Abstract:  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 Lorenz systems with 𝜌 just above 𝜌𝐻 when all equilibria in each isolated system 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.

Abstract:  We study collective dynamics of networks of mutually coupled identical Lorenz oscillators near subcritical Hopf bifurcation. Such systems exhibit induced multistable behavior with interesting spatio-temporal dynamics including synchronization, desynchronization and chimera states. For analysis, we first consider a ring topology with nearest neighbour 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 behaviour of the intermittent laminar phases is characterized by the characteristic time spent in the synchronization manifold, which decays as power law. Such intermittent dynamics is quite general and is also observed in an ensemble of large number of oscillators arranged in variety of network topologies including nonlocal, scale-free, random, and small-world networks.

Abstract: We discuss a simple yet powerful control technique called "Linear Augmentation" (LA) for nonlinear dynamical systems. The linear augmentation can be perceived as a type of interaction that may occur naturally in dynamical systems as an environmental effect, or can be explicitly added to a system in order to control its collective dynamical behavior. LA has been known to effectively regulate resulting dynamics of various dynamical systems and can be used as a powerful control strategy in various applications. Examples include targeting attractor(s), regulating multistable dynamics, suppression of extreme events, and controlling chimera states in the nonlinear dynamical systems.

Abstract: The Internet of Medical Things (IoMT) is the most pervasive era of the Internet of Things (IoT), gaining more researcher attention exponentially daily due to its wide applicability in smart healthcare systems (SHS). Due to the current pandemic situation, it is very dangerous for individuals to visit the doctor for any minor problem. With the IoMT device, you can easily monitor your daily health data and take the first preventive measures yourself. IoMT is vulnerable to many attacks, including denial of service (DoS), malware, and eavesdropping. Moreover, IoMT is exposed to various vulnerabilities, such as security, privacy, and confidentiality. Despite multiple security threats, new cryptographic techniques, such as access control, identity authentication, and data encryption, can help improve the security and reliability of IoMT devices. In this paper, we will talk about cyber security in IoMT. We have reviewed different works done to secure IoMT using machine learning and also discussed the challenges and limitations of IoMT. And also reviewed the techniques/algorithms used and compared the results of this research.

Abstract: We study synchronization in networks of delay-coupled electronic oscillators, so-called phase-locked loops (PLLs). Using a phase-model description, we study the collective dynamics of mutually coupled PLLs and report the phenomenon of heterogeneity-induced synchronization. This phenomenon refers to the observation that heterogeneity in the system’s parameters can induce synchronization by stabilizing the states which are unstable without such heterogeneity. In systems where component heterogeneity can be tuned and controlled, we show how the complex collective self-organized dynamics can be guided towards synchronized states with specific operational frequencies and phase relations. This is of importance for the technical applicability of self- organized dynamics. In electrical engineering, for example, where components can be strongly heterogeneous, our theoretical framework can inform the design process for networks of spatially distributed PLLs. The results presented here are also useful in understanding the collective dynamics in ensembles of phase oscillators with time-delayed interactions, inertia, and heterogeneity.

Significance of the work: In this work, we report and experimentally demonstrate the occurrence of a peculiar phenomenon--stabilization of synchronized states through parameter-heterogeneity. Since heterogeneity is typically expected to create hindering effects on synchrony, this counterintuitive outcome opens up prospect for further analysis to get insight into such curious phenomena. We introduce parameter-heterogeneity in mutually delay-coupled electronic clocks for controlling the collective dynamics and show how heterogeneity can be used to tune such systems into specific synchronized states. This analysis is helpful in understanding the mutual synchronization of electronic clocks, which has direct applications in the design of distributed electronic systems with potential to bring significant improvement to currently available state-of-the-art technology. 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.

Abstract: The production process of integrated electronic circuitry inherently leads to large heterogeneities on the component level. For electronic clock networks this implies detuned intrinsic frequencies and differences in coupling strength and the characteristic time delays associated with signal transmission, processing, and feedback. Using a phase-model description, we study the effects of such component heterogeneity on the dynamical properties of synchronization in networks of mutually delay-coupled Kuramoto oscillators. We test the theory against experimental results and circuit-level simulations in a prototype system of mutually delay-coupled electronic clocks, so-called phase-locked loops. Interestingly, our results show that heterogeneity in the system can actually enhance the stability of synchronized states. That means that beyond the optimizations that can be achieved by tuning homogeneous coupling strengths, time delays, and loop-filter cut-off frequencies, hetero- geneities in these system parameters enable much better optimization of perturbation decay rates, stabilization of synchronous states, and tuning of phase differences between the clocks. Our theory enables the design of custom-fit synchronization layers according to the specific requirements and properties of electronic systems, such as operational frequencies, phase relations, and, e.g., transmission delays. These results are not restricted to electronic systems, because signal transmission, processing, and feedback delays are common to networks of spatially distributed and coupled autonomous oscillators.

