Exactly solvable Stuart-Landau models in arbitrary dimensions.

We use Clifford's geometric algebra to extend the Stuart-Landau system to dimensions D>2 and give an exact solution of the oscillator equations in the general case. At the supercritical Hopf bifurcation marked by a transition from stable fixed-point dynamics to oscillatory motion, the Jacobian matrix evaluated at the fixed point has N=⌊D/2⌋ pairs of complex conjugate eigenvalues which cross the imaginary axis simultaneously. For odd D there is an additional purely real eigenvalue that does the same.

Phase slips in coupled oscillator systems.

Phase slips are a typical dynamical behavior in coupled oscillator systems: the route to phase synchrony is characterized by intervals of constant phase difference interrupted by abrupt changes in the phase difference. Qualitatively similar to stick-slip phenomena, analysis of phase slip has mainly relied on identifying remnants of saddle-nodes or "ghosts." We study sets of phase oscillators and by examining the dynamics in detail, offer a more precise, quantitative description of the phenomenon.

Transition from inhomogeneous limit cycles to oscillation death in nonlinear oscillators with similarity-dependent coupling.

We consider a system of coupled nonlinear oscillators in which the interaction is modulated by a measure of the similarity between the oscillators. Such a coupling is common in treating spatially mobile dynamical systems where the interaction is distance dependent or in resonance-enhanced interactions, for instance. For a system of Stuart-Landau oscillators coupled in this manner, we observe a novel route to oscillation death via a Hopf bifurcation.

Intermingled attractors in an asymmetrically driven modified Chua oscillator.

Understanding the asymptotic behavior of a dynamical system when system parameters are varied remains a key challenge in nonlinear dynamics. We explore the dynamics of a multistable dynamical system (the response) coupled unidirectionally to a chaotic drive. In the absence of coupling, the dynamics of the response system consists of simple attractors, namely, fixed points and periodic orbits, and there could be chaotic motion depending on system parameters.

A stochastic model of homeostasis: The roles of noise and nuclear positioning in deciding cell fate.

We study a population-based cellular model that starts from a single stem cell that divides stochastically to give rise to either daughter stem cells or differentiated daughter cells. There are three main components in the model: nucleus position, the underlying gene-regulatory network, and stochastic segregation of transcription factors in the daughter cells. The proportion of self-renewal and differentiated cell lines as a function of the nucleus position which in turn decides the plane of cleavage is studied.

Transition and identification of pathological states in p53 dynamics for therapeutic intervention.

We study a minimal model of the stress-driven p53 regulatory network that includes competition between active and mutant forms of the tumor-suppressor gene p53. Depending on the nature and level of the external stress signal, four distinct dynamical states of p53 are observed. These states can be distinguished by different dynamical properties which associate to active, apoptotic, pre-malignant and cancer states.

Dynamical effects of breaking rotational symmetry in counter-rotating Stuart-Landau oscillators.

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).

Coupled Lorenz oscillators near the Hopf boundary: Multistability, intermingled basins, and quasiriddling.

We investigate the dynamics of coupled identical chaotic Lorenz oscillators just above the subcritical Hopf bifurcation. In the absence of coupling, the motion is on a strange chaotic attractor and the fixed points of the system are all unstable. With the coupling, the unstable fixed points are converted into chaotic attractors, and the system can exhibit a multiplicity of coexisting attractors.

Synchronization properties of coupled chaotic neurons: The role of random shared input.

Spike-time correlations of neighbouring neurons depend on their intrinsic firing properties as well as on the inputs they share. Studies have shown that periodically firing neurons, when subjected to random shared input, exhibit asynchronicity. Here, we study the effect of random shared input on the synchronization of weakly coupled chaotic neurons.

Driving-induced multistability in coupled chaotic oscillators: Symmetries and riddled basins.

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.

Erratum: Driving-induced bistability in coupled chaotic attractors [Phys. Rev. E 87, 042909 (2013)].

This corrects the article DOI: 10.1103/PhysRevE.87.

Local properties of vigilance states: EMD analysis of EEG signals during sleep-waking states of freely moving rats.

Understanding the inherent dynamics of the EEG associated to sleep-waking can provide insights into its basic neural regulation. By characterizing the local properties of the EEG using power spectrum, empirical mode decomposition (EMD) and Hilbert-spectral analysis, we can examine the dynamics over a range of time-scales. We analyzed rat EEG during wake, NREMS and REMS using these methods.

