Two important themes in nanoscale physics in the last two decades are correlations between electrons and mesoscopic fluctuations. Here we review our recent work on the intersection of these two themes. The setting is the Kondo effect, a paradigmatic example of correlated electron physics, in a nanoscale system with mesoscopic fluctuations; in particular, we consider a small quantum dot coupled to a finite reservoir (which itself may be a large quantum dot). We discuss three aspects of this problem. First, in the high-temperature regime, we argue that a Kondo temperature $T_K$ which takes into account the mesoscopic fluctuations is a relevant concept: for instance, physical properties are universal functions of $T/T_K$. Secondly, when the temperature is much less than the mean level spacing due to confinement, we characterize a natural cross-over from weak to strong coupling. This strong coupling regime is itself characterized by well-defined single-particle levels, as one can see from a Nozières Fermi-liquid theory argument. Finally, using a mean-field technique, we connect the mesoscopic fluctuations of the quasiparticles in the weak coupling regime to those at strong coupling.

We discuss recent findings about properties of quantum nonequilibrium steady states. In particular we focus on transport properties. It is shown that the time-dependent density matrix renormalization method can be used successfully to find a stationary solution of Lindblad master equation. Furthermore, for a specific model an exact solution is presented.

Recent experiments with Bose–Einstein condensates (BEC) in traps and speckle potentials have explored the dynamical regime in which the evolving BEC clouds localize due to the influence of classical dynamics. The growth of their mean energy is effectively arrested. This is in contrast with the well-known localization phenomena that originate due to quantum interferences. We show that classically induced localization can also be obtained in a classically chaotic, non-interacting system. In this work, we study the classical and quantum dynamics of non-interacting particles in a double-barrier structure. This is essentially a non-KAM system and, depending on the parameters, can display chaotic dynamics inside the finite well between the barriers. However, for the same set of parameters, it can display nearly regular dynamics above the barriers. We exploit this combination of two qualitatively different classical dynamical features to obtain saturation of energy growth. In the semiclassical regime, this classical mechanism strongly influences the quantum behaviour of the system.

We study the interplay of topology and dynamics of excitable nodes on random networks. Comparison is made between systems grown by purely random (Erd˝os–Rényi) rules and those grown by the Achlioptas process. For a given size, the growth mechanism affects both the thresholds for the emergence of different structural features as well as the level of dynamical activity supported on the network.

Sounds in the natural environment form an important class of biologically relevant nonstationary signals. We propose a dynamic spectral measure to characterize the spectral dynamics of such non-stationary sound signals and classify them based on rate of change of spectral dynamics. We categorize sounds with slowly varying spectral dynamics as simple and those with rapidly changing spectral dynamics as complex. We propose rate of spectral dynamics as a possible scheme to categorize sounds in the environment.

We study the effect of learning dynamics on network topology. Firstly, a network of discrete dynamical systems is considered for this purpose and the coupling strengths are made to evolve according to a temporal learning rule that is based on the paradigm of spike-time-dependent plasticity (STDP). This incorporates necessary competition between different edges. The final network we obtain is robust and has a broad degree distribution. Then we study the dynamics of the structure of a formal neural network. For properly chosen input signals, there exists a steady state with a residual network. We compare the motif profile of such a network with that of the real neural network of 𝐶. elegans and identify robust qualitative similarities. In particular, our extensive numerical simulations show that this STDP-driven resulting network is robust under variations of model parameters.

The matrix model of (simplified) RNA folding with an external linear interaction in the action of the partition function is reviewed. The important results for structure combinatorics of the model are discussed and analysed in terms of the already existing models.

Many natural and engineered complex networks have intricate mesoscopic organization, e.g., the clustering of the constituent nodes into several communities or modules. Often, such modularity is manifested at several different hierarchical levels, where the clusters defined at one level appear as elementary entities at the next higher level. Using a simple model of a hierarchical modular network, we show that such a topological structure gives rise to characteristic time-scale separation between dynamics occurring at different levels of the hierarchy. This generalizes our earlier result for simple modular networks, where fast intramodular and slow intermodular processes were clearly distinguished. Investigating the process of synchronization of oscillators in a hierarchical modular network, we show the existence of as many distinct time-scales as there are hierarchical levels in the system. This suggests a possible functional role of such mesoscopic organization principle in natural systems, viz., in the dynamical separation of events occurring at different spatial scales.

