Classical Mechanics
Classical mechanics from the time of Newton till the birth of quantum mechanics is discussed.
The NLD Lab is located in Room 233 in the School of Physical Sciences, JNU.
Our group works on a number of current problems in nonlinear science, ranging from a study of time delay dynamics to dynamical chimeric states, to problems of computational neuroscience, systems biology, and nonequilibrium statistical physics.
The accompanying Wordle image was made out of the titles of papers published over the years, and this should give you a fair idea of what we are mainly interested in. For greater details visit our research page.
Prof. Ram Ramaswamy has been in the School of Physical Sciences at JNU since 1986, coming here after a few years at the TIFR, Mumbai. Since 2001, he has also held a concurrent position in the School of Computational and Integrative Sciences at JNU.
Our research interests are diverse: here is a list of publications from the group.
We work in the broad area of nonlinear science, the Wordle image gives a sample of the words in titles of our recent papers.
Over the past few years, our interests have been in applying concepts of nonlinear dynamics to various systems of current interest such as models of coupled oscillators, synchronization, time-delay dynamics, and so on.
Systems under the effect of parametric modulation - periodic, quasiperiodic, chaotic, or even stochastic forcing - exhibit many interesting phenomena and different dynamical properties that are obtained in the unforced systems are modified, leading to the bifurcations and attractors with novel dynamical properties.
Over the years we have examined a number of such dynamical systems, particularly a variety of quasiperiodically driven oscillators. One interest was the phenomenon of strange nonchaotic motion on attractors (SNAs) that were seen in nonlinear systems driven by aperiodic forcing. Such motion is strange, in the sense of being on fractal attractors, but nonchaotic, in the sense of having non-positive Lyapunov exponents.
We have studied a number of issues relating to SNAs, ranging from the mechanisms of their formation and destruction, to applications in synchronization, in creating stable aperiodic motion, in their robustness to noise and so on. Some of the main results have been summarised in a couple of review articles on SNAs and their applications.
In most natural systems, the transmission of signals takes place at finite velocity. This makes time delay coupling important in a proper model of such situations. Several studies have explored the manner in which this type of coupling affects complex phenomena (such as synchronization for example) in nonlinear dynamical systems. Time delay increases the dimensionality of the system and often also analytically intractable. Our recent work on time-delay coupled systems address specific features such as amplitude or oscillation death, changes in the modes of synchronization and in the phase-flip transition. Some of the main work has been summarized in two reviews.
We have also been looking at how systems are coupled to one another, when the coupling terms are not of the simple, diffusive form, or when "unlike" variables are coupled to each other. This latter situation often occurs in experimental systems. We have termed this the conjugate coupling, when dissimilar varables are coupled. Examples range from laser systems to electrochemical cells, to ecological models of predator-prey dynamics...
Biochemical systems are commonly modelled by differential equations or simulated by stochastic algorithms---the macroscopic and microscopic descriptions respectively. The macroscopic dynamics can be justified only when the participating molecule numbers are high enough to be replaced by concentrations. For small systems, intrinsic fluctuations comes into play som owing to the random creation and decay of individual molecules: a microscopic description of such disordered molecular motion is physically more correct.
Synchrony or concerted dynamical behaviour is observed in a wide variety of natural systems. Such behaviour is also often robust, namely, systems with large stochastic fluctuations, possessing a range of internal time--scales are nevertheless capable of exhibiting sustained correlated dynamics over long times. In one of our recent works, we examined mechanisms by which two (or more) stochastic systems can be microscopically coupled so as to result in the phase synchronization of their dynamical variables. Study of model systems, that produce sustained oscillations shows that a suitable coupling of different networks can lead to such correlated behaviour, and that both in--phase and anti--phase synchronization can occur when the coupling is time delayed.
We have two main interests in this area: How is rhythmic (or oscillatory) behaviour created, and what affects sycnhrony in neuron models. We have studied a number of automaton models to examine the major features - in terms of topology and individual "neuron" dynamics - that contribute to the creation of sustained rhythms. We have also studied a number of (more realistic) dynamical models of neurons in order to see how the coupling induces synchrony, and also how noise can either assist or deter synchronous dynamics. We are also interested in devising new methods for deciphering inherent dynamics in neuronal signals (from, say, EEG data or from local field potential measurements) through an Empirical Mode Decomposition (EMD) analysis.
Our interests in Computational Biology fall in two classes: the analysis of genomic sequence data using bioinformatics methods, and the applications of dynamical systems theory and modeling to problems of biological interest.
We have analysed a variety of genomic sequences using various tools from mathematics/statistics and signal processing to identify features of biological interest such as genes, transposable elements, lateral gene transfer events, noncoding genes and so on. The tools and techniques used range from Fourier and Wavelet transforms to the use of Markov models, Shannon entropy and Support Vector Machines.
In cellular and subcellular processes, the dynamics is greatly affected by the small numbers of participating entities. This makes fluctuations very drastic, and a source of intrinsic noise. Thus the modeling of such phenomena proceeds via the Master equation, and approximations such as the Langevin equation. In the past several years we have been trying to understand the nature of synchrony in such intrinsically stochastic systems and the consequences for regulation at the cellular level.
Here you have a list of publications.
We report the scaling behaviour of rotational energy transfer moments. The quantum moments exhibit a polynomial scaling behaviour in the variable j,(_j, + 1 ), whereas the classical moments scale as a polynomial in 1 f , where j, is the initial rotational quantum number or action. Applications are made to Li:-rare gas collisions, as welpas to a classical planar-rotor collision model. The scaling theory allows an accurate interpolation and extrapolation of experimental scattering data.
