Tuning limit cycles with a noise: Survival and collapse.

We consider a general class of limit cycle oscillators driven by an additive Gaussian white noise. Based on the separation of timescales, we construct the equation of motion for slow dynamics after appropriate averaging over the fast motion. The equation for slow motion whose coefficients are modified by noise characteristics is solved to obtain the analytic solution in the long time limit. We show that with increase of noise strength, the loop area of the limit cycle decreases until a critical value is reached, beyond which the limit cycle collapses. We determine the noise threshold from the condition for removal of secular divergence of the perturbation series and work out two explicit examples of Van der Pol and Duffing-Van der Pol oscillators for corroboration between the theory and numerics.

Phase diffusion and fluctuations in a dissipative Bose-Josephson junction.

We analyze the phase diffusion, quantum fluctuations and their spectral features of a one-dimensional Bose-Josephson junction (BJJ) nonlinearly coupled to a bosonic heat bath. The phase diffusion is considered by taking into account of random modulations of the BJJ modes causing a phase loss of initial coherence between the ground and excited states, whereby the frequency modulation is incorporated in the system-reservoir Hamiltonian by an interaction term linear in bath operators but nonlinear in system (BJJ) operators. We examine the dependence of the phase diffusion coefficient on the on-site interaction and temperature in the zero- and π-phase modes and demonstrate its phase transition-like behavior between the Josephson oscillation and the macroscopic quantum self-trapping (MQST) regimes in the π-phase mode. Based on the thermal canonical Wigner distribution, which is the equilibrium solution of the associated quantum Langevin equation for phase, coherence factor is calculated to study phase diffusion for the zero- and π-phase modes. We investigate the quantum fluctuations of the relative phase and population imbalance in terms of fluctuation spectra which capture an interesting shift in Josephson frequency induced by frequency fluctuation due to nonlinear system-reservoir coupling, as well as the on-site interaction-induced splitting in the weak dissipative regime.

Method for direct analytic solution of the nonlinear Langevin equation using multiple timescale analysis: Mean-square displacement.

We consider a class of nonlinear Langevin equations with additive, Gaussian white noise. Because of nonlinearity, the calculation of moments poses a serious problem for any direct solution of the Langevin equation. Based on multiple timescale analysis we introduce a scheme for directly solving the equations. We first derive the equations for the fast and slow dynamics, in the spirit of the Blekhman perturbation method in vibrational mechanics, the fast motion being described by the Brownian motion of a harmonic oscillator whose effect is subsumed in the slow motion resulting in a parametrically driven nonlinear oscillator. The multiple timescale perturbation theory is then used to obtain a secular divergence-free analytic solution for the slow nonlinear dynamics for calculation of the moments. Our analytical results for mean-square displacement are corroborated with direct numerical simulation of Langevin equations.

Universality in bio-rhythms: A perspective from nonlinear dynamics.

Bio-rhythms are ubiquitous in all living organisms. A prototypical bio-rhythm originates from the chemical oscillation of intermediates or metabolites around the steady state of a thermodynamically open bio-chemical reaction network with autocatalysis and feedback and is often described by minimal kinetics with two state variables. It has been shown that notwithstanding the diverse nature of the underlying bio-chemical and biophysical processes, the associated kinetic equations can be mapped into the universal form of the Lie´nard equation which admits of mono-rhythmic and bi-rhythmic solutions. Several examples of bio-kinetic schemes are examined to illustrate this universality.

Subharmonics and superharmonics of the weak field in a driven two-level quantum system: Vibrational resonance enhancement.

We consider a quantum two-level system in bichromatic classical time-periodic fields, the frequency of one of which far exceeds that of the other. Based on systematic separation of timescales and averaging over the fast motion a reduced quantum dynamics in the form of a nonlinear forced Mathieu equation is derived to identify the stable oscillatory resonance zones intercepted by unstable zones in the frequency-amplitude plot. We show how this forcing of the dressed two-level system may generate the subharmonics and superharmonics of the weak field in the stable region, which can be amplified by optimization of the strength of the high frequency field. We have carried out detailed numerical simulations of the driven quantum dynamics to corroborate the theoretical analysis.

Microscopic quantum generalization of classical Liénard oscillators.

Based on a system-reservoir model and an appropriate choice of nonlinear coupling, we have explored the microscopic quantum generalization of classical Liénard systems. Making use of oscillator coherent states and canonical thermal distributions of the associated c numbers, we have derived the quantum Langevin equation of the reduced system which admits single or multiple limit cycles. It has been shown that detailed balance in the form of the fluctuation-dissipation relation preserves the dynamical stability of the attractors even in the case of vacuum excitation. The quantum versions of Rayleigh, van der Pol, and several other variants of Liénard oscillators are derived as special cases in our theoretical scheme within a mean-field description.

