Complex dynamics in simple mechanical systems: Similarities to neuronal bursting.

We present an overview of our studies on some simple mechanical systems including the ‘simple’ nonlinear pendulum and its variants. We show that these systems exhibit numerous types of regular bursting oscillations which are seen in biological neurons. In particular, we discuss bow-tie shaped bursts which we found in a driven pendulum with linear velocity damping, under constant torque and dynamic feedback. Similar bursts of identical bow-tie shape have been reported by us previously in a system of two resistively coupled Josephson junctions in a certain parameter regime under certain conditions. We discuss the bifurcation mechanism producing some of these burst

Dynamics of bow-tie shaped bursting: Forced pendulum with dynamic feedback

A detailed study is performed on the parameter space of the mechanical system of a driven pendulum with damping and constant torque under feedback control. We report an interesting bow-tie shaped bursting oscillatory behaviour, which is exhibited for small driving frequencies, in a certain parameter regime, which has not been reported earlier in this forced system with dynamic feedback. We show that the bursting oscillations are caused because of a transition of the quiescent state to the spiking state by a saddle-focus bifurcation, and because of another saddle-focus bifurcation, which leads to cessation of spiking, bringing the system back to the quiescent state. The resting period between two successive bursts (Trest) is estimated analytically

Oscillatory dynamics of a charged microbubble under ultrasound

Nonlinear oscillations of a bubble carrying a constant charge and suspended in a fluid, undergoing periodic forcing due to incident ultrasound are studied. The system exhibits period-doubling route to chaos and the presence of charge has the effect of advancing these bifurcations. The minimum magnitude of the charge Qmin above which the bubble's radial oscillations can occur above a certain velocity c1 is found to be related by a simple power law to the driving frequency omega of the acoustic wave. We find the existence of a critical frequency omega_H above which uncharged bubbles necessarily have to oscillate at velocities below c1. We further find that this critical frequency crucially depends upon the amplitude Ps of the driving acoustic pressure wave. The temperature of the gas within the bubble is calculated. A critical value P_{tr} of Ps equalling the upper transient threshold pressure demarcates two distinct regions of omega dependence of the maximal radial bubble velocity v_{max} and maximal internal temperature T_{max}. Above this pressure, T_{max} and v_{max} decrease with increasing omega while below P_{tr}, they increase with omega. The dynamical effects of the charge and of the driving pressure and frequency of ultrasound on the bubble are discussed

Predicting climatic tipping points

Increased levels of greenhouse gases in the atmosphere, especially carbon dioxide, are leading contributors to a significant increase in the global temperature, and the consequent global climatic changes are more noticeable in recent years than in the past. A persistent increased growth of such gases might lead to an irreversible transition or tipping of the Earth’s climatic system to a new dynamical state. A change of regimes in CO2 buildup being correlated to one in global climate patterns, predicting this tipping point becomes crucially important. We propose here an innovative conceptual model, which does just this. Using the idea of rate-induced bifurcations, we show that a sufficiently rapid change in the system parameters beyond a critical value tips the system over to a new dynamical state. Our model when applied to real-world data detects tipping points, enables calculation of tipping rates and predicts their future values, and identifies thresholds beyond which tipping occurs. The model well captures the growth in time of the total global atmospheric fossil-fuel CO2 concentrations, identifying regime shift changes through measurable parameters and enabling prediction of future trends based on past data. Our model shows two distinct routes to tipping. We predict that with the present trend of variation of atmospheric greenhouse gas concentrations, the Earth’s climatic system would move over to a new stable dynamical regime in the year 2022. We determine a limit of 10.62 GtC at the start of 2022 for global CO2 emissions in order to avoid this tipping. The Earth’s climate has seen many changes over the years, affecting the physical environment (be it terrestrial, marine, or the atmosphere). These major changes or regime shifts from one stable dynamical state of the physical environment to another, each of which may persist for several years, produce major shifts in natural ecosystems involving trophic structures, changes in composition, and abundance of species. The climatic system moves over to a new regime once it crosses a climatic tipping point—a threshold crossed irreversibly by the system’s dynamics. Anthropogenic influences brought about in the physical environment invariably contribute in a substantial way to climate change globally as the dynamics of the climatic system is governed by the coupling between the land, the atmosphere, and the oceans. An increase in levels of greenhouse gases in the atmosphere mainly caused by human activities, especially carbon dioxide, has been one of the important contributing factors leading to climate change in the last few decades. We present here a theoretical model that well captures the rate of increase of the total global concentrations of carbon dioxide, the major contributing greenhouse gas in the atmosphere. We then employ the concept of rate-induced bifurcations to demonstrate that it is possible to determine the climatic tipping points from our model. This way, we predict that the climatic system would relocate to a new stable state early in the year 2022. It has been widely accepted that tipping point mechanisms can be used to study climate change. In this paper, we shall introduce and apply a rate-induced tipping model to global fossil-fuel emissions data. Our model shows two distinct routes in which tipping can occur, and the parameters describing these can be calculated from data and are physically measurable. Through the application of this model, we identify crucial tipping points, which lead to climate change and quantify exact boundaries crossed that induce tipping. Control can be exercised over the parameters describing tipping, if desired, such that tipping can be prevented. The methods developed can further be applied to any growth curve that may have undergone rate-induced tipping

