This thesis involves the study of four problems in the area of nonlinear vibrations, using the asymptotic method of multiple scales(MMS). Accordingly, it consists of four sequentially arranged parts. In the first part of this thesis we study some nonlinear dynamics related to the amplitude control of a lightly damped, resonantly forced, harmonic oscillator. The slow flow equations governing the evolution of amplitude and phase of the controlled system are derived using the MMS. Upon choice of a suitable control law, the dynamics is represented by three coupled ,nonlinear ordinary differential equations involving a scalar free parameter. Preliminary study of this system using the bifurcation analysis package MATCONT reveals the presence of Hopf bifurcations, pitchfork bifurcations, and limit cycles which seem to approach a homoclinic orbit. However, close approach to homoclinic orbit is not attained using MATCONT due to an inherent limitation of time domain-based continuation algorithms. To continue the limit cycles closer to the homoclinic point, a new algorithm is proposed. The proposed algorithm works in phase space with an ordered set of points on the limit cycle, along with spline interpolation. The algorithm incorporates variable stretching of arclength based on local curvature, through the use of an auxiliary index-based variable. Several numerical examples are presented showing favorable comparisons with MATCONT near saddle homoclinic points. The algorithm is also formulated with infinitesimal parameter increments resulting in ordinary differential equations, which gives some advantages like the ability to handle fold points of periodic solution branches upon suitable re-parametrization. Extensions to higher dimensions are outlined as well. With the new algorithm, we revisit the amplitude control system and continue the limit cycles much closer to the homoclinic point. We also provide some independent semi-analytical estimates of the homoclinic point, and mention an a typical property of the homoclinic orbit. In the second part of this thesis we analytically study the classical van der Pol oscillator, but with an added fractional damping term. We use the MMS near the Hopf bifurcation point. Systems with (1)fractional terms, such as the one studied here, have hitherto been largely treated numerically after suitable approximations of the fractional order operator in the frequency domain. Analytical progress has been restricted to systems with small fractional terms. Here, the fractional term is approximated by a recently pro-posed Galerkin-based discretization scheme resulting in a set of ODEs. These ODEs are then treated by the MMS, at parameter values close to the Hopf bifurcation. The resulting slow flow provides good approximations to the full numerical solutions. The system is also studied under weak resonant forcing. Quasiperiodicity, weak phase locking, and entrainment are observed. An interesting observation in this work is that although the Galerkin approximation nominally leaves several long time scales in the dynamics, useful MMS approximations of the fractional damping term are nevertheless obtained for relatively large deviations from the nominal bifurcation point. In the third part of this thesis, we study a well known tool vibration model in the large delay regime using the MMS. Systems with small delayed terms have been studied extensively as perturbations of harmonic oscillators. Systems with (1) delayed terms, but near Hopf points, have also been studied by the method of multiple scales. However, studies on systems with large delays are few in number. By “large” we mean here that the delay is much larger than the time scale of typical cutting tool oscillations. The MMS up to second order, recently developed for such large-delay systems, is applied. The second order analysis is shown to be more accurate than first order. Numerical integration of the MMS slow flow is much faster than for the original equation, yet shows excellent accuracy. A key point is that although certain parameters are treated as small(or, reciprocally, large), the analysis is not restricted to infinitesimal distances from the Hopf bifurcation. In the present analysis, infinite dimensional dynamics is retained in the slow flow, while the more usual center manifold reduction gives a planar phase space. Lower-dimensional dynamical features, such as Hopf bifurcations and families of periodic solutions, are also captured by the MMS. The strong sensitivity of the slow modulation dynamics to small changes in parameter values, peculiar to such systems with large delays, is seen clearly. In the last part of this thesis, we study the weakly nonlinear whirl of an asymmetric, overhung rotor near its gravity critical speed using a well known two-degree of freedom model. Gravity critical speeds of rotors have hitherto been studied using linear analysis, and ascribed to rotor stiffness asymmetry. Here we present a weakly nonlinear study of this phenomenon. Nonlinearities arise from finite displacements, and the rotor’s static lateral deflection under gravity is taken as small. Assuming small asymmetry and damping, slow flow equations for modulations of whirl amplitudes are developed using the MMS. Inertia asymmetry appears only at second order. More interestingly, even without stiffness asymmetry, the gravity-induced resonance survives through geometric nonlinearities. The gravity resonant forcing does not influence the resonant mode at leading order, unlike typical resonant oscillations. Nevertheless, the usual phenomena of resonances, namely saddle-node bifurcations, jump phenomena and hysteresis, are all observed. An unanticipated periodic solution branch is found. In the three dimensional space of two modal coefficients and a detuning parameter, the full set of periodic solutions is found to be an imperfect version of three mutually intersecting curves: a straight line, a parabola, and an ellipse. To summarize, the first and fourth problems, while involving routine MMS involve new applications with rich dynamics. The second problem demonstrated a semi-analytical approach via the MMS to study a fractional order system. Finally, the third problem studied a known application in a hitherto less-explored parameter regime through an atypical MMS procedure. In this way, a variety of problems that showcase the utility of the MMS have been studied in this thesis.