We have studied the collective behaviour of a one-dimensional ring of cells for conditions when the individual uncoupled cells show stable, bistable and oscillatory dynamics. We show that the global dynamics of this model multicellular system depends on the system size, coupling strength and the intrinsic dynamics of the cells. The intrinsic variability in dynamics of the constituent cells are suppressed to stable dynamics, or modified to intermittency under different conditions. This simple model study reveals that cell–cell communication, system size and intrinsic cellular dynamics can lead to evolution of collective dynamics in structured multicellular biological systems that is significantly different from its constituent single-cell behaviour.

We show how novel behaviour can emerge in complex systems at the global level through synchronization of the activities of their constituent units. Two mechanisms are suggested for the emergence, namely non-diffusive coupling and time delays. In this way, simple units can synchronize to display complex dynamics, or conversely, simple dynamics may arise from complex constituents.

The paper investigates synchronization in unidirectionally coupled dynamical systems wherein the influence of drive on response is cumulative: coupling signals are integrated over a time interval 𝜏. A major consequence of integrative coupling is that the onset of the generalized and phase synchronization occurs at higher coupling compared to the instantaneous $(\tau = 0)$ case. The critical coupling strength at which synchronization sets in is found to increase with 𝜏. The systems explored are the chaotic Rössler and limit cycle (the Landau–Stuart model) oscillators. For coupled Rössler oscillators the region of generalized synchrony in the phase space is intercepted by an asynchronous region which corresponds to anomalous generalized synchronization.

The strength and stability properties of hierarchical load-bearing networks and their strengthened variants have been discussed in a recent work. Here, we study the avalanche time distributions on these load-bearing networks. The avalanche time distributions of the V-lattice, a unique realization of the networks, show power-law behaviour when tested with certain fractions of its trunk weights. All other avalanche distributions show Gaussian peaked behaviour. Thus the V-lattice is the critical case of the network. We discuss the implications of this result.

The dynamics of indirectly coupled Lorenz circuits is investigated experimentally. The in-phase and anti-phase synchronization of indirectly coupled chaotic oscillators reported in Phys. Rev. E 81, 046216 (2010) is verified by physical experiments with electronic circuits. Two chaotic systems coupled through a common dynamic environment shows the verity of synchronization behaviours such as anti-phase synchronization, in-phase synchronization, identical synchronization, anti-synchronization, etc.

We present a mechanism for the synchronization of one-way coupled nonlinear systems in which the coupling uses a variable delay, that is reset at finite intervals. Here the delay varies in the same way as the system in time and so the coupling function remains constant for the reset interval at the end of which it is reset to the value at that time. This leads to a novel and discrete error dynamics and the resulting general stability analysis is applicable to chaotic or hyperchaotic systems. We apply this method to standard chaotic systems and hyperchaotic time delay systems. The results of the detailed numerical analysis agree with the results from stability analysis in both cases. This method has the advantage that it is cost-effective since information from the driving system is needed only at intervals of reset. Further, in the context of time delay systems, optimization among the different time-scales depending upon the application is possible due to the flexibility among the four different time-scales in our method, viz. delay in the driving system, anticipation in the response system, system delay time and reset time. We suggest a bi-channel scheme for implementing this method in communication field with enhanced security

We investigate phase-locked solutions of a continuum field of nonlocally coupled identical phase oscillators with distance-dependent propagation delays. Equilibrium relations for both synchronous and travelling wave solutions in the parameter space characterizing the nonlocality and time delay are delineated. For the synchronous states a comprehensive stability diagram is presented that provides a heuristic synchronization condition as well as an analytic relation for the marginal stability curve. The relation yields simple stability expressions in the limiting cases of local and global coupling of phase oscillators.

We employ Jacobi’s last multiplier (JLM) to study planar differential systems. In particular, we examine its role in the transformation of the temporal variable for a system of ODEs originally analysed by Calogero–Leyvraz in course of their identification of isochronous systems. We also show that JLM simplifies to a great extent the proofs of isochronicity for the Liénard-type equations.