Transport through a random medium in a external field is modelled by particles performing biased random walks on the infinite cluster above the percolation threshold. Steps are more likely in the direction of the field-say downward-than against. A particle is allowed to move only onto an empty site (particles interact via hard core exclusion). Branches that predominantly point downwards and backbends-backbone segments on which particles must move upwards-act as traps. We have studied the movement of interacting random walkers in branches and backbends by Monte Carlo simulations and also analytically. In the full network, the trap-limited current flows primarily through the part of the backbone composed of paths with the smallest backbends and its magnitude in high fields is estimated. Unlike in the absence of interactions, the drift velocity does not vanish in finite fields. However, it continues to show a non-monotonic dependence on the field over a sizeable range of density and percolation probability.
The existence of elaborate control mechanisms for the various biochemical processes inside and within living cells is responsible for the coherent behaviour observed in its spatio-temporal organisation. Stability and sensitivity are both necessary properties of living systems and these are achieved through negative and positive feedback loops as in other control systems. We have studied a three-step reaction scheme involving a negative and a positive feedback loop in the form of end-product inhibition and allosteric activation. The variety of behaviour exhibited by this system, under different conditions, includes steady state, simple limit cycle oscillations, complex oscillations and period bifurcations leading to random oscillations or chaos. The system also shows the existence of two distinct chaotic regimes under the variation of a single parameter. These results, in comparison with single biochemical control loops, show that new behaviours can be exhibited in a more complex network which are not seen in the single control loops. The results are discussed in the light of a diverse variety of cellular functions in normal and altered cells indicating the role of controlled metabolic network as the underlying basis for cellular behaviour.
Bound-state eigenfunctions for a (classically) nonintegrable two degrees of freedom Hamiltonian system are studied. Between the de Broglie wavelength and a localization length, the probability density has a statistically fractal structure in some eigenstates. This novel characterization of eigenstates is intrinsically basis-set and coordinate independent and might therefore provide an objective approach to the question of quantum-chaotic behaviour.
The repressor-mediated repression process in bacteria is modelled using a gene-enzyme-endproduct control unit. A combined analytical-numerical study shows that the system, though stable normally, becomes unstable for super-repressing strains even at low values of the cooperativity of repression, provided demand for the endproduct saturates at large endproduct concentrations. In addition the system also shows bistability, i.e., the co-existence of a stable steady-state and a stable limit cycle. The tryptophan operon is used as a model system and the results are discussed in the light of differential regulation of gene expression in lower organisms, especially in mutant strains.
Practical schemes of determining the energy eigenvalue spectrum of multimode resonant systems are suggested; these include an adaptation of the scaling (polynomial) interpolation technique, as well as an adiabatic-switching method.
It has been conjectured that the unpredictability of climatic systems is due to strange attractors (SAs) in the configuration space dynamics. One climatic record, the oxygen isotope ratio data from deep-sea cores that pertains to long periods on the order of one million years and provides direct correlation with the glaciation-deglaciation periods, seemed to indicate (under earlier analysis) a low dimensional attractor of correlation dimension D2 ≈ 3.1. Our present reanalysis of this data in light of recent methods suggested by Broomhead and King (BK) is at variance with that result. Two (model) four-variable systems that support chaotic strange attractors are examined using an analysis similar to BK to investigate the practical drawbacks of using a short time-series vis-a-vis the estimation of attractor dimension.
In this Brief Report we analyze the limit of very weak damping of a quantum-mechanical Brownian oscillator. It is shown that the propagator for the reduced density operator of the oscillator can be written as a double path integral of the same form as that obtained in the high-temperature limit. As a direct consequence, we can write a Fokker-Planck equation for the reduced density operator of the weakly damped oscillator (at any temperature) involving only the damping and a generalized diffusion constant in momentum space.
We study the level statistics of parity-selected electronic states of atomic uranium (including autoionisation levels), obtained from recent photoionisation experiments. The spacings distribution which reflects short-range structure appears to be Poisson but spectral fluctuation measures reveal rigidity, and are consistent with a superposition of GOE sequences as is typically seen in nuclear spectra.
We define a variant of the model of Bak, Tang, and Wiesenfeld of self-organized critial behavior by introducing a preferred direction. We characterize the critical state and, by establishing equivalence to a voter model, determine the critical exponents exactly in arbitrary dimension d. The upper critical dimension for this model is three. In two dimensions the model is equivalent to an earlier solved special case of directed percolation.
We extend an adaptive control algorithm recently suggested by Huberman and Lumer to multi-parameter and higherdimensional nonlinear systems. This control mechanism is remarkably effective in returning a system to its original dynamics after a sudden perturbation in the system parameters changes the dynamical behaviour. We find that in all cases, the recovery time is linearly proportional to the inverse of control stiffness (for small stiffness). In higher dimensions there is an additional optimization problem since increasing stiffness beyond a certain value can retard recovery. The control of fixed point dynamics in systems capable of a wide variety of dynamical behaviour is demonstrated. We further suggest methods by which periodic motion such as limit cycles can be adaptively controlled, and demonstrate the robustness of the procedure in the presence of (additive) background noise.
From the viewpoint of eigenvalue level statistics, harmonic-oscillator systems are unusual. Although integrable, these systems are nongeneric, and a spacing distribution does not exist even as the number of levels N→∞. The origins of this pathological behavior are explored using methods of number theory and ergodic analysis. However, such nongenericity is extremely fragile, and the smallest nonlinearity asymptotically restores generic behavior. These results are of relevance to the study of molecular spectra, as well as to the quasienergy spectra of integrable quantum maps.
The power spectrum of the potential energy fluctuation of liquid water is examined and found to yield so-called l/f frequency dependence (f is frequency). This is in sharp contrast to spectra of simple liquids (e.g., liquid argon), which exhibit a near white spectrum. This indicates that there exists an extended multiplicity of hydrogen bond network relaxations in liquid water. A simple model of cellular dynamics is proposed to explain this frequency dependence. On the other hand, the cluster dynamics of argon also involves energy fluctuations of a l/f type, resulting from various relaxation processes at core and surface.