Significance of the work: In this work, we report and experimentally demonstrate the occurrence of a peculiar phenomenon--stabilization of synchronized states through parameter-heterogeneity. Since heterogeneity is typically expected to create hindering effects on synchrony, this counterintuitive outcome opens up prospect for further analysis to get insight into such curious phenomena. We introduce parameter-heterogeneity in mutually delay-coupled electronic clocks for controlling the collective dynamics and show how heterogeneity can be used to tune such systems into specific synchronized states. This analysis is helpful in understanding the mutual synchronization of electronic clocks, which has direct applications in the design of distributed electronic systems with potential to bring significant improvement to currently available state-of-the-art technology. 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.

Abstract: 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.

Abstract: Stuart-Landau oscillators can be coupled so as to either preserve or destroy the rotational symmetry that the uncoupled system possesses. We examine some of the simplest cases of such couplings for a system of two nonidentical oscillators. When the coupling breaks the rotational invariance, there is a qualitative difference between oscillators wherein the phase velocity has the same sign (termed co-rotation) or opposite signs (termed counter-rotation). In the regime of oscillation death the relative sense of the phase rotations plays a major role. In particular, when rotational invariance is broken, counter-rotation or phase velocities of opposite signs appear to destabilize existing fixed points, thereby preserving and possibly extending the range of oscillatory behavior. The dynamical “frustration” induced by counter-rotations can thus suppress oscillation quenching when coupling breaks the symmetry.

Significance of the work: Here, we studied the collective dynamics of coupled counter-rotating Stuart–Landau oscillators. In this system, we study the phenomenon of amplitude death (AD), where the oscillators drag each other towards a fixed point and stop oscillating. This phenomenon was first reported in the oscillator-ensembles with mismatched parameters. Later, it was observed that, with time-delayed coupling, AD can occur in identical oscillators as well. Since then, the phenomenon has been observed/achieved by different strategies including the use of conjugate coupling and phase-frustration parameter. With this work, we hope to understand its mechanism better and propose that symmetry-breaking might be a general criterion for the occurrence of AD connecting all these methods.

Abstract: Chimeras, namely coexisting desynchronous and synchronized dynamics, are formed in an ensemble of identically coupled identical chaotic oscillators when the coupling induces multiple stable attractors, and further when the basins of the different attractors are intertwined in a complex manner. When there is coupling-induced multistability, an ensemble of identical chaotic oscillators—with global coupling, or also under the influence of common noise or an external drive (chaotic, periodic, or quasiperiodic)—inevitably exhibits chimeric behavior. Induced multistability in the system leads to the formation of distinct subpopulations, one or more of which support synchronized dynamics, while in others the motion is asynchronous or incoherent. We study the mechanism for the emergence of such chimeric states, and we discuss the generality of our results.

Abstract: We study the multistability that results when a chaotic response system that has an invariant symmetry is driven by another chaotic oscillator. We observe that there is a transition from a desynchronized state to a situation of multistability. In the case considered, there are three coexisting attractors, two of which are synchronized and one is desynchronized. For large coupling, the asynchronous attractor disappears, leaving the system bistable. We study the basins of attraction of the system in the regime of multistability. The three attractor basins are interwoven in a complex manner, with extensive riddling within a sizeable region of (but not the entire) phase space. A quantitative characterization of the riddling behavior is made via the so–called uncertainty exponent, as well as by evaluating the scaling behavior of tongue–like structures emanating from the synchronization manifold.

Significance of the work: In this work, we carried out a detailed analysis to understand how multistability can be induced in an ensemble of chaotic oscillators through a common drive (including periodic, quasi-periodic, chaotic or even noisy signals) or mutual coupling. For chaotic ensembles, our analysis showed that multistable attractors thus obtained have different dynamical properties: oscillators settling on some attractors show synchronized dynamics, whereas motion on other attractors can be incoherent. This is an interesting property upon which our next study was based. We used these multistable attractors with different synchronization properties to obtain ‘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 such heterogeneities.

Abstract: We study the dynamics of nonlocally coupled phase oscillators in a modular network. The interactions include a phase lag, α. Depending on the various parameters the system exhibits a number of different dynamical states. In addition to global synchrony there can also be modular synchrony when each module can synchronize separately to a different frequency. There can also be multicluster frequency chimeras, namely coherent domains consisting of modules that are separately synchronized to different frequencies, coexisting with modules within which the dynamics is desynchronized. We apply the Ott-Antonsen ansatz in order to reduce the effective dimensionality and thereby carry out a detailed analysis of the different dynamical states.

