Possible Effects of Anthropogenically-Increased CO2 on the Dynamics of Climate: Implications for Ice-Age Cycles

A dynamical model, developed to account for the observed major variations of global ice mass and atmospheric CO2 during the late Cenozoic, is used to provide a quantitative demonstration of the possibility that the anthropogenically-forced increase of atmospheric CO2, if maintained over a long period of time (perhaps by tectonic forcing), could displace the climatic system from an unstable regime of oscillating ice ages into a more stable regime representative of the pre-Pleistocene. This stable regime is characterized by orbitally-forced oscillations that are of much weaker amplitude than prevailed during the Pleistocene.

Hard transition to chaotic dynamics in Alfven wave fronts

The derivative nonlinear Schrodinger DNLS equation, describing propagation of circularly polarized Alfven waves of finite amplitude in a cold plasma, is truncated to explore the coherent, weakly nonlinear, cubic coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. In a reduced three-wave model equal dampings of daughter waves, three-dimensional flow for two wave amplitudes and one relative phase, no matter how small the growth rate of the unstable wave there exists a parametric domain with the flow exhibiting chaotic relaxation oscillations that are absent for zero growth rate. This hard transition in phase-space behavior occurs for left-hand LH polarized waves, paralleling the known fact that only LH time-harmonic solutions of the DNLS equation are modulationally unstable, with damping less than about unstable wave frequency 2/4 x ion cyclotron frequency. The structural stability of the transition was explored by going into a fully 3-wave model different dampings of daughter waves,four-dimensional flow; both models differ in significant phase-space features but keep common features essential for the transition.

Reduced three-wave model to study the hard transition to chaotic dynamics in Alfven wave-fronts

The derivative nonlinear Schrödinger (DNLS) equation, describing propagation of circularly polarized Alfven waves of finite amplitude in a cold plasma, is truncated to explore the coherent, weakly nonlinear, cubic coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. In a reduced three-wave model (equal damping of daughter waves, three-dimensional flow for two wave amplitudes and one relative phase), no matter how small the growth rate of the unstable wave there exists a parametric domain with the flow exhibiting chaotic dynamics that is absent for zero growth-rate. This hard transition in phase-space behavior occurs for left-hand (LH) polarized waves, paralelling the known fact that only LH time-harmonic solutions of the DNLS equation are modulationally unstable.

Fully 3-wave model to study the hard transition to chaotic dynamics in alfven wave-fronts

The derivative nonlinear Schrödinger (DNLS) equation, describing propagation of circularly polarized Alfven waves of finite amplitude in a cold plasma, is truncated to explore the coherent, weakly nonlinear coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. No matter how small the growth rate of the unstable wave, the four-dimensional flow for the three wave amplitudes and a relative phase, with both resistive damping and linear Landau damping, exhibits chaotic relaxation oscillations that are absent for zero growth-rate. This hard transition in phase-space behavior occurs for left-hand (LH) polarized waves, paralleling the known fact that only LH time-harmonic solutions of the DNLS equation are modulationally unstable. The parameter domain developing chaos is much broader than the corresponding domain in a reduced 3-wave model that assumes equal dampings of the daughter waves

Asymptotic and computational analysis of mistuning effects on the dynamics of bladed disks

Es bien conocido que las pequeñas imperfecciones existentes en los álabes de un rótor de turbomaquinaria (conocidas como “mistuning”) pueden causar un aumento considerable de la amplitud de vibración de la respuesta forzada y, por el contrario, tienen típicamente un efecto beneficioso en el flameo del rótor. Para entender estos efectos se pueden llevar a cabo estudios numéricos del problema aeroelástico completo. Sin embargo, el cálculo de “mistuning” usando modelos de alta resolución es una tarea difícil de realizar, ya que los modelos necesarios para describir de manera precisa el componente de turbomáquina (por ejemplo rotor) tienen, necesariamente, un número muy elevado de grados de libertad, y, además, es necesario hacer un estudio estadístico para poder explorar apropiadamente las distribuciones posibles de “mistuning”, que tienen una naturaleza aleatoria. Diferentes modelos de orden reducido han sido desarrollados en los últimos años para superar este inconveniente. Uno de estos modelos, llamado “Asymptotic Mistuning Model (AMM)”, se deriva de la formulación completa usando técnicas de perturbaciones que se basan en que el “mistuning” es pequeño. El AMM retiene sólo los modos relevantes para describir el efecto del mistuning, y permite identificar los mecanismos clave involucrados en la amplificación de la respuesta forzada y en la estabilización del flameo. En este trabajo, el AMM se usa para estudiar el efecto del “mistuning” de la estructura y de la amortiguación sobre la amplitud de la respuesta forzada. Los resultados obtenidos son validados usando modelos simplificados del rotor y también otros de alta definición. Además, en el marco del proyecto europeo FP7 "Flutter-Free Turbomachinery Blades (FUTURE)", el AMM se aplica para diseñar distribuciones de “mistuning” intencional: (i) una que anula y (ii) otra que reduce a la mitad la amplitud del flameo de un rotor inestable; y las distribuciones obtenidas se validan experimentalmente. Por último, la capacidad de AMM para predecir el comportamiento de flameo de rotores con “mistuning” se comprueba usando resultados de CFD detallados. Abstract It is well known that the small imperfections of the individual blades in a turbomachinery rotor (known as “mistuning”) can cause a substantial increase of the forced response vibration amplitude, and it also typically results in an improvement of the flutter vibration characteristics of the rotor. The understanding of these phenomena can be attempted just by performing numerical simulations of the complete aeroelastic problem. However, the computation of mistuning cases using high fidelity models is a formidable task, because a detailed model of the whole rotor has to be considered, and a statistical study has to be carried out in order to properly explore the effect of the random mistuning distributions. Many reduced order models have been developed in recent years to overcome this barrier. One of these models, called the Asymptotic Mistuning Model (AMM), is systematically derived from the complete bladed disk formulation using a consistent perturbative procedure that exploits the smallness of mistuning to simplify the problem. The AMM retains only the essential system modes that are involved in the mistuning effect, and it allows to identify the key mechanisms of the amplification of the forced response amplitude and the flutter stabilization. In this work, AMM methodolgy is used to study the effect of structural and damping mistuning on the forced response vibration amplitude. The obtained results are verified using a one degree of freedom model of a rotor, and also high fidelity models of the complete rotor. The AMM is also applied, in the frame of the European FP7 project “Flutter-Free Turbomachinery Blades (FUTURE)”, to design two intentional mistuning patterns: (i) one to complete stabilize an unstable rotor, and (ii) other to approximately reduce by half its flutter amplitude. The designed patterns are validated experimentally. Finally, the ability of AMM to predict the flutter behavior of mistuned rotors is checked against numerical, high fidelity CFD results.

Nonlinear dynamics of cilia and flagella

Cilia and flagella are hairlike extensions of eukaryotic cells which generate oscillatory beat patterns that can propel micro-organisms and create fluid flows near cellular surfaces. The evolutionary highly conserved core of cilia and flagella consists of a cylindrical arrangement of nine microtubule doublets, called the axoneme. The axoneme is an actively bending structure whose motility results from the action of dynein motor proteins cross-linking microtubule doublets and generating stresses that induce bending deformations. The periodic beat patterns are the result of a mechanical feedback that leads to self-organized bending waves along the axoneme. Using a theoretical framework to describe planar beating motion, we derive a nonlinear wave equation that describes the fundamental Fourier mode of the axonemal beat. We study the role of nonlinearities and investigate how the amplitude of oscillations increases in the vicinity of an oscillatory instability. We furthermore present numerical solutions of the nonlinear wave equation for different boundary conditions. We find that the nonlinear waves are well approximated by the linearly unstable modes for amplitudes of beat patterns similar to those observed experimentally.

Lattice approach to the dynamics of phase-coded soliton trains

We address the collective dynamics of a soliton train propagating in a medium described by the nonlinear Schrödinger equation. Our approach uses the reduction of train dynamics to the discrete complex Toda chain (CTC) model for the evolution of parameters for each train constituent: such a simplification allows one to carry out an approximate analysis of the dynamics of positions and phases of individual interacting pulses. Here, we employ the CTC model to the problem which has relevance to the field of fibre optics communications where each binary digit of transmitted information is encoded via the phase difference between the two adjacent solitons. Our goal is to elucidate different scenarios of the train distortions and the subsequent information garbling caused solely by the intersoliton interactions. First, we examine how the structure of a given phase pattern affects the initial stage of the train dynamics and explain the general mechanisms for the appearance of unstable collective soliton modes. Then we further discuss the nonlinear regime concentrating on the dependence of the Lax scattering matrix on the input phase distribution; this allows one to classify typical features of the train evolution and determine the distance where the soliton escapes from its slot. In both cases, we demonstrate deep mathematical analogies with the classical theory of crystal lattice dynamics.