Abstract here
Irregular scattering in molecular inelastic collision is analyzed classical mechanically by a novel method called “decoupling surface analysis.” Effective Hamiltonian of this analysis provides a phase space view of collision processes analogous to the PoincarC section of coupled-oscillator systems. In this phase space view irregular scattering occurs in a stochastic layer formed around separatrix connected to resonance structure of the effective Hamiltonian. This circumstance is parallel to that in the coupled-oscillator systems, in which stochastic motion is known to be connected to nonlinear resonance. The resonance structure in collision indicates trapping of classical trajectories in a certain dynamical well. The decoupling surface analysis suggests that the dynamical well is formed by a dip of stability exponents of trajectories as a function of time. By using a prototypical model exhibiting irregular scattering, a formal theoretical treatment is developed to analyze the structure of the fractal, termed icicle structure, observed in the plot of final vibrational action against the initial vibrational phase angle.
We show, using semiclassical methods, that as a symmetry is broken, the transition between universality classes for the spectral correlations of quantum chaotic systems is governed by the same parametrization as in the theory of random matrices. The theory is quantitatively verified for the kicked rotor quantum map. We also provide an explicit substantiation of the random matrix hypothesis, namely that in the symmetry-adapted basis the symmetry-violating operator is random.
Potential energy fluctuations in small atomic clusters have long-ranged temporal correlations, which lead to l/f noise in the power spectra. The relaxation dynamics in clusters has a hierarchical organization, resulting from different processes at the surface and core. A cellular dynamical model is proposed to understand the origin of such fluctuations.
The signatures of classical chaos and the role of periodic orbits in the wave-mechanical eigenvalue spectra of two-dimensional billiards are studied experimentally in microwave cavities. The survival probability for all the chaotic cavity data shows a ‘‘correlation hole,’’ in agreement with theory, that is absent for the integrable cavity. The spectral rigidity Δ3(L), which is a measure of long-range correlation, is shown to be particularly sensitive to the presence of marginally stable periodic orbits. Agreement with random-matrix theory is achieved only after excluding such orbits, which we do by constructing a special geometry, the Sinai stadium. Pseudointegrable geometries are also studied, and are found to display intermediate behavior.
We review the problem of particle transport in random media in the presence of an external field. The random medium is modeled by the infinite cluster above the percolation threshold. The field imposes a preferred direction of motion along which diffusing particles (random walkers are more likely to move than against. Two kinds of traps occur - branches pointing in the direction of the field, and backbends, in which particles must move against the field. For noninteracting particles, the drift velocity is a nonmonotonic function of the biasing field, and the two kinds of traps make the current vanish above a threshold value of the bias. If there is hard-core repulsion between the particles, branches get filled up and eventually cease to be traps. Below the directed percolation threshold, transport is rate-limited by backbends, and the particle current flows predominantly along those paths on the percolation backbone on which the length of every backbend is bounded. The current is a nonmonotonic function of the biasing field. We also consider a different sort of interparticle interaction which leads to levels of particles equalising near backbend bottoms. The motion along a typical path is then described by 'drop-push' dynamics: between backbends, particles drop down, assisted by the field, and push those on the next backbend, possibly leading to a cascade of overflows. Drop-push dynamics has interesting connections with other lattice gas automata, and Monte Carlo simulations show that the model supports kinematic waves and exhibits interesting behaviour of time-dependent correlations.
Potential energy fluctuations in the liquid phase of small atomic clusters (e.g. Ar13) have been seen to have long-range temporal correlations. This is manifest in a power-law decay for the power spectrum, which has a characteristic 1 / f dependence on the frequency,f. (More precisely, the dependence is l/f”, with a = 1 .) In order to understand the origin of this behavior, we study the melting of mixed rare-gas clusters ArlzXe and XelzAr (via molecular dynamics simulations). Substitution of atomic impurities introduces widely differing time scales in the dynamics, and we show that long-lived memory-effects have their origins in hierarchical relaxation processes arising in the motion of the atoms from the surface to the core and vice versa.
We study the dynamics of a lattice gas with two species of "charged" diffusing particles in the presence of an external field. There are several time scales characterizing the approach of the steady state. At short times diffusion smears out inhomogeneities. For fields such that the density is below the virtual (size-dependent) threshold of jamming, i.e. in the flow phase, this is the main relaxation mechanism. In addition, there are kinematic waves arising from the non-linear dependence of the current on the density. Above the threshold there is an instability leading to a multistrip structure of blockages: in each strip one type of particle prevents the other type from following the direction of the imposed field. The strips coarsen logarithmically slowly, until finally a single blockage remains.
We study small clusters of atomic argon, Ar7, Ar&3, and Ar55, in the temperature range vrhere they undergo a transition from a solidlike phase to a liquidlike phase. The signature of the phase transition is clearly seen as a dramatic increase in the largest I yapunov exponent as the cluster "melts. "
We study small clusters of atomic argon, Ar7, Ar», and Ar», in the temperature range when they undergo a transition from a solidlike phase to a liquidlike phase. The power spectra of potential energy fluctuations of tagged atoms in the liquid state show 1/f behavior over a wide range of frequency f, unlike either the solid phase or bulk liquid, and suggest a new experimental means of detecting cluster melting. The origin of this temporal scale invariance is explored by studying the individual potential energy distributions, which are observed to become multimodal when the clusters melt.