The lengthening pendulum: Adiabatic invariance and bursting solutions

The adiabatic invariance of the action variable of a length varying pendulum is investigated in terms of the two different time scales that are associated with the problem. A length having a general polynomial variation in time is studied; an analytical solution for a pendulum with length which varies quadratically in time is obtained in the small angle approximation. We find that for length with quadratic time variation, the action neither converges (as it does for linear time variation), nor diverges (as it does for exponential time variation), but rather shows oscillatory behaviour with constant amplitude. It is then shown that for a pendulum length which has a combination of periodic and linear time variations, the action is no longer an adiabatic invariant and shows jumps with time. In the case in which the length varies sinusoidally in time, we demonstrate that the full nonlinear system exhibits bursting oscillations

The flight of the hornbill: drift and diffusion in arboreal avian movement

Capturing movement of animals in mathematical models has long been a keenly pursued direction of research1 . Any good model of animal movement is built upon information about the animal’s environment and the available resources including whether prey is in abundance or scarce, densely distributed or sparse2 . Such an approach could enable the identification of certain quantities or measures from the model that are species-specific characteristics. We propose here a mechanistic model to describe the movement of two species of Asian hornbills in a resource-abundant heterogenous landscape which includes degraded forests and human settlements. Hornbill telemetry data was used to this end. The birds show a bias both towards features of attraction such as nesting and roosting sites as well as possible bias away from points of repulsion such as human presence. These biases are accounted for with suitable potentials. The spatial patterns of movement are analyzed using the Fokker–Planck equation, which helps explain the variation in movement of different individuals. Search times to target locations were calculated using first passage time equations dual to the Fokker–Planck equations. We also find that the diffusion coefficients are larger for breeding birds than for non-breeding ones—a manifestation of repeated switching of directions to move back to the nest from foraging sites. The degree of directedness towards nests and roosts is captured by the drift coefficients. Non-breeding hornbills show similar values of the ratio of the two coefficients irrespective of the fact that their movement data is available from different seasons. Therefore, the ratio of drift to diffusion coefficients is indicative of an individual’s breeding status, as seen from available data. It could possibly also characterize different species. For all individuals, first passage times increase with proximity to human settlements, in agreement with the premise that anthropogenic activities close to nesting/roosting sites are not desirable

An improved effective potential for electroweak phase transitions

It is shown that improved potentials and corrected mass terms can be introduced by using a quadratic source term in the path integral construction for the effective action. The advantage of doing things this way is that we avoid ever having to deal with complex propagators in the loop expansion. The resulting effective action for electroweak phase transitions is similar to the usual results.Comment: 16 pages, NCL93-TP16, (REVTEX

Burst mechanisms and burst synchronization in a system of coupled type-I and type-II neurons

The rich dynamics of a system comprising of a Type-I neuron coupled to a Type-II neuron via an electrical synapse (gap junction) are explored in this paper. Diverse dynamical behaviour ranging from quiescence and periodic spiking, to bursting and burst synchronization, were observed for different coupling schemes. The bifurcation mechanisms underlying the various bursts observed were identified. We report a unique burst mechanism, based on a focus/node bifurcation, occurring for bidirectionally coupled neurons. We attempt to understand the physical basis for the transitions from one burst pattern to another and also between the different aforementioned forms of dynamical behaviour observed on varying the coupling strength, in both unidirectionally and bidirectionally coupled neurons. The various dynamical regimes of the coupled system are exhaustively studied and demarcated through parameter plots. Type-I and type-II neurons exhibit mutually phase synchronized burst patterns at large values of the coupling which tend towards complete synchronization on increasing the coupling strength. Such collective dynamical behaviour can have important implications in biological systems