Pinaki Pal

Oscillation death in diffusively coupled oscillators by local repulsive link.

A death of oscillation is reported in a network of coupled synchronized oscillators in the presence of additional repulsive coupling. The repulsive link evolves as an averaging effect of mutual interaction between two neighboring oscillators due to a local fault and the number of repulsive links grows in time when the death scenario emerges. Analytical condition for oscillation death is derived for two coupled Landau-Stuart systems. Numerical results also confirm oscillation death in chaotic systems such as a Sprott system and the Rössler oscillator. We explore the effect in large networks of globally coupled oscillators and find that the number of repulsive links is always fewer than the size of the network.

Lag synchronization and scaling of chaotic attractor in coupled system

We report a design of delay coupling for lag synchronization in two unidirectionally coupled chaotic oscillators. A delay term is introduced in the definition of the coupling to target any desired lag between the driver and the response. The stability of the lag synchronization is ensured by using the Hurwitz matrix stability. We are able to scale up or down the size of a driver attractor at a response system in presence of a lag. This allows compensating the attenuation of the amplitude of a signal during transmission through a delay line. The delay coupling is illustrated with numerical examples of 3D systems, the Hindmarsh-Rose neuron model, the Rössler system, a Sprott system, and a 4D system. We implemented the coupling in electronic circuit to realize any desired lag synchronization in chaotic oscillators and scaling of attractors.

Diverse routes of transition from amplitude to oscillation death in coupled oscillators under additional repulsive links.

We report the existence of diverse routes of transition from amplitude death to oscillation death in three different diffusively coupled systems, which are perturbed by a symmetry breaking repulsive coupling link. For limit-cycle systems the transition is through a pitchfork bifurcation, as has been noted before, but in chaotic systems it can be through a saddle-node or a transcritical bifurcation depending on the nature of the underlying dynamics of the individual systems. The diversity of the routes and their dependence on the complex dynamics of the coupled systems not only broadens our understanding of this important phenomenon but can lead to potentially new practical applications.

Transition from amplitude to oscillation death in a network of oscillators

We report a transition from a homogeneous steady state (HSS) to inhomogeneous steady states (IHSSs) in a network of globally coupled identical oscillators. We perturb a synchronized population of oscillators in the network with a few local negative or repulsive mean field links. The whole population splits into two clusters for a certain number of repulsive mean field links and a range of coupling strength. For further increase of the strength of interaction, these clusters collapse into a HSS followed by a transition to IHSSs where all the oscillators populate either of the two stable steady states. We analytically determine the origin of HSS and its transition to IHSS in relation to the number of repulsive mean-field links and the strength of interaction using a reductionism approach to the model network. We verify the results with numerical examples of the paradigmatic Landau-Stuart limit cycle system and the chaotic Rössler oscillator as dynamical nodes. During the transition from HSS to IHSSs, the network follows the Turing type symmetry breaking pitchfork or transcritical bifurcation depending upon the system dynamics.

Pattern dynamics near inverse homoclinic bifurcation in fluids.

We report for the first time the pattern dynamics in the vicinity of an inverse homoclinic bifurcation in an extended dissipative system. We observe, in direct numerical simulations of three dimensional Rayleigh-Bénard convection, a spontaneous breaking of a competition of two mutually perpendicular sets of oscillating cross rolls to one of two possible sets of oscillating cross rolls as the Rayleigh number is raised above a critical value. The time period of the cross-roll patterns diverges, and shows scaling behavior near the bifurcation point. This is an example of a transition from nonlocal to local pattern dynamics near an inverse homoclinic bifurcation. We also present a simple four-mode model that captures the pattern dynamics quite well.

Dynamics of zero-Prandtl number convection near onset.

We present a detailed bifurcation scenario of zero-Prandtl number Rayleigh-Bénard convection using direct numerical simulations (DNS) and a 27-mode low-dimensional model containing the most energetic modes of DNS. The bifurcation analysis reveals a rich variety of convective flow patterns and chaotic solutions, some of which are common to that of the 13-mode model of Pal et al. [EPL 87, 54003 (2009)]. We also observed a set of periodic and chaotic wavy rolls in DNS and in the model similar to those observed in experiments and numerical simulations. The time period of the wavy rolls is closely related to the eigenvalues of the stability matrix of the Hopf bifurcation points at the onset of convection. This time period is in good agreement with the experimental results for low-Prandtl number fluids. The chaotic attractor of the wavy roll solutions is born through a quasiperiodic and phase-locking route to chaos.