This thesis reports an experimental characterization of the ultrafast gain and refractive index dynamics of a novel InAs/InGaAsP/InP quantum-dot (QD) semiconductor optical amplifier (SOA) operating near 1.55-µm wavelengths, assessing its high-speed performance characteristics for the first time. The thesis also studies the influence of the degree of quantum confinement on the dynamics of SOAs by comparing the zero-dimensional (0-D) QD's dynamics to those in 1-D InAs/InAlGaAs/InP quantum-dash (QDash), and 2-D InGaAsP/InGaAsP/InP quantum-well (QW) SOAs, both of which also operate near 1.55-µm wavelengths, and are made with matching or similar materials and structures. The ultrafast (around 1 ps) and long-lived (up to 2 ns) amplitude and phase dynamics of the SOAs are characterized via advanced heterodyne pump-probe measurements with 150-femtosecond resolution. It is found that the QD SOA has an 80-picosecond amplitude, and 110-picosecond phase recovery lifetime in the gain regime, 4-6 times faster than the QDash and QW recovery lifetimes, as well as reduced ultrafast transients, giving it the best properties for high-speed (> 100 Gb/s) all-optical signal processing in the important telecommunications wavelength bands. An impulse response model is developed and used to analyze the dynamics, facilitating a comparison of the gain compression factors, time-resolved linewidth enhancement factors (alpha-factors), and instantaneous dynamic coefficients (two-photon absorption and nonlinear refractive-index coefficients) amongst the three structures. The quantum-dot device is found to have the lowest effective alpha-factor, 2-10, compared to 8-16 in the QW, as well as time-resolved alpha-factors lower than in the QW—promising for reduced-phase-transient operation at high bitrates. Significant differences in the alpha-factors of lasers with the same structure are found, due to the differences between gain changes that are induced optically or through the electrical bias. The relative contributions of stimulated transitions and free-carrier absorption to the total carrier heating dynamics in SOAs of varying dimensionality are also reported for the first time. Examining the QD electroluminescence and linear gain spectra in combination with the carrier dynamics also brings about conclusions on the nature of the quantum confinement, dot energy-level structure, and density of states—aspects of the material that have not been previously well understood.