We study coupled maps on a Cayley tree, with local (nearest-neighbor) interactions, and with a variety of boundary conditions. The homogeneous state (where every lattice site has the same value) and the node-synchronized state (where sites of a given generation have the same value) are both shown to occur for particular values of the parameters and coupling constants. We study the stability of these states and their domains of attraction. Since the number of sites that become synchronized is much higher compared to that on a regular lattice, control is easier to achieve. A general procedure is given to deduce the eigenvalue spectrum for these states. Perturbations of the synchronized state lead to different spatiotemporal structures. We find that a mean-field-like treatment is valid on this (effectively infinite dimensional) lattice.
We present a new technique for circumventing the problem of the zero-point leak in classical trajectories by extending the action-billiard approach of de Aguiar and Ozorio de Almeida @Nonlinearity 5, 523 ~1992!#. In addition to demonstrating its utility in a model problem, we examine the application of various methods of overcoming the zero-point leak in the case of collinear He1H2 1 collisions. We also show that not neglecting leaky trajectories gives, on an average, good agreement with quantal results for collinear as well as 3-dimensional collisions. © 1995 American Institute of Physics.
We study the dynamics of a hamiltonian system with two degrees of freedom coupled to a No&-Hoover thermostat. In the absence of the thermostat, the system is quasi-integrable: at low energies, most of the motion is on twodimensional tori, while at higher energies, the motion is mainly chaotic. Upon coupling to the thermostat the system becomes more chaotic, as evidenced by the magnitude of the largest Lyapunov exponent. In contrast to the case of isotropic oscillator systems coupled to thermostats, there is no evidence for a regime of integrable behaviour, even for large values of Q.
We cast a model of biological resource management as a problem of adaptive control in a nonlinear dynamical system. Optimisation of harvest, while ensuring that the resource population persists, is achieved through a simple algorithmic procedure which is remarkably robust under a variety of perturbations.
We characterize the steady state of a driven diffusive lattice gas in which each site holds several particles, and the dynamics is activated and asymmetric. Using a quantum Hamiltonian formalism, we show that for arbitrary transition rates the model has product invariant measure. In the steady state, a pairwise balance condition is shown to hold. Configurations n" and n' leading respectively into and out of a given configuration n are matched in pairs so that the flux of transitions from n" to n is equal to the flux from n to n'. Pairwise balance is more general than the condition of detailed balance and holds in the non-equilibrium steady state of a number of stochastic models.
The quasibound spectrum of the transition state in collinear ~He, H2 1! collisions is obtained from time-dependent wave packet calculations. Examination of short- and long-range correlations in the eigenvalue spectra through a study of the nearest neighbor spacing distribution, P(s), and the spectral rigidity, D3(L), reveals signatures of quantum chaotic behavior. Analysis in the time domain is carried out by computing the survival probability ^^P(t)&& averaged over initial states and Hamiltonian. All these indicators show intermediate behavior between regular and chaotic. A quantitative comparison of ^^P(t)&& with the results of random matrix theory provides an estimate of the fraction of phase space exhibiting chaotic behavior, in reasonable agreement with the classical dynamics. We also analyse the dynamical evolution of coherent Gaussian wave packets located initially in different regions of phase space and compute the survival probability, power spectrum and the volume of phase space over which the wave packet spreads and illustrate the different behaviors.
We study the variation of Lyapunov exponents of simple dynamical systems near attractorwidening and attractor-merging crises. The largest Lyapunov exponent has universal behavior, showing abrupt variation as a function of the control parameter as the system passes through the crisis point, either in the value itself, in the case of an attractor-widening crisis, or in the slope, for an attractor-merging crisis. The distribution of local Lyapunov exponents is very diferent for the two cases: the fluctuations remain constant through a merging crisis, but there is a dramatic increase in the fluctuations at a widening crisis.
We study a directed coupled map lattice model in d52 dimensions, with two degrees of freedom associated with each lattice site. The two freedoms are coupled at a fraction c of lattice bonds acting as quenched random defects. The system is driven ~by adding ‘‘energy,’’ say! in one of the degrees of freedom at the top of the lattice, and the relaxation rules depend on the local difference between the two variables at a lattice site. In the case of conservative dynamics, at any concentration of defects the system reaches a self-organized critical state with universal critical exponents close to the mean-field values t t51, t s52/3, and t n51/2, for the integrated distributions of avalanche durations (t), size (s), and released energy (n), respectively. The probability distributions follow the general scaling form P(X,L)5L2aP(XL2DX), where a'1 is the scaling exponent for the distribution of avalanche lengths, X stands for t, s, or n, and DX is the ~independently determined! fractal dimension with respect to X. The distribution of current through the system is, however, nonuniversal, and does not show any apparent scaling form. In the case of nonconservative dynamics—obtained by incomplete energy transfer at the defect bonds—the system is driven out of the critical state. In the scaling region close to c50 the probability distributions exhibit the general scaling form P(X,c,L)5X2t XP@X/j X(c),XL2DX#, where t X5a/DX and the corresponding coherence length j X(c) depends on the concentration of defect bonds c as j X(c);c2DX. [S1063-651X~96!01609-1]
We study the stability, energetics and dynamics of small model hydrogen fluoride clusters (HF) n using isoergic molecular dynamics simulations. The largest Lyapunov exponent is computed over the energy range when the clusters melt, and is found to be more useful in defining the onset of melting than Lindemann's index. We also examine the power spectrum of potential energy fluctuations of clusters in the liquid state, which show 1/f dependence over a smaller frequency range than rare-gas clusters of comparable size.
We study the jam phase of the deterministic traffic model in two dimensions. Within the jam phase, there is a phase transition, from a self-organized jam (formed by initial synchronization followed by jamming), to a random-jam structure. The backbone of the jam is defined and used to analyse self-organization in the jam. The fractal dimension and interparticle correlations on the backbone indicate a continous phase transition at density with critical exponent , which are characterized through simulations.