Lattice approach to the dynamics of phase-coded soliton trains

We address the collective dynamics of a soliton train propagating in a medium described by the nonlinear Schrödinger equation. Our approach uses the reduction of train dynamics to the discrete complex Toda chain (CTC) model for the evolution of parameters for each train constituent: such a simplification allows one to carry out an approximate analysis of the dynamics of positions and phases of individual interacting pulses. Here, we employ the CTC model to the problem which has relevance to the field of fibre optics communications where each binary digit of transmitted information is encoded via the phase difference between the two adjacent solitons. Our goal is to elucidate different scenarios of the train distortions and the subsequent information garbling caused solely by the intersoliton interactions. First, we examine how the structure of a given phase pattern affects the initial stage of the train dynamics and explain the general mechanisms for the appearance of unstable collective soliton modes. Then we further discuss the nonlinear regime concentrating on the dependence of the Lax scattering matrix on the input phase distribution; this allows one to classify typical features of the train evolution and determine the distance where the soliton escapes from its slot. In both cases, we demonstrate deep mathematical analogies with the classical theory of crystal lattice dynamics.

Dropout dynamics in pulsed quantum dot lasers due to mode jumping

We examine the response of a pulse pumped quantum dot laser both experimentally and numerically. As the maximum of the pump pulse comes closer to the excited-state threshold, the output pulse shape becomes unstable and leads to dropouts. We conjecture that these instabilities result from an increase of the linewidth enhancement factor α as the pump parameter comes close to the excitated state threshold. In order to analyze the dynamical mechanism of the dropout, we consider two cases for which the laser exhibits either a jump to a different single mode or a jump to fast intensity oscillations. The origin of these two instabilities is clarified by a combined analytical and numerical bifurcation diagram of the steady state intensity modes.

Spatio-temporal intensity dynamics of passively mode-locked fiber laser

We present recent results on measurements of intensity spatio-temporal dynamics in passively mode-locked fibre laser. We experimentally uncover distinct, dynamic and stable spatio-temporal generation regimes of various stochasticity and periodicity properties in though-to-be unstable laser. We present a method to distinguish various types of generated coherent structures, including rogue and shock waves, within the radiation by means of introducing of intensity ACF evolution map. We also discuss how the spectral dynamics could be measured in fiber lasers generating irregular train of pulses of quasi-CW generation via combination of heterodyning and intensity spatio-temporal measurement concept.

Characterizing the transition dynamics for multi-pulsing in mode-locked lasers

We consider experimentally and theoretically a refined parameter space near the transition to multi-pulse modelocking. Near the transition, the onset of instability is initiated by a Hopf (periodic) bifurcation. As cavity energy is increased, the band of unstable, oscillatory modes generates a chaotic behavior between single- and multi-pulse operation. Both theory and experiment are in good qualitative agreement and they suggest that the phenomenon is of a universal nature in mode-locked lasers at the onset of multi-pulsing from N to N + 1 pulses per round trip. This is the first theoretical and experimental characterization of the transition behavior, made possible by a highly refined tuning of the gain pump level. © 2010 Copyright SPIE - The International Society for Optical Engineering.

Lattice approach to the dynamics of the DPSK-encoded soliton trains

We study the dynamical properties of the RZ-DPSK encoded sequences, focusing on the instabilities in the soliton train leading to the distortions of the information transmitted. The problem is reformulated within the framework of complex Toda chain model which allows one to carry out the simplified description of the optical soliton dynamics. We elucidate how the bit composition of the pattern affects the initial (linear) stage of the train dynamics and explain the general mechanisms of the appearance of unstable collective soliton modes. Then we discuss the nonlinear regime using asymptotic properties of the pulse stream at large propagation distances and analyze the dynamical behavior of the train classifying different scenarios for the pattern instabilities. Both approaches are based on the machinery of Hermitian and non-Hermitian lattice analysis. © 2010 IEEE.