This thesis reports an experimental characterization of the ultrafast gain and refractive index dynamics of a novel InAs/InGaAsP/InP quantum-dot (QD) semiconductor optical amplifier (SOA) operating near 1.55-µm wavelengths, assessing its high-speed performance characteristics for the first time. The thesis also studies the influence of the degree of quantum confinement on the dynamics of SOAs by comparing the zero-dimensional (0-D) QD's dynamics to those in 1-D InAs/InAlGaAs/InP quantum-dash (QDash), and 2-D InGaAsP/InGaAsP/InP quantum-well (QW) SOAs, both of which also operate near 1.55-µm wavelengths, and are made with matching or similar materials and structures. The ultrafast (around 1 ps) and long-lived (up to 2 ns) amplitude and phase dynamics of the SOAs are characterized via advanced heterodyne pump-probe measurements with 150-femtosecond resolution. It is found that the QD SOA has an 80-picosecond amplitude, and 110-picosecond phase recovery lifetime in the gain regime, 4-6 times faster than the QDash and QW recovery lifetimes, as well as reduced ultrafast transients, giving it the best properties for high-speed (> 100 Gb/s) all-optical signal processing in the important telecommunications wavelength bands. An impulse response model is developed and used to analyze the dynamics, facilitating a comparison of the gain compression factors, time-resolved linewidth enhancement factors (alpha-factors), and instantaneous dynamic coefficients (two-photon absorption and nonlinear refractive-index coefficients) amongst the three structures. The quantum-dot device is found to have the lowest effective alpha-factor, 2-10, compared to 8-16 in the QW, as well as time-resolved alpha-factors lower than in the QW—promising for reduced-phase-transient operation at high bitrates. Significant differences in the alpha-factors of lasers with the same structure are found, due to the differences between gain changes that are induced optically or through the electrical bias. The relative contributions of stimulated transitions and free-carrier absorption to the total carrier heating dynamics in SOAs of varying dimensionality are also reported for the first time. Examining the QD electroluminescence and linear gain spectra in combination with the carrier dynamics also brings about conclusions on the nature of the quantum confinement, dot energy-level structure, and density of states—aspects of the material that have not been previously well understood.

Soft condensed matter (SCM) systems are ubiquitous in nature. SCM systems contain mesoscopic structures in the size range 10 nm to 1 am that are held together by weak entropic forces. These materials are therefore easily perturbed by external fields such as shear, gravity and electric and magnetic fields and are novel systems for studying non-equilibrium phenomena. The elastic constants of these materials are ≈ 109 times smaller than conventional atomic fluids and hence it is possible to measure the viscoelastic response of these materials using commercial instruments such as rheometers. The relaxation time in SCM systems are of the order of milliseconds as compared to atomic systems where relaxation times are of the order of picoseconds. It is easy to study the effect of shear on SCM, as the shear rates attainable by commercial rheometers are of the order of the inverse of their relaxation times. The dynamics of SCM systems and their local rheological properties obtained using the method of probe diffusion can be quantified through dynamic light scattering experiments. The structure of SCM systems can be quantified using diffraction techniques such as small angle x-ray scattering. In this thesis we report experimental studies on the linear and nonlinear rheology and the dynamics of surfactant cetyltrimethylammonium tosylate (CTAT), which forms cylindrical wormlike micelles, studied using bulk rheology and dynamic light scattering (DLS) technique, respectively. We have also studied the phase behaviour of the ternary system formed by cetyltrimethylammonium 3-hydroxy-napthalene 2-carboxylate (CTAHN), sodium bromide (NaBr) and water using small angle x-ray scattering (SAXS). In Chapter 1, we discuss why SCM systems are suitable for studying non-equilibrium phenomena such as the effect of shear on the structure and dynamics of condensed matter. This is followed by a discussion on the chemical structure, phase behaviour and self assembling properties of the amphiphilic molecules in water. We then discuss the intermacromolecular forces such as van der Waals interaction, the screened Coulomb repulsion and hydrophobic and hydration forces. The systems that have been the subject of our experimental studies, viz. CTAT and CTAHN/NaBr/water have also been discussed in detail. This is followed by a theoretical background of linear and nonlinear rheology, dynamic light scattering and small angle x-ray scattering techniques. Next we describe the stress relaxation mechanisms in wormlike micelles. This is followed by a discussion on some standard techniques of nonlinear time series analysis, in particular the evaluation of the delay time L, the embedding dimension m, the correlation dimension ν and the Lyapunov exponent λ. We have also mentioned a few examples of experimental systems where chaos has been observed. We have also discussed in detail the various routes to chaos namely, the period-doubling route, the quasiperiodic route and the intermittency route. The concluding part of this chapter summarises the main results of the thesis. Chapter 2 discusses the experimental apparatus used in our studies. We have