We calculate the maximal Lyapunov exponent in constant-energy molecular dynamics simulations at the melting transition for finite clusters of 6 to 13 particles (model rare–gas and metallic systems) as well as for bulk rare– gas solid. For clusters, the Lyapunov exponent generally varies linearly with the total energy, but the slope changes sharply at the melting transition. In the bulk system, melting corresponds to a jump in the Lyapunov exponent, and this corresponds to a singularity in the variance of the curvature of the potential energy surface. In these systems there are two mechanisms of chaos – local instability and parametric instability. We calculate the contribution of the parametric instability towards the chaoticity of these systems using a recently proposed formalism. The contribution of parametric instability is a continuous function of energy in small clusters but not in the bulk where the melting corresponds to a decrease in this quantity. This implies that the melting in small clusters does not lead to enhanced local instability.
We study the power spectrum of fluctuations in the potential energy of atoms in small rare-gas clusters. At temperatures when the cluster is in a liquid-like state the spectra have a “1/f” dependence over a wide range of frequency f. This behavior is distinctly different from both the solid phase of clusters or bulk liquid, and is indicative of long-range temporal correlations. The origins of this phenomenon is explored by studying the individual potential-energy distributions in pure and mixed rare-gas clusters, Xe55 and ArXe54, via molecular dynamics simulations. Substitution of atomic impurities acts as an effective probe of the dynamics, and we observe that long-lived memory effects have their origins in hierarchical relaxation processes arising in the motion of the atoms from the surface to the core and vice-versa.
DNA sequence analysis has emerged as one of the major disiplines of theoretical biology and has become an essential tool for study in molecular biology. In this article we review various methods currently available for the analysis of genomes and large-scale DNA sequences in order to detect potential genes out of sequence information.
We study three models of driven sandpile-type automata in the presence of quenched random defects. When the dynamics is conservative, all these models, termed the random sites (A), random bonds (B), and random slopes (C), self-organize into a critical state. For model C the concentration-dependent exponents are nonuniversal. In the case of nonconservative defects, the asymptotic state is subcritical. Possible defect-mediated nonequilibrium phase transitions are also discussed.
Small clusters of Ni atoms are known to exhibit a phase transition from solidlike phase to a liquidlike phase. We perform classical molecular dynamics simulations of clusters of 6 and 13 atoms, using the many-body Gupta potential and compute the Lyapunov exponent of the system as a function of temperature. The Lyapunov exponent, which characterises the exponential divergence of trajectories in the phase space, has a crossover at the melting point, indicating that there is a dynamical transition from one chaotic regime to another when the cluster melts.
Finite clusters of atoms or molecules, typically composed of about 50 particles (and often as few as 13 or even less) have proved to be useful prototypes of systems undergoing phase transitions. Analogues of the solid-liquid melting transition, surface melting, structural phase transitions and the glass transition have been observed in cluster systems. The methods of nonlinear dynamics can be applied to systems of this size, and these have helped elucidate the nature of the microscopic dynamics, which, as a function of internal energy (or ‘temperature’) can be in a solidlike, liquidlike, or even gaseous state. The Lyapunov exponents show a characteristic behaviour as a function of energy, and provide a reliable signature of the solid-liquid melting phase transition. The behaviour of such indices at other phase transitions has only partially been explored. These and related applications are reviewed in the present article.
The collinear atom-diatom collision system provides one of the simplest instances of chaotic or irregular scattering. Classically, irregular scattering is manifest in the sensitive dependence of post-collision variables on initial conditions, and quantally, in the appearance of a dense spectrum of dynamical resonances. We examine the influence of kinematic factors on such dynamical resonances in collinear (He, H2+) collisions by computing the transition state spectra for collinear (He, HD+) and (He, DH+) collisions using the time-dependent quantum mechanical approach. The nearest neighbor spacing distribution P(s) and the spectral rigidity Δ3(L) for these resonances suggest that the dynamics is predominantlyirregular for collinear (He, HD+) and predominantlyregular for collinear (He, DH+). These findings are reinforced by a significantly larger "correlation hole" in ensemble averaged survival probability «P(t)» values for collinear (He, HD+) than for collinear (He, DH+). In addition we have also examined measures of classical chaos through the dependence of the final vibrational action, nf, on the initial vibrational phase φi of the diatom, and Poincaré surfaces-of-section. They show that (He, HD+) collisions are partly chaotic over the entire energy range (0-2.78 eV) while (He, DH+) collisions, in contrast, are highly regular at collision energies below the classical threshold for reaction. Above the threshold, the scattering remains regular for initial vibrational states ν=0 and 1 of DH+.
The spectrum of instantaneous normal mode (INM) frequencies of finite Lennard-Jones clusters is studied as a function of the extent of quantum delocalization. Configurations are sampled from the equilibrium distribution by a Fourier path integral Monte Carlo procedure. The INM spectra, average force constants and Einstein frequencies are shown to be interesting dynamical markers for the quantum delocalization-induced cluster solid–liquid transition. Comparisons are made with INM spectra of quantum and classical Lennard-Jones liquids. The methodology used here suggests a general strategy to obtain quantal analogs of various classical dynamical quantities.
Motivation: The major signal in coding regions of genomic sequences is a three-base periodicity. Our aim is to use fourier techniques to analyse this periodicity, and thereby to develop a tool to recognize coding regions in genomic DNA. Result: The three-base periodicity in the nucleotide arrangement is evidenced as a sharp peak at frequency f = 1/3 in the Fourier (or power) spectrum. From extensive spectral analysis of DNA sequences of total length over 5.5 million base pairs from a wide variety or organisms (including the human genome), and by separately examining coding and non-coding sequences, we find that the relative height of the peak at f = 1/3 in the Fourier spectrum is a good discriminator of coding potential. This feature is utilized by us to detect probable coding regions in DNA sequences, by examining the local signal-to-noise ratio of the peak within a sliding window. While the overall accuracy is comparable to that of other techniques currently in use, the measure that is presently proposed is independent of training sets or existing database information, and can thus find general application. Availability: A computer program GeneScan which locates coding open reading frames and exonic regions in genomic sequences has been developed, and is available on request.