Transition from homogeneous to inhomogeneous steady states in oscillators under cyclic coupling

Abstract We report a transition from homogeneous steady state to inhomogeneous steady state in coupled oscillators, both limit cycle and chaotic, under cyclic coupling and diffusive coupling as well when an asymmetry is introduced in terms of a negative parameter mismatch. Such a transition appears in limit cycle systems via pitchfork bifurcation as usual. Especially, when we focus on chaotic systems, the transition follows a transcritical bifurcation for cyclic coupling while it is a pitchfork bifurcation for the conventional diffusive coupling. We use the paradigmatic Van der Pol oscillator as the limit cycle system and a Sprott system as a chaotic system. We verified our results analytically for cyclic coupling and numerically check all results including diffusive coupling for both the limit cycle and chaotic systems.

Role of uniform horizontal magnetic field on convective flow

The effect of uniform magnetic field applied along a fixed horizontal direction in Rayleigh-Bénard convection in low-Prandtl-number fluids has been studied using a low dimensional model. The model shows the onset of convection (primary instability) in the form of two dimensional stationary rolls in the absence of magnetic field, when the Rayleigh number R is raised above a critical value Rc. The flow becomes three dimensional at slightly higher values of Rayleigh number via wavy instability. These wavy rolls become chaotic for slightly higher values of R in low-Prandtl-number (Pr) fluids. A uniform magnetic field along horizontal plane strongly affects all kinds of convective flows observed at higher values of R in its absence. As the magnetic field is raised above certain value, it orients the convective rolls in its own direction. Although the horizontal magnetic field does not change the threshold for the primary instability, it affects the threshold for secondary (wavy) instability. It inhibits the onset of wavy instability. The critical Rayleigh number Ro(Q,Pr) at the onset of wavy instability, which depends on Chandrasekhar’s number Q and Pr, increases monotonically with Q for a fixed value of Pr. The dimensionless number Ro(Q,Pr) / (RcQPr) scales with Q as Q−1. A stronger magnetic field suppresses chaos and makes the flow two dimensional with roll pattern aligned along its direction.

Bursting dynamics in a population of oscillatory and excitable Josephson junctions.

We report an emergent bursting dynamics in a globally coupled network of mixed population of oscillatory and excitable Josephson junctions. The resistive-capacitive shunted junction (RCSJ) model of the superconducting device is considered for this study. We focus on the parameter regime of the junction where its dynamics is governed by the saddle-node on invariant circle (SNIC) bifurcation. For a coupling value above a threshold, the network splits into two clusters when a reductionism approach is applied to reproduce the bursting behavior of the large network. The excitable junctions effectively induce a slow dynamics on the oscillatory units to generate parabolic bursting in a broad parameter space. We reproduce the bursting dynamics in a mixed population of dynamical nodes of the Morris-Lecar model.

Effect of Prandtl number on wavy rolls in Rayleigh–Bénard convection

We study the effect of the Prandtl number on wavy rolls in three-dimensional (3D) Rayleigh–Benard convection (RBC) with free-slip boundary conditions. Direct numerical simulation (DNS) of the 3D RBC with free-slip boundary conditions is performed in a rectangular box of size for this purpose. Dynamics of the wavy rolls are investigated for different values of the horizontal aspect ratio in the range . For DNS shows interesting time-dependent wavy rolls near the onset of convection. A low-dimensional model is constructed from DNS data to understand the origin of different wavy rolls in low Prandtl number (Pr) fluids including quasiperiodic and chaotic ones. The investigation of the model using the tools of dynamical systems reveals that periodic wavy rolls are generated via supercritical Hopf bifurcation of the stationary two-dimensional rolls for . These periodic wavy rolls remain stable up to above the onset of convection for fluids with Pr in the range of . However, for lower Prandtl numbers () the periodic wavy rolls undergo a pair of Neimark–Sacker (NS) bifurcations of which one is forward and the other is backward. These NS bifurcations are found to be responsible for the apperance of quasiperiodic wavy rolls from a periodic one and vice versa near the onset of convection. Chaotic wavy rolls are also found to appear in between quasiperiodic wavy rolls for very low Prandtl number fluids. These results of the model are consistent with the DNS results for . For larger values of Γ, the bifurcation structures are found to be modified significantly.