Parametric instability-induced synchronization in chemical oscillations and spatiotemporal patterns.

We consider a model reaction-diffusion system with two coupled layers in which one of the components in a layer is parametrically driven by a periodic force. On perturbation of a homogeneous stable steady state, the system exhibits parametric instability inducing synchronization in temporal oscillation at half the forcing frequency in absence of diffusion and spatiotemporal patterns in presence of diffusion, when strength of parametric forcing and the strength of coupling are kept above their critical thresholds. We have formulated a general scheme to derive analytically the critical thresholds and dispersion relation to locate the unstable spatial modes lying between the tilted Arnold tongue in the amplitude-frequency plot. Full numerical simulations on Gierer-Meinhardt activator-inhibitor model corroborate our theoretical analysis on parametric instability-induced antiphase synchronization in chemical oscillation and spatiotemporal pattern formation, between the two layers.

Spatiotemporal antiresonance in coupled reaction-diffusion systems.

We present a theoretical study of the spatiotemporal antiresonance in a system of two diffusively coupled chemical reactions, one of which is driven by an external periodic forcing. Although antiresonance is well known in various physical systems, the phenomenon in coupled chemical reactions has largely been overlooked. Based on the linearized dynamics around the steady state of the two-component coupled reaction-diffusion systems we have derived the general analytical expressions for the amplitude-frequency response functions of the driven and undriven components of the system. Our theoretical analysis is well corroborated by detailed numerical simulations on coupled Gray-Scott reaction-diffusion systems exhibiting antiresonance dip in the amplitude-frequency response curve as a result of destructive interference between the coupling and the periodic external forcing imparting differential stability of the two subsystems. This leads to the emergence of spatiotemporal patterns in an undriven subsystem, while the driven one settles down to a homogeneously stable steady state.

Clusterization of self-propelled particles in a two-component system.

We consider a mixture of active solute molecules in a suspension of passive solvent particles comprising a thermal bath. The solute molecules are considered to be extended objects with two chemically distinct heads, one head of which having chemical affinity towards the solvent particles. The coupled Langevin equations for the solvent particles along with the equations governing the dynamics of active molecules are numerically simulated to show how the active molecules self-assemble to form clusters which remain in dynamic equilibrium with the free solute molecules. We observe an interesting crossover at an intermediate time in the variation of the order parameter with time when the temperature of the bath is changed signifying the differential behavior of clusterization below and above the crossover time associated with a transition between a thermodynamic and a quasithermodynamic regime. Enthalpy-entropy compensation in the formation of clusters below the crossover is demonstrated.

Vibrational antiresonance in nonlinear coupled systems.

We examine the response of a system of coupled nonlinear oscillators driven by a rapidly varying field, to a low frequency weak periodic excitation of one of the oscillators. The response amplitude of the weak field-driven oscillator at an optimal strength of the rapidly varying field exhibits a strong suppression accompanied by a large negative shift in its oscillation phase. The minimum can be identified as vibrational antiresonance in between the two maxima corresponding to vibrational resonance. This vibrational antiresonance can be observed only in nonlinear coupled systems and not in linearly coupled systems or in a single nonlinear oscillator, under similar physical condition. We discuss the underlying dynamical mechanism, the role of nonlinearity and high frequency in characterizing this counter-resonance effect. Our theoretical analysis is corroborated by detailed numerical simulations.

Brownian dynamics of self-regulated particles with additional degrees of freedom: Symmetry breaking and homochirality.

We consider the Brownian motion of a collection of particles each with an additional degree of freedom. The degree of freedom of a particle (or, in general, a molecule) can assume distinct values corresponding to certain states or conformations. The time evolution of the additional degree of freedom of a particle is guided by those of its neighbors as well as the temperature of the system. We show that the local averaging over these degrees of freedom results in emergence of a collective order in the dynamics in the form of selection or dominance of one of the isomers leading to a symmetry-broken state. Our statistical model captures the basic features of homochirality, e.g., autocatalysis and chiral inhibition.

Nonequilibrium transition and pattern formation in a linear reaction-diffusion system with self-regulated kinetics.

We consider a reaction-diffusion system with linear, stochastic activator-inhibitor kinetics where the time evolution of concentration of a species at any spatial location depends on the relative average concentration of its neighbors. This self-regulating nature of kinetics brings in spatial correlation between the activator and the inhibitor. An interplay of this correlation in kinetics and disparity of diffusivities of the two species leads to symmetry breaking non-equilibrium transition resulting in stationary pattern formation. The role of initial noise strength and the linear reaction terms has been analyzed for pattern selection.

Differential-flow-induced transition of traveling wave patterns and wave splitting.

We have analyzed the differential flow-induced instability in the presence of diffusive transport in a reaction-diffusion system following activator-inhibitor kinetics. The conspicuous interaction of differential flow and differential diffusivity that leads to pattern selection during transition of the traveling waves from stripes to rotating spots propagating in hexagonal arrays subsequent to wave splitting has been explored on the basis of a few-mode Galerkin scheme.

Parametric spatiotemporal oscillation in reaction-diffusion systems.

We consider a reaction-diffusion system in a homogeneous stable steady state. On perturbation by a time-dependent sinusoidal forcing of a suitable scaling parameter the system exhibits parametric spatiotemporal instability beyond a critical threshold frequency. We have formulated a general scheme to calculate the threshold condition for oscillation and the range of unstable spatial modes lying within a V-shaped region reminiscent of Arnold's tongue. Full numerical simulations show that depending on the specificity of nonlinearity of the models, the instability may result in time-periodic stationary patterns in the form of standing clusters or spatially localized breathing patterns with characteristic wavelengths. Our theoretical analysis of the parametric oscillation in reaction-diffusion system is corroborated by full numerical simulation of two well-known chemical dynamical models: chlorite-iodine-malonic acid and Briggs-Rauscher reactions.

Noisy-flow-induced instability in a reaction-diffusion system.

We consider a generic reaction-diffusion-advection system where the flow velocity of the advection term is subjected to dichotomous noise with zero mean and Ornstein-Zernike correlation. A general condition for noisy-flow-induced instability is derived in the flow velocity-correlation rate parameter plane. Full numerical simulations on Gierer-Meinhardt model with activator-inhibitor kinetics have been performed to show how noisy differential flow can lead to symmetry breaking of a homogeneous stable state in the presence of noise resulting in traveling waves.

Landauer's blowtorch effect as a thermodynamic cross process: Brownian cooling.

The local heating of a selected region in a double-well potential alters the relative stability of the two wells and gives rise to an enhancement of population transfer to the cold well. We show that this Landauer's blowtorch effect may be considered in the spirit of a thermodynamic cross process linearly connecting the flux of particles and the thermodynamic force associated with the temperature difference and consequently ensuring the existence of a reverse cross effect. This reverse effect is realized by directing the thermalized particles in a double-well potential by application of an external bias from one well to the other, which suffers cooling.

Landauer's blow-torch effect in systems with entropic potential.

We consider local heating of a part of a two-dimensional bilobal enclosure of a varying cross section confining a system of overdamped Brownian particles. Since varying cross section in higher dimension results in an entropic potential in lower dimension, local heating alters the relative stability of the entropic states. We show that this blow-torch effect modifies the entropic potential in a significant way so that the resultant effective entropic potential carries both the features of variation of width of the confinement and variation of temperature along the direction of transport. The reduced probability distribution along the direction of transport calculated by full numerical simulations in two dimensions agrees well with our analytical findings. The extent of population transfer in the steady state quantified in terms of the integrated probability of residence of the particles in either of the two lobes exhibits interesting variation with the mean position of the heated region. Our study reveals that heating around two particular zones of a given lobe maximizes population transfer to the other.

Rayleigh-type parametric chemical oscillation.

We consider a nonlinear chemical dynamical system of two phase space variables in a stable steady state. When the system is driven by a time-dependent sinusoidal forcing of a suitable scaling parameter at a frequency twice the output frequency and the strength of perturbation exceeds a threshold, the system undergoes sustained Rayleigh-type periodic oscillation, wellknown for parametric oscillation in pipe organs and distinct from the usual forced quasiperiodic oscillation of a damped nonlinear system where the system is oscillatory even in absence of any external forcing. Our theoretical analysis of the parametric chemical oscillation is corroborated by full numerical simulation of two well known models of chemical dynamics, chlorite-iodine-malonic acid and iodine-clock reactions.

Nonlinear vibrational resonance.

We examine the nonlinear response of a bistable system driven by a high-frequency force to a low-frequency weak field. It is shown that the rapidly varying temporal oscillation breaks the spatial symmetry of the centrosymmetric potential. This gives rise to a finite nonzero response at the second harmonic of the low-frequency field, which can be optimized by an appropriate choice of vibrational amplitude of the high-frequency field close to that for the linear response. The potential implications of the nonlinear vibrational resonance are analyzed.

Chemical oscillator as a generalized Rayleigh oscillator.

We derive the conditions under which a set of arbitrary two dimensional autonomous kinetic equations can be reduced to the form of a generalized Rayleigh oscillator which admits of limit cycle solution. This is based on a linear transformation of field variables which can be found by inspection of the kinetic equations. We illustrate the scheme with the help of several chemical and bio-chemical oscillator models to show how they can be cast as a generalized Rayleigh oscillator.