We calculate the maximal Lyapunov exponent in constant-energy molecular-dynamics simulations at the melting transition for finite clusters of 6-13 particles (model rare-gas and metallic systems) as well as for bulk rare-gas solids. For clusters, the Lyapunov exponent generally varies linearly with the total energy, but the slope changes sharply at the melting transition. In the bulk system, melting corresponds to a jump in the Lyapunov exponent, and this corresponds to a singularity in the variance of the curvature of the potential-energy surface. In these systems there are two mechanisms of chaos-local instability and parametric instability. We calculate the contribution of the parametric instability toward the chaoticity of these systems using a recently proposed formalism. The contribution of parametric instability is a continuous function of energy in small clusters but not in the bulk where the melting corresponds to a decrease in this quantity. This implies that the melting in small clusters does not lead to enhanced local instability.
Strange nonchaotic attractors (SNA) arise in quasiperiodically driven systems in the neighborhood of a saddle-node bifurcation whereby a strange attractor is replaced by a periodic (torus) attractor. This transition is accompanied by Type-I intermittency. The largest nontrivial Lyapunov exponent Λ is a good order parameter for this route from chaos to SNA to periodic motion: the signature is distinctive and unlike that for other routes to SNA. In particular, Λ changes sharply at the SNA to torus transition, as does the distribution of finite-time or N-step Lyapunov exponents, P(ΛN).
Strange nonchaotic attractors (SNAs), which are realized in many quasiperiodically driven nonlinear systems, are strange (geometrically fractal) but nonchaotic (the largest nontrivial Lyapunov exponent is negative). Two such identical independent systems can be synchronized by in-phase driving: Because of the negative Lyapunov exponent, the systems converge to a common dynamics, which, because of the strangeness of the underlying attractor, is aperiodic. This feature, which is robust to external noise, can be used for applications such as secure communication. A possible implementation is discussed and its performance is evaluated. The use of SNAs rather than chaotic attractors can offer some advantages in experiments involving synchronization with aperiodic dynamics.
Different mechanisms for the creation of strange nonchaotic dynamics in the quasiperiodically forced logistic map are studied. These routes to strange nonchaos are characterized through the behavior of the largest nontrivial Lyapunov exponent, as well as through the characteristic distributions of finite-time Lyapunov exponents. Strange nonchaotic attractors can be created at a saddle-node bifurcation when the dynamics shows type-I intermittency; this intermittent transition, which is studied in detail, is characterized through scaling exponents. Band-merging crises through which dynamics remains nonchaotic are also studied, and correspondence is made with analogous behavior in the unforced logistic map. Robustness of these phenomena with respect to additive noise is investigated.
We describe adaptive control algorithms whereby a chaotic dynamical system can be steered to a target state with desired characteristics. A specific implementation considered has the objective of directing the system to a state which is more chaotic or mixed than the uncontrolled one. This methodology is easy to implement in discrete or continuous dynamical systems. It is robust and efficient, and has the additional advantage that knowledge of the detailed behavior of the system is not required.
We study the structural and dynamical aspects of 13–atom binary rare-gas clusters of Ar and Xe using constant–energy molecular dynamics simulations. The ground state geometry for ArnXe13−n, n=1−12, remains near-icosahedral, with an Ar atom occupying the central position. The thermodynamic properties of these clusters are significantly different from the pure 13-atom Ar or Xe clusters and for Xe–dominated compositions, melting is preceded by a surface–melting stage. Slow oscillations of the short-time-averaged (STA) temperature are observed both for surface–melting and complete melting stage, suggesting dynamical coexistence between different phases. At the complete melting stage, the oscillations in the STA temperature and the species of the central atom are correlated.
The performance of the GeneScan algorithm for gene identification has been improved by incorporation of a directed iterative scanning procedure. Application is made here to the cases of bacterial and organnellar genomes. The sensitivity of gene identification was 100% in Plasmodium falciparum plastid-like genome (35 kb) and in 98% in the Mycoplasma genitalium genome (∼580 kb) and the Haemophilus influenzae Rd genome (∼1.8 Mb). Sensitivity was found to improve in both the Open Reading Frames (ORFs) which have been identified as genes (by homology or by other methods) and those that are classified as hypothetical. False positive assignments (at the nucleotide level) were 0.25% in H. influenzae genome and 0.3% in M. genitalium. There were no false positive assignments in the plastid-like genome. The agreement between the GeneScan predictions and GeneMark predictions of putative ORFs was 97% in M. genitalium genome and 86% in H. influenzae genome. In terms of an exact match between predicted genes/ORFs and the annotation in the databank, GeneScan performance was evaluated to be between 72% and 90% in different genomes. We predict five putative ORFs that were not annotated earlier in the GenBank files for both M. genitalium and H. influenzae genomes. Our preliminary analysis of the newly sequenced G+C rich genome of Mycobacterium tuberculosis H37Rv also shows comparable sensitivity (99%).
We study the probability densities of finite-time or local Lyapunov exponents in low-dimensional chaotic systems. While the multifractal formalism describes how these densities behave in the asymptotic or long-time limit, there are significant finite-size corrections, which are coordinate dependent. Depending on the nature of the dynamical state, the distribution of local Lyapunov exponents has a characteristic shape. For intermittent dynamics, and at crises, dynamical correlations lead to distributions with stretched exponential tails, while for fully developed chaos the probability density has a cusp. Exact results are presented for the logistic map, x⃗ 4x(1−x). At intermittency the density is markedly asymmetric, while for “typical” chaos, it is known that the central limit theorem obtains and a Gaussian density results. Local analysis provides information on the variation of predictability on dynamical attractors. These densities, which are used to characterize the nonuniform spatial organization on chaotic attractors, are robust to noise and can, therefore, be measured from experimental data.
Strange nonchaotic attractors (SNAs) can be created due to the collision of an invariant curve with itself. This novel "homoclinic" transition to SNAs occurs in quasiperiodically driven maps which derive from the discrete Schrödinger equation for a particle in a quasiperiodic potential. In the classical dynamics, there is a transition from torus attractors to SNAs, which, in the quantum system, is manifest as the localization transition. This equivalence provides new insight into a variety of properties of SNAs, including its fractal measure. Further, there is a symmetry breaking associated with the creation of SNAs which rigorously shows that the Lyapunov exponent is nonpositive. We show that these characteristics associated with the appearance of SNA are robust and occur in a large class of systems.
The results of the experimental study of the dynamics of a shallow fluidized bed are reported. The behavior of granular material is controlled by the interplay of two factors—levitation due to the upward airflow, and sliding back due to gravity. Near the threshold of instability, the system shows critical behavior with remarkably long transient dynamics. The experimental observations are compared with a simple cellular automata model.
A number of parasite genome projects are under way, and large amounts of nucleotide sequence data are becoming available for analysis. There is an urgent need for development of theoretical tools to analyze the genome data, including identification of protein-coding sequences. The majority of the methods developed to date require prior information about the genome before accurate predictions can be made. Because such information is not available for many parasites, these methods cannot be directly applied. In this article, Alok Bhattacharya and colleagues describe some of the gene-prediction methods commonly in use, and a new method, GeneScan, that they have developed for the analysis of parasite genomes.
Different mechanisms for the creation of strange nonchaotic attractors (SNAs) are studied in a two-frequency parametrically driven Duffing oscillator. We focus on intermittency transitions in particular, and show that SNAs in this system are created through quasiperiodic saddle-node bifurcations (type-I intermittency) as well as through a quasiperiodic subharmonic bifurcation (type-III intermittency). The intermittent attractors are characterized via a number of Lyapunov measures including the behavior of the largest nontrivial Lyapunov exponent and its variance, as well as through distributions of finite-time Lyapunov exponents. These attractors are ubiquitous in quasiperiodically driven systems; the regions of occurrence of various SNAs are identified in a phase diagram of the Duffing system.
We have studied the adiabatic instantaneous normal modes (INMs) for a mixed 19-particle cluster, Ar9Xe10 as a function of temperature. In finite clusters, the INM frequencies, which are well-separated, do not mix as a consequence of the noncrossing rule. The frequencies of the lowest few modes of the system progressively soften as the temperature is increased, and prior to melting, the lowest few modes become unstable: these INM frequencies become imaginary. Eigenvectors corresponding to the lowest modes that appear to be involved in the actual melting process are identified.
We discuss several bifurcation phenomena that occur in the quasiperiodically driven logistic map. This system can have strange nonchaotic attractors (SNAs) in addition to chaotic and regular attractors; on SNAs the dynamics is aperiodic, but the largest Lyapunov exponent is nonpositive. There are a number of different transitions that occur here, from periodic attractors to SNAs, from SNAs to chaotic attractors, etc. We describe some of these transitions by examining the behavior of the largest Lyapunov exponent, distributions of finite time Lyapunov exponents and the invariant densities in the phase space.
Aperiodic dynamics which is nonchaotic is realized on Strange Nonchaotic Attractors (SNAs). Such attractors are generic in quasiperiodically driven nonlinear systems, and like strange attractors, are geometrically fractal. The largest Lyapunov exponent is zero or negative: trajectories do not show exponential sensitivity to initial conditions. In recent years, SNAs have been seen in a number of diverse experimental situations ranging from quasiperiodically driven mechanical or electronic systems to plasma discharges. An important connection is the equivalence between a quasiperiodically driven system and the Schrödinger equation for a particle in a related quasiperiodic potential, showing a correspondence between the localized states of the quantum problem with SNAs in the related dynamical system. In this review we discuss the main conceptual issues in the study of SNAs, including the different bifurcations or routes for the creation of such attractors, the methods of characterization, and the nature of dynamical transitions in quasiperiodically forced systems. The variation of the Lyapunov exponent, and the qualitative and quantitative aspects of its local fluctuation properties, have emerged as an important means of studying fractal attractors, and this analysis finds useful application here. The ubiquity of such attractors, in conjunction with their several unusual properties, suggests novel applications.
We show that it is possible to devise a large class of skew-product dynamical systems which have strange nonchaotic attractors (SNAs): the dynamics is asymptotically on fractal attractors and the largest Lyapunov exponent is non-positive. Furthermore, we show that quasiperiodic forcing, which has been a hallmark of essentially all hitherto known examples of such dynamics is not necessary for the creation of SNAs.
Localized states of Harper's equation correspond to strange nonchaotic attractors in the related Harper mapping. In parameter space, these fractal attractors with nonpositive Lyapunov exponents occur in fractally organized tongue-like regions which emanate from the Cantor set of eigenvalues on the critical line ε=1. A topological invariant characterizes wave functions corresponding to energies in the gaps in the spectrum. This permits a unique integer labeling of the gaps and also determines their scaling properties as a function of potential strength.
We apply a recently introduced method for global optimization to determine the ground state energy and configuration for model metallic clusters. The global minimum for a given N–atom cluster is found by following the damped dynamics of the N particle system on an evolving potential energy surface. In this application, the time dependent interatomic potential interpolates adiabatically between the Lennard–Jones (LJ) and the Sutton–Chen (SC) forms. Starting with an ensemble of initial conditions corresponding to the ground state configuration of the Lennard–Jones cluster, the system asymptotically reaches the ground state of the Sutton–Chen cluster. We describe the method and present results for specific cluster size N = 15, when the ground state symmetry of LJN and SCN differ.