The Gross–Pitaevskii equation (GPE) describing the evolution of the Bose–Einstein condensate (BEC) order parameter for weakly interacting bosons supports dark solitons for repulsive interactions and bright solitons for attractive interactions. After a brief introduction to BEC and a general review of GPE solitons, we present our results on solitons that arise in the BEC of hard-core bosons, which is a system with strongly repulsive interactions. For a given background density, this system is found to support both a dark soliton and an antidark soliton (i.e., a bright soliton on a pedestal) for the density profile. When the background has more (less) holes than particles, the dark (antidark) soliton solution dies down as its velocity approaches the sound velocity of the system, while the antidark (dark) soliton persists all the way up to the sound velocity. This persistence is in contrast to the behaviour of the GPE dark soliton, which dies down at the Bogoliubov sound velocity. The energy–momentum dispersion relation for the solitons is shown to be similar to the exact quantum low-lying excitation spectrum found by Lieb for bosons with a delta-function interaction.

A brief review of a recent work on a novel collisionless scheme for stopping electron current pulse in plasma is presented. This scheme relies on the inhomogeneity of the plasma medium. This mechanism can be used for heating an overdense regime of plasma where lasers cannot penetrate. The method can ensure efficient localized heating at a desired location. The suitability of the scheme to the frontline fast ignition laser fusion experiment has been illustrated.

We aim to study the nonlinear optical phenomena with ultra-broadband radiation in photonic crystal fibre (PCF). While PCFs with cores made from different glasses are well studied in previous works, in this paper, it is planned to investigate the dynamics of nonlinear processes of supercontinuum generation (SCG) in liquid-filled PCF (LCPCF) to understand the physical phenomena of femtosecond pulse propagation, particularly, the temporal and spectral changes of the pulse propagating through specific PCFs. Since the CS₂-filled LCPCF has complex nonlinear phenomena, we intend to analyse the role of saturable nonlinear response and slow nonlinear response on SCG in detail. For the physical explanation, soliton fission and modulational instability techniques will be implemented to investigate the impact of slow nonlinear response and saturable nonlinear response respectively, in SCG process.

The study of the dynamics of 1D chains with both harmonic and nonlinear interactions, as in the Fermi–Pasta–Ulam (FPU) and related problems, has played a central role in efforts to identify the broad consequences of nonlinearity in these systems. Here we study the dynamics of highly localized excitations, or discrete breathers, which are known to be initiated by the quasistatic stretching of bonds between adjacent particles. We show via dynamical simulations that acoustic waves introduced by the harmonic term stabilize the discrete breather by suppressing the breather’s tendency to delocalize and disperse. We conclude that the harmonic term, and hence acoustic waves, are essential for the existence of localized breathers in these systems.

We study theoretically the effect of truncating the nonlinear restoring force (exp $(\Phi)−1 = \Sum^{\infty}_{n=1} \Phi^n/n!$) in the bistability pattern of the periodically driven, damped one-degree-of-freedom Toda oscillator that originally exhibits soft-spring bistability with counterclockwise hysteresis cycle. We observe that if the truncation is made third order, the harmonic bistability changes to hard-spring type with a clockwise hysteresis cycle. In contrast, for the fourth-order truncation, the bistability again becomes soft-spring type, overriding the effect of third-order nonlinearity. Furthermore, each higher odd-order truncation attempts to introduce hard-spring nature while each even-order truncation turns to soft-spring type of bistability. Overall, the hard-spring effect of every odd-order nonlinear term is weaker in comparison to the soft-spring effect of the next even-order nonlinear term. As a consequence, higher-order approximations ultimately converge to the soft-spring nature. Similar approximate analysis of Toda lattice has in recent past revealed remarkably similar flip-flop pattern between stochasticity (chaotic behaviour) and regularity (integrability).

We present computational data on the thermal conductivity of nonlinear waves in disordered chains. Disorder induces Anderson localization for linear waves and results in a vanishing conductivity. Cubic nonlinearity restores normal conductivity, but with a strongly temperature-dependent conductivity $\kappa (T)$. We find indications for an asymptotic low-temperature $\kappa \sim T^4$ and intermediate temperature $\kappa \sim T^2$ laws. These findings are in accord with theoretical studies of wave packet spreading, where a regime of strong chaos is found to be intermediate, followed by an asymptotic regime of weak chaos (Laptyeva et al, Europhys. Lett. 91, 30001 (2010)).