discussed the different components of the MCR-300 stress-controlled rheometer (Paar Physica, Germany). The rheo-small angle light scattering experiments and the direct visualisation experiments done using a home-made shear cell are also discussed. Next we describe the various experiments that can be done using a commercial rheometer. The frequency response and flow experiments have been discussed with some examples from our own work on entangled, cylindrical micelles. This is followed by a discussion on the various components of our dynamic light scattering (DLS) setup (Brookhaven Instruments, USA). Particle sizing of submicrometer colloidal spheres using our DLS setup has been discussed with an example of an angle-resolved DLS study of 0.05µm polystyrene colloids. Next we describe the various components of the SAXS setup (Hecus M. Braun, Austria). As an example application of SAXS we have quantified the structure of the lamellar phase formed by the surfactant CTAHN/water. We finally describe the sample preparation methods employed by us for the different experiments. Our nonlinear rheology experiments on viscoelastic gels of surfactant CTAT (cCT AT= 2wt%) in the presence of salt sodium chloride (NaCl) at various concentrations has been discussed in Chapter 3. We observe a plateau in the measured flow curve and this is attributed to a mechanical instability of the shear banding type. The slope of this plateau can be tuned by the addition of salt NaCl. This slope is due to a concentration difference between the shear bands arising from a Helfand-Fredrickson mechanism. This is confirmed by the presence of a “Butterfly” light scattering pattern in SALS experiments performed simultaneously with rheological measurements. We have carried out experiments at six different salt concentrations 10mM < cN aCl<1M, which yield plateau slopes (α) ranging from 0.07 < α < 0.4. We find that a minimum slope of 0.12, corresponding to a salt concentration of 25mM NaCl, is essential to see a “Butterfly” pattern indicating the onset of flow-concentration coupling at this α value. After this we turn our attention to stress/shear rate relaxation experiments. The remainder of this chapter is split in four parts. We show in Part-I that the routes to rheochaos in stress relaxation experiments is via Type-II intermittency. Interestingly in shear rate relaxation, the route is via Type-III intermittency. We also show that flow-concentration coupling is essential to see the route to rheochaos. This section also brings out the crucial role played by orientational ordering of the nematics during rheochaos using SALS measurements performed simultaneously with rheological measurements. In part-II, we study the spatio-temporal dynamics of the shear induced band en route to rheochaos. Our direct visualisation experiments show that the complex dynamics observed in stress/shear rate relaxation measurements during the route to rheochaos is a manifestation of the spatio-temporal dynamics of the high shear band. In part-III, we describe the results of our stress/shear rate relaxation measurements at a fixed shear rate/stress with temperature as the control parameter and thereby control the micellar length. We see the Type-II intermittency route to rheochaos in stress relaxation measurements and the Type-III intermittency route to rheochaos in shear rate relaxation measurements. We conclude this section by showing the results of linear rheology measurements carried out at different temperatures. We estimate the mean micellar length ¯L, reptation time τrepand the breaking time τbreak. We show that L¯ increases by ≈ 58%, as the sample goes through the route to rheochaos. In Part-I of this chapter we had only qualitatively discussed the correlations between the measured time series of stress and the VH scattered intensity during the Type-II intermittency route to rheochaos. In part-IV we have attempted to quantify the correlations between the two time series using the technique of linear and nonlinear Granger causality. We have also studied the phase space dynamics of the two time series using the technique of Cross Recurrence Plots. We show that there exists a causal feedback mechanism between the stress and the VH intensity with the latter having a stronger causal effect. We have also shown that the bivariate time series share similar phase space dynamics using the method of Cross Recurrence Plots. In chapter 4, we have studied the dynamics of wormlike micellar gels of surfactant CTAT using the DLS technique. We report an interesting result in the dynamics of these systems: concentration fluctuations in semidilute wormlike-micelle solutions of the cationic surfactant Cetyltrimethylammonium Tosylate (CTAT) at wavenumber q have a mean decay rate α qz, with z -̃1.8, for a wide range of surfactant concentrations just above the overlap value c∗. The process we are seeing is thus superdiffusive, like a L´evy flight, relaxing on a length scale L in a time of order less than L2 . The rheological behaviour of this system is highly non-Maxwellian and indicates that the micelle-recombination kinetics is diffusion-controlled (DC) (micelles recombine with their original partners). With added salt (100mM NaCl) the rheometric behaviour turns Maxwellian, indicating a crossover to a mean-field (MF) regime (micelles can recombine with any other micellar end). The concentration fluctuations, correspondingly, show normal diffusive behaviour. The stress relaxation time, moreover is about twenty times slower without salt than with 100mM NaCl. Towards the end of this chapter, we propose an explanