservative systems, these $PT$-symmetric systems show many interesting characteristics and applications that cannot be seen in the usual dissipative and conservative ones. Among the variety of applications, we here focus on the applicability of $PT$-symmetric systems in unidirectional light transport. This particular application clearly evidences the usefulness of the non-reciprocal nature and symmetry broken phase of a PT-symmetric system and importantly the role of nonlinearities in optics.

We recall six different analytical methods, namely extended Prelle–Singer procedure, Lie point symmetries, $\lambda$-symmetries, adjoint symmetries, Jacobi last multiplier and Darboux polynomial methods which are used to identify integrability quantifiers of nonlinear ordinary differential equations (ODEs).We point out how these methods are interconnected in the case of two-coupled first-order nonlinear ODEs.We demonstrate the interconnections with an example.

The existence of inter-dependence between multiple networks imparts an additional scale of complexity to such systems often referred to as ‘network of networks’ (NONs).We have investigated the robustness of NONs torandom breakdown of their components, as well as targeted attacks, as a function of the relative proportion of intra and inter-dependence among the constituent networks. We focus on bi-layer networks with two layers comprisingdifferent numbers of nodes in general and where the ratio of intra-layer to inter-layer connections, r, can be varied, keeping the total number of nodes and overall connection density invariant. We observe that while the responsesof different networks to random breakdown of nodes are similar, dominantly intra-dependent networks ($r \ll 1$) are robust with respect to attacks that target nodes having the highest degree but when nodes are removed on the basis of the highest betweenness centrality (CB), they exhibit a sharp decrease in the size of the largest connected component (LCC) (resembling a first-order phase transition) followed by a more gradual decrease as more nodes are removed (akin to a second-order transition). As r is increased resulting in the network becoming strongly interdependent ($r \gg 1$), we observe that this hybrid nature of the transition in the size of the LCC in response to targeted node removal (based on the highest CB) changes to a purely continuous or second-order transition.We also explorethe role of layer size heterogeneity on robustness, finding that for a given r having layers comprising very different numbers of nodes results in a bimodal degree distribution. For dominantly inter-dependent networks, this resultsin the nodes of the smaller layer becoming structurally central. Selective removal of these nodes, which constitute a relatively small fraction of the network, leads to breakdown of the entire system – making the inter-dependent networks even more fragile to targeted attacks than scale-free networks having power-law degree distribution.

Boltzmann sampling based on the Metropolis algorithm has been extensively used for simulating a canonical ensemble. An estimate of a mechanical property, like energy, of an equilibrium system, can be made by averaging over a large number of microstates generated by Boltzmann Monte Carlo methods. However, a thermalproperty like entropy is not easily accessible to these methods. The reason is simple. We can assign a numerical value for energy to each microstate. But we cannot make such an assignment for entropy. Entropy is not a propertyassociated with any single microstate. It is a collective property of all the microstates. Towards calculating entropy and other thermal properties, a non-Boltzmann Monte Carlo technique called Umbrella sampling was proposedin the mid-seventies (of the last century). Umbrella sampling has since undergone several metamorphoses and we have now, multicanonical Monte Carlo, entropic sampling, flat histogram methods, Wang–Landau algorithm etc. This class of methods generates non-Boltzmann ensembles which are unphysical. However, physical quantities can be calculated by un-weighting the microstates of the entropic ensemble, followed by re-weighting to the desired physical ensemble.In this review we shall tell you of the Metropolis algorithm for estimating the mechanical properties and of the Wang–Landau algorithm for estimating both mechanical and thermal properties of an equilibrium system. We shall demonstrate the utility of non-Boltzmann Monte Carlo methods by calculating Landau free energy in a model system consisting of $q$-state Potts spins on a two-dimensional square lattice. The model exhibits a first-order phase transition for $q > 4$ and a second-order phase transition for $q \leq 4$. We report results on the Potts spin model for $q = 8$ (first-order phase transition) and for $q = 2$ (second-order phase transition). We also present the results on a more realistic problem of temperature-induced unbinding or denaturation of a hairpin DNA. The transition from a closed phase to an open phase is found to be first order.We shall attempt to make this review as pedagogical and self-contained as possible.

Reaction–diffusion equations are ubiquitous in population dynamics, laser physics, bacterial growth, domain wall kinetics, order parameter relaxation and a host of other problems, cutting across disciplines. The occurrence of nonlinearity adds further complexity in terms of bifurcation-solutions, phase transitions, fractalgrowth, etc. In most cases the non-transient, asymptotic solutions of these equations lead to patterns, the nature of which depends on certain symmetry properties of the underlying variable(s). In this paper we discuss a few suchequations with applications to domain motion of different kinds in ferroelectrics, multiferroics and their switching characteristics because of the underlying nonlinearities, and glucose-induced fractal colony growth of Bacillus thuringiensis.

‘Generalized synchronization (GS)’ was proposed by Rulkov et al. (Phys. Rev. E 51, 980 (1995)) to explain synchronization in unidirectionally coupled systems. This concept has been effective in providing a deeper understanding of synchronization between several non-identical nonlinear systems. The study of GS has been extended to various coupling schemes ranging from bidirectionally coupled systems to complex networks. We review the major ideas that were involved in the field of GS. The characterizing tools to detect generalizedsynchrony and its types along with their relative merits are discussed. We outline some of the interesting results that were published in the last two decades along with the inter-relatedness of different dynamical phenomena reported in coupled dynamical systems.

Recent sequencing of genome of northern white-cheeked gibbon (Nomascus leucogenys) has provided important insight into fast evolution of gibbons and signatures relevant to gibbon biology.Mobile-genetic elements (MGEs) seem to play a major role in gibbon evolution. Here, we report that most of the gibbon genome is occupied by MGEs such as Alus, MIRs, LINE1, LINE2, LINE3, ERVL, ERV-classI, ERV-classII and other DNA elements which include hAT Charlie and TcMar tigger.We provide detailed description and genome-wide distribution of allthe MGEs present in the gibbon genome. Previously, it was reported that gibbon-specific retrotransposon (LAVA) tends to insert into chromosome segregation genes and alter transcription by providing a premature terminationsite, suggesting a possible molecular mechanism for the genome plasticity of the gibbon lineage. We show that insertion sites of LAVA elements present atypical signals/patterns which are different from typical signals present at the insertion sites of Alu elements. This suggests the possibility of a distinct insertion mechanism used by LAVA elements for their insertions. We also find similarity in the signals of LAVA element insertion sites with atypical signals present at the Alus/L1s insertion sites disrupting the genes leading to diseases such as cancer and Duchenne muscular dystrophy. This suggests the role of LAVA in premature transcription termination.

We present an overview of our studies on some simple mechanical systems including the ‘simple’ nonlinear pendulum and its variants. We show that these systems exhibit numerous types of regular bursting oscillations which are seen in biological neurons. In particular, we discuss bow-tie shaped bursts which we foundin a driven pendulum with linear velocity damping, under constant torque and dynamic feedback. Similar bursts of identical bow-tie shape have been reported by us previously in a system of two resistively coupled Josephson junctions in a certain parameter regime under certain conditions. We discuss the bifurcation mechanism producing some of these bursts.

We review how coaxial carbon nanotubes (CNTs) and vesicular nanotubes exhibit nonlinear oscillations and how theoretical models match with experimental observations. In particular, we discuss how coaxial CNTs may be modelled by mechanical analogues with interactions mediated by nonlinear spring forces with weakening elastic constants in addition to a weak van der Waals-like interaction. The model’s predictions are in remarkable agreement with quantum mechanical calculations for the system. Predicted oscillatory frequencies are also in agreement with those reported in the literature. We then discuss our theoretical work on nanotubes in a biological system: Nanotubes that are drawn out from micrometre-scale vesicles and which exhibit very interesting dynamics.Our theoretical model reproduces all aspects of the force–extension curves reported in the experimental literature and completely explains the dynamics of vesicular nanotubulation and force fluctuations. The serrations seen inthe force–extension curves are explained to be a consequence of stick–slip dynamics.

The purpose of this review paper is to summarize progress on a recently introduced topic: Sample space reducing (SSR) stochastic processes. The concept of SSR has been coined to offer an alternative route that can explain the origin of power law features associated with complex systems. Understanding the emergence of scale-free behavior remains a topic of continuing interest, specially in the statistical physics community, due to its appearance in diverse systems, besides the existence of different explanations. One crucial aspect of theSSR is that the problem can be solved analytically. This develops the topic even more attracting and offers a possibility to provide a deeper understanding. Eventually numerous generalization of the SSR has been advanced,and a shift in the focus was also seen from the scale-free statistics to different physically relevant forms. We here primarily discuss the following points: (i) Various ways to visualize the SSR and its applications, (ii) mention, fora comparison, other mechanisms that reveal the origin of the power law distribution, (iii) different generalizations with their importance and (iv) a brief discussion with future perspectives.

Excitation waves in two-dimensional media form various travelling wave patterns such as spiral and target waves. These waves can interact with heterogeneities in the tissue. Spiral waves can attach and form stable pinned waves in heterogeneous excitable media. These spirals can be unpinned by delivering a carefully timed electric stimulus, delivered very close to the core. We study the spiral wave unpinning when a wave is attached to two obstacles at the same time. We show that the unpinning window decreases as the distance between theobstacles increases, and beyond a critical distance, this window completely vanishes. Our study implies that the distribution of heterogeneities can play a critical role in developing the low-energy defibrillation methods.