In my PhD, we started with the projects where we studied the mechanism of the phenomenon called ‘phase-flip’ transition, namely, the abrupt change from inphase synchronized dynamics to antiphase synchronized dynamics in limit-cycle as well as chaotic oscillators coupled with symmetric and asymmetric time-delays. We learned that this phenomenon is associated with frequency discontinuity and avoided-crossing of the Lyapunov exponents at the transition point. ( Phys. Rev. E 82, 046219 (2010); Phys. Rev. E 85, 046204 (2012); AIP Conf. Proc. 1582, 158 (2014) )

The following work was done in collaboration with Prof. F. M. Atay at Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany. In this work, we considered a network of phase-oscillators coupled in a bipartite topology with general nonlinear coupling and distributed time-delay. We studied properties of the synchronized solutions that are observed in this system, and analyzed how the emergence of different branches of synchronized-solutions lead to different phase-synchronized scenarios and multistable behavior in such networks. ( Phys. Rev. E 91, 042906 (2015) )

In this next work, we analyzed different phase-locked regimes that exist in a delay-coupled oscillator network with global coupling and study how transitions between different sync-states may occur in this system. ( CHAOS 24, 043111 (2014) )

In this project, we observe and analyze the occurrence of delay-induced ‘remote synchronization’ (synchronization of indirectly coupled oscillators in the network) and ‘chimeras’ (coexistence of synchronized and incoherent dynamics) in mismatched phase oscillators on a bipartite network. Along with the phase-synchrony observed in such delay-coupled oscillators, we find that the bipartite geometry and the intrinsic-frequency mismatch may lead the system to show various other interesting synchronization scenarios including remote synchronization, chimera-states and individual-synchrony. ( Phys. Rev. E 91, 022922 (2015) )

In this work, we studied the effects of nonlocal coupling on modular network of phase oscillators and observed, an interesting spatio-temporal state called multi-cluster frequency chimera, namely, when multiple coherent domains with different modular frequencies coexist with the domains of modules within which the dynamics is desynchronized. We learned that three competing factors in our system --- (i) phase-lag (frustration) parameter, which favours disorder in the system, (ii) inter-modular nonlocal coupling, supporting synchronization of modules and (iii) intra-modular interactions, which support intra-modular synchronization --- may lead the system to show  dynamical scenarios such as modular synchrony (MS) and multi-cluster frequency chimeras (MFC). ( Phys. Rev. E 93, 012207 (2016) )

Here, in an ensemble of identical chaotic oscillators, we investigated how drive from external signals or mutual couplings may lead to the formation of multiple stable attractors in the system. We learned when multistability is induced using such coupling strategies, the basins of different attractors are intertwined in a complex manner. These results also led to our next work, which is related to the observation of ‘chimera’ states, namely, coexistence of synchronized and incoherent dynamics. Here, we use the fact that these multistable attractors have different dynamical properties: the oscillators settling on some attractors show synchronized dynamics, whereas the motion on other attractors can be incoherent. Such multistable attractors with different synchronization properties can be used to induce ‘chimera’ states in an ensemble of chaotic oscillators. Since chimera states typically appear when some degree of non-uniformity (in the coupling, topology or parameters) is introduced in the system, this is an interesting novel method where chimera states can be obtained without introducing such heterogeneities. ( CHAOS 26, 063111 (2016); Phys. Rev. E 95, 032203 (2017) )

In this study, we explore the dynamical effects of breaking rotational-symmetry in counter-rotating Stuart–Landau oscillators. We analyze the manner in which symmetry and relative phase velocity in the system affects the collective dynamics of coupled oscillators, particularly the phenomenon of oscillation quenching. In a system of counter-rotating oscillators, we learned how different coupling strategies, which either preserve or destroy the symmetry, may lead to suppression (oscillation quenching) or revival of oscillations. ( Phys. Rev. E 98, 022212 (2018) )

One of the key research objective in this project --- which was done at MPI-PKS in collaboration with Dr. Lucas Wetzel at MPI-PKS and Center for Advancing Electronics Dresden (cfaed) --- is to investigate synchronization properties of a delay-coupled system of electronic clocks called phase-locked loops (PLLs). Arrays of such electronic oscillators are used in, for example, mobile and radio communications for synchronizing antenna arrays. For this part, we examine how heterogeneity affects the properties of synchronized states and their stability in such systems. Using a phase model description, we analytically study the effects of heterogeneous components on the synchronized dynamics. We validate our findings through circuit-level simulations and experiments using prototype devices. The next part of the project is to test the robustness of these results in practical scenarios and to explore how this approach can be implemented in large-scale real-life applications. The broader intent of the project is to efficiently design such electronic devices with desired synchronization properties which can then be used in industrial applications. ( Phys. Rev. E 106, L052201 (2022); Phys. Rev. E 106, 054216 (2022) )

In this work, we show how ‘chimera states’, namely, the dynamical situation when synchronized and desynchronized domains coexist in an oscillator-ensemble, can be controlled through Linear Augmentation (LA) technique. Specifically, in the networks of coupled chaotic oscillators, we obtain chimera states through induced multistability and demonstrate how LA can be used to control the size and spatial location of the incoherent and coherent populations in the ensemble. We examine basins of attraction of the system to analyze the effects of LA on its multistable behavior and thus on chimera states. Stability of the synchronized dynamics is analyzed through Master Stability Function (MSF). We find that these results are independent of system’s initial conditions and the strategy is applicable to the networks of globally, locally as well as nonlocally coupled oscillators. Our results suggest that LA control can be an effective method to control chimera states and to realize a desired collective dynamics in such ensembles. ( Phys. Rev. E 103, 042202 (2021) )

We study collective dynamics of networks of mutually coupled identical Lorenz oscillators near a subcritical Hopf bifurcation. Such systems exhibit induced multistable behavior with interesting spatiotemporal dynamics including synchronization, desynchronization, and chimera states. For analysis, we first consider a ring topology with nearest-neighbor coupling and find that the system may exhibit intermittent behavior due to the complex basin structures and dynamical frustration, where temporal dynamics of the oscillators in the ensemble switches between different attractors. Consequently, different oscillators may show a dynamics that is intermittently synchronized (or desynchronized), giving rise to intermittent chimera states. The behavior of the intermittent laminar phases is characterized by the characteristic time spent in the synchronization manifold, which decays as a power law. Such intermittent dynamics is quite general and is also observed in an ensemble of a large number of oscillators arranged in variety of network topologies including nonlocal, scale-free, random, and small-world networks. [ Phys. Rev. E 109, 034208 (2024) ]

In the chaotic Lorenz system, a Hopf bifurcation occurs when the parameter ρH ≃ 24.74, the other parameters being maintained at σ = 10 and β = 8/3. We study the collective dynamics of two mutually coupled oscillators with ρ just above ρH when all equilibria in each isolated oscillator are unstable. The Lorenz systems are coupled through both x and z variables which have different symmetry properties, and this leads to induced multistability in the dynamics. In addition to the existing strange attractor, two fixed points are stabilized, and in addition there are two new chaotic attractors that are smaller (in size) than the familiar butterfly-shaped attractor. The coupled dynamics has some similarities to that of the uncoupled system below the Hopf bifurcation, although it displays richer patterns. For suitable coupling strength there can be as many as six distinct stable states. We describe the basins of attraction of these coexisting states and their boundaries using various measures and establish that the basins are intermingled.