Analysis of Molecular Dynamics Simulations of Protein Folding

Microsecond long Molecular Dynamics (MD) trajectories of biomolecular processes are now possible due to advances in computer technology. Soon, trajectories long enough to probe dynamics over many milliseconds will become available. Since these timescales match the physiological timescales over which many small proteins fold, all atom MD simulations of protein folding are now becoming popular. To distill features of such large folding trajectories, we must develop methods that can both compress trajectory data to enable visualization, and that can yield themselves to further analysis, such as the finding of collective coordinates and reduction of the dynamics. Conventionally, clustering has been the most popular MD trajectory analysis technique, followed by principal component analysis (PCA). Simple clustering used in MD trajectory analysis suffers from various serious drawbacks, namely, (i) it is not data driven, (ii) it is unstable to noise and change in cutoff parameters, and (iii) since it does not take into account interrelationships amongst data points, the separation of data into clusters can often be artificial. Usually, partitions generated by clustering techniques are validated visually, but such validation is not possible for MD trajectories of protein folding, as the underlying structural transitions are not well understood. Rigorous cluster validation techniques may be adapted, but it is more crucial to reduce the dimensions in which MD trajectories reside, while still preserving their salient features. PCA has often been used for dimension reduction and while it is computationally inexpensive, being a linear method, it does not achieve good data compression. In this thesis, I propose a different method, a nonmetric multidimensional scaling (nMDS) technique, which achieves superior data compression by virtue of being nonlinear, and also provides a clear insight into the structural processes underlying MD trajectories. I illustrate the capabilities of nMDS by analyzing three complete villin headpiece folding and six norleucine mutant (NLE) folding trajectories simulated by Freddolino and Schulten [1]. Using these trajectories, I make comparisons between nMDS, PCA and clustering to demonstrate the superiority of nMDS. The three villin headpiece trajectories showed great structural heterogeneity. Apart from a few trivial features like early formation of secondary structure, no commonalities between trajectories were found. There were no units of residues or atoms found moving in concert across the trajectories. A flipping transition, corresponding to the flipping of helix 1 relative to the plane formed by helices 2 and 3 was observed towards the end of the folding process in all trajectories, when nearly all native contacts had been formed. However, the transition occurred through a different series of steps in all trajectories, indicating that it may not be a common transition in villin folding. The trajectories showed competition between local structure formation/hydrophobic collapse and global structure formation in all trajectories. Our analysis on the NLE trajectories confirms the notion that a tight hydrophobic core inhibits correct 3-D rearrangement. Only one of the six NLE trajectories folded, and it showed no flipping transition. All the other trajectories get trapped in hydrophobically collapsed states. The NLE residues were found to be buried deeply into the core, compared to the corresponding lysines in the villin headpiece, thereby making the core tighter and harder to undo for 3-D rearrangement. Our results suggest that the NLE may not be a fast folder as experiments suggest. The tightness of the hydrophobic core may be a very important factor in the folding of larger proteins. It is likely that chaperones like GroEL act to undo the tight hydrophobic core of proteins, after most secondary structure elements have been formed, so that global rearrangement is easier. I conclude by presenting facts about chaperone-protein complexes and propose further directions for the study of protein folding.

Sensitisation waves in a bidomain fire-diffuse-fire model of intracellular Ca²⁺ dynamics

We present a bidomain threshold model of intracellular calcium (Ca²⁺) dynamics in which, as suggested by recent experiments, the cytosolic threshold for Ca²⁺ liberation is modulated by the Ca²⁺ concentration in the releasing compartment. We explicitly construct stationary fronts and determine their stability using an Evans function approach. Our results show that a biologically motivated choice of a dynamic threshold, as opposed to a constant threshold, can pin stationary fronts that would otherwise be unstable. This illustrates a novel mechanism to stabilise pinned interfaces in continuous excitable systems. Our framework also allows us to compute travelling pulse solutions in closed form and systematically probe the wave speed as a function of physiologically important parameters. We find that the existence of travelling wave solutions depends on the time scale of the threshold dynamics, and that facilitating release by lowering the cytosolic threshold increases the wave speed. The construction of the Evans function for a travelling pulse shows that of the co-existing fast and slow solutions the slow one is always unstable.