By calculating the conditional entropy of two different chaotic time series, converted into symbolic sequences through the application of prescribed (but otherwise arbitrary) rules, it can be determined whether or not these originate from the same underlying dynamics. We show that by comparing the conditional entropy of a sequence, obtained by coarse-graining of a chaotic time series, with respect to shifted copies of itself, time-delays that may be inherent in the dynamics can be found. Application is made to time-series obtained from dynamical systems such as Mackey–Glass equation and Ikeda equation. The method appears equally effective in determining the dynamical coupling of climatic time signals. Our results are robust to additive noise, and can thus be applied even when the conversion from a time series to a symbolic sequence has a small proportion of errors.
We compare the annotation of three complete genomes using theab initio methods of gene identification GeneScan and GLIMMER. The annotation given in GenBank, the standard against which these are compared, has been made using GeneMark. We find a number of novel genes which are predicted by both methods used here, as well as a number of genes that are predicted by GeneMark, but are not identified by either of the nonconsensus methods that we have used. The three organisms studied here are all prokaryotic species with fairly compact genomes. The Fourier measure forms the basis for an efficient non-consensus method for gene prediction, and the algorithm GeneScan exploits this measure. We have bench-marked this program as well as GLIMMER using 3 complete prokaryotic genomes. An effort has also been made to study the limitations of these techniques for complete genome analysis. GeneScan and GLIMMER are of comparable accuracy insofar as gene-identification is concerned, with sensitivities and specificities typically greater than 0.9. The number of false predictions (both positive and negative) is higher for GeneScan as compared to GLIMMER, but in a significant number of cases, similar results are provided by the two techniques. This suggests that there could be some as-yet unidentified additional genes in these three genomes, and also that some of the putative identifications made hitherto might require re-evaluation. All these cases are discussed in detail.
Phase order, namely the average direction of sequential iterations, is studied in the family of unimodal maps x→1−μ|x|z on the interval [−1,1]. The average phase order or “magnetization” M is sensitive to local changes in the dynamics. At merging crises, this quantity increases from zero with the scaling behaviour M∼(μ−μc)1/z, while at exterior crises, M decreases, also having the same scaling exponent. We find that the exponent z is governed by the singularities of the invariant density ρ(x) at the edges of the interval: as x→±1, ρ(x)∼(1−|x|z)−β with β=1−1/z.
Genomic DNA is fragmented into segments using the Jensen-Shannon divergence. Use of this criterion results in the fragments being entropically homogeneous to within a predefined level of statistical significance. Application of this procedure is made to complete genomes of organisms from archaebacteria, eubacteria, and eukaryotes. The distribution of fragment lengths in bacterial and primitive eukaryotic DNAs shows two distinct regimes of power-law scaling. The characteristic length separating these two regimes appears to be an intrinsic property of the sequence rather than a finite-size artifact, and is independent of the significance level used in segmenting a given genome. Fragment length distributions obtained in the segmentation of the genomes of more highly evolved eukaryotes do not have such distinct regimes of power-law behavior.
By using the Jensen-Shannon divergence, genomic DNA can be divided into compositionally distinct domains through a standard recursive segmentation procedure. Each domain, while significantly different from its neighbors, may, however, share compositional similarity with one or more distant (non-neighboring) domains. We thus obtain a coarse-grained description of the given DNA string in terms of a smaller set of distinct domain labels. This yields a minimal domain description of a given DNA sequence, significantly reducing its organizational complexity. This procedure gives a new means of evaluating genomic complexity as one examines organisms ranging from bacteria to human. The mosaic organization of DNA sequences could have originated from the insertion of fragments of one genome (the parasite) inside another (the host), and we present numerical experiments that are suggestive of this scenario.
Locating the global minimum of a complex potential energy surface is facilitated by considering a homotopy, namely, a family of surfaces that interpolate continuously from an arbitrary initial potential to the system under consideration. Different strategies can be used to follow the evolving minima. It is possible to enhance the probability of locating the global minimum through a heuristic choice of interpolation schemes and parameters, and the continuously evolving potential landscape reduces the probability of trapping in local minima. In application to a model problem, finding the ground-state configuration and the energy of rare-gas (Lennard-Jones) atomic clusters, we demonstrate the utility and the efficacy of this method.
Integrable dynamical systems, namely those having as many independent conserved quantities as freedoms, have all Lyapunov exponents equal to zero. Locally, the instantaneous or finite time Lyapunov exponents are nonzero, but owing to a symmetry, their global averages vanish. When the system becomes nonintegrable, this symmetry is broken. A parallel to this phenomenon occurs in mappings which derive from quasiperiodic Schrödinger problems in 1-dimension. For values of the energy such that the eigenstate is extended, the Lyapunov exponent is zero, while if the eigenstate is localized, the Lyapunov exponent becomes negative. This occurs by a breaking of the quasiperiodic symmetry of local Lyapunov exponents, and corresponds to a breaking of a symmetry of the wavefunction in extended and critical states.
Prof. Ramaswamy is currently teaching the second semester MSc course on Classical mechanics.
The lab also holds Saturday talks where either the current research interests of the lab are discussed or an outside speaker is invited to discuss his area of research.
Classical mechanics from the time of Newton till the birth of quantum mechanics is discussed.
Computational Systems Biology was discussed.
Techniques of Mathematical Physics was discussed.
Theoretical and Computational models in comparative systems biology are discussed.
Concepts of networks in application to biology are discusses. Machine learning for modeling of biological systems is discussed.
A peek into the lab 233 in the School of Physical Sciences of JNU would introduce you to us and our research.