Amplitude death phenomena in delay-coupled Hamiltonian systems.

Hamiltonian systems, when coupled via time-delayed interactions, do not remain conservative. In the uncoupled system, the motion can typically be periodic, quasiperiodic, or chaotic. This changes drastically when delay coupling is introduced since now attractors can be created in the phase space.

Nature of weak generalized synchronization in chaotically driven maps.

Weak generalized synchrony in a drive-response system occurs when the response dynamics is a unique but nondifferentiable function of the drive, in a manner that is similar to the formation of strange nonchaotic attractors in quasiperiodically driven dynamical systems. We consider a chaotically driven monotone map and examine the geometry of the limit set formed in the regime of weak generalized synchronization. The fractal dimension of the set of zeros is studied both analytically and numerically.

Driving-induced bistability in coupled chaotic attractors.

We examine the effects of symmetry-preserving and -breaking interactions in a drive-response system where the response has an invariant symmetry in the absence of the drive. Subsequent to the onset of generalized synchronization, we find that there can be more than one stable attractor. Numerical as well as analytical results establish the presence of phase synchrony in such coexisting attractors.

Distribution of MGEs and their insertion sites in the Macaca mulatta genome.

Mobile genetic elements (MGEs) are fragments of DNA that can move around within the genome through retrotransposition. These are responsible for various important events such as gene inactivation, transduction, regulation of gene expression and genome expansion. The present work involves the identification and study of the distribution of Alu and L1 retrotransposons in the genome of Macaca mulatta, an extensively used organism in biomedical studies.

Frequency discontinuity and amplitude death with time-delay asymmetry.

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.

Relaying phase synchrony in chaotic oscillator chains.

We study the manner in which the effect of an external drive is transmitted through mutually coupled response systems by examining the phase synchrony between the drive and the response. Two different coupling schemes are used. Homogeneous couplings are via the same variables while heterogeneous couplings are through different variables.

Phase-flip transition in relay-coupled nonlinear oscillators.

We study the dynamics of oscillators that are coupled in relay; namely, through an intermediary oscillator. From previous studies it is known that the oscillators show a transition from in-phase to out-of-phase oscillations or vice versa when the interactions involve a time delay. Here we show that, in the absence of time delay, relay coupling through conjugate variables has the same effect.

Genome-wide analysis of mobile genetic element insertion sites.

Mobile genetic elements (MGEs) account for a significant fraction of eukaryotic genomes and are implicated in altered gene expression and disease. We present an efficient computational protocol for MGE insertion site analysis. ELAN, the suite of tools described here uses standard techniques to identify different MGEs and their distribution on the genome.

Nature of the phase-flip transition in the synchronized approach to amplitude death.

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.

Quasiperiodic forcing of coupled chaotic systems.

We study the manner in which the effect of quasiperiodic modulation is transmitted in a coupled nonlinear dynamical system. A system of Rössler oscillators is considered, one of which is subject to driving, and the dynamics of other oscillators which are, in effect, indirectly forced is observed. Strange nonchaotic dynamics is known to arise only in quasiperiodically driven systems, and thus the transmitted effect is apparent when such motion is seen in subsystems that are not directly modulated.

Synchronization regimes in conjugate coupled chaotic oscillators.

Nonlinear oscillators that are mutually coupled via dissimilar (or conjugate) variables display distinct regimes of synchronous behavior. In identical chaotic oscillators diffusively coupled in this manner, complete synchronization occurs only by chaos suppression when the coupled subsystems drive each other into a regime of periodic dynamics. Furthermore, the coupling does not vanish but acts as an "internal" drive.

Design strategies for the creation of aperiodic nonchaotic attractors.

Parametric modulation in nonlinear dynamical systems can give rise to attractors on which the dynamics is aperiodic and nonchaotic, namely, with largest Lyapunov exponent being nonpositive. We describe a procedure for creating such attractors by using random modulation or pseudorandom binary sequences with arbitrarily long recurrence times. As a consequence the attractors are geometrically fractal and the motion is aperiodic on experimentally accessible time scales.

Coexisting attractors in periodically modulated logistic maps.

We consider the logistic map wherein the nonlinearity parameter is periodically modulated. For low periods, there is multistability, namely two or more distinct dynamical attractors coexist. The case of period 2 is treated in detail, and it is shown how an extension of the kneading theory for one-dimensional maps can be applied in order to analyze the origin of bistability, and to demarcate the principal regions of bistability in the phase space.