Abstract: We study synchronization in bipartite networks of phase oscillators with general nonlinear coupling and distributed time delays. Phase-locked solutions are shown to arise, where the oscillators in each partition are perfectly synchronized among themselves but can have a phase difference with the other partition, with the phase difference necessarily being either zero or π radians. Analytical conditions for the stability of both types of solutions are obtained and solution branches are explicitly calculated, revealing that the network can have several coexisting stable solutions. With increasing value of the mean delay, the system exhibits hysteresis, phase flips, final state sensitivity, and an extreme form of multistability where the numbers of stable in-phase and antiphase synchronous solutions with distinct frequencies grow without bound. The theory is applied to networks of Landau-Stuart and Rössler oscillators and shown to accurately predict both in-phase and antiphase synchronous behavior in appropriate parameter ranges.

Abstract: We study a system of mismatched oscillators on a bipartite topology with time-delay coupling, and analyze the synchronized states. For a range of parameters, when all oscillators lock to a common frequency, we find solutions such that systems within a partition are in complete synchrony, while there is lag synchronization between the partitions. Outside this range, such a solution does not exist and instead one observes scenarios of remote synchronization—namely, chimeras and individual synchronization, where either one or both of the partitions are synchronized independently. In the absence of time delay such states are not observed in phase oscillators.

Significance of the work: Here, for the first time, we report the occurrence of ‘remote synchronization’ in phase oscillators. Remote synchronization is the phenomenon where indirectly coupled oscillators, interacting through a relaying unit, can be synchronized while the mediating oscillator remains incoherent. One of the necessary conditions for this to happen is amplitude variations in relaying oscillators: this leads to the criterion that remote synchrony can not be observed in phase oscillators. However, this work establishes that in the presence of time-delay coupling, one can observe this phenomenon in phase-oscillators as well.

Abstract: For an ensemble of globally coupled oscillators with time-delayed interactions, an explicit relation for the frequency of synchronized dynamics corresponding to different phase behaviors is obtained. One class of solutions corresponds to globally synchronized in-phase oscillations. The other class of solutions have mixed phases, and these can be either randomly distributed or can be a splay state, namely with phases distributed uniformly on a circle. In the strong coupling limit and for larger networks, the in-phase synchronized configuration alone remains. Upon variation of the coupling strength or the size of the system, the frequency can change discontinuously, when there is a transition from one class of solutions to another. This can be from the in-phase state to a mixed-phase state, but can also occur between two inphase configurations of different frequency. Analytical and numerical results are presented for coupled Landau–Stuart oscillators, while numerical results are shown for Rössler and FitzHugh-Nagumo systems.

Abstract: Here we extend a recent review (Physics Reports 521, 205 (2012)) of amplitude death, namely the suppression of oscillations due to the coupling interactions between nonlinear dynamical systems. This is an important emergent phenomenon that is operative under a variety of scenarios. We summarize results of recent studies that have significantly added to our understanding of the mechanisms that underlie the process, and also discuss the phase–flip transition, a characteristic and unusual effect that occurs in the transient dynamics as the oscillations die out.

Abstract: We consider oscillators coupled with asymmetric time delays, namely, when the speed of information transfer is direction dependent. As the coupling parameter is varied, there is a regime of amplitude death within which there is a phase-flip transition. At this transition the frequency changes discontinuously, but unlike the equal delay case when the relative phase difference changes by π , here the phase difference changes by an arbitrary value that depends on the difference in delays. We consider asymmetric delays in coupled Landau-Stuart oscillators and Rössler oscillators. Analytical estimates of phase synchronization frequencies and phase differences are obtained by separating the evolution equations into phase and amplitude components. Eigenvalues and eigenvectors of the Jacobian matrix in the neighborhood of the transition also show an “avoided crossing,” as has been observed in previous studies with symmetric delays.

Significance of the work: In this work, we went beyond limit-cycle oscillators and extended our study to explore the mechanism of the phenomenon ‘phase-flip’ transition in coupled chaotic oscillators. We not only established the generality of our proposed mechanism but also learned how qualitative behavior of a complex multi-delayed system can be traced through a simplified (reduced) system with single-delay.

Abstract: We study the dynamics of time-delay coupled limit-cycle oscillators in the amplitude death regime. Through a detailed analysis of the Jacobian at the fixed point, we show that the phase-flip transition, namely, the abrupt change from in-phase synchronized dynamics to antiphase synchronized dynamics, is associated with an interchange of the imaginary parts of complex pairs of eigenvalues at an “avoided crossing” of Lyapunov exponents as a parameter is varied. An order parameter for the transition is constructed through the eigenvectors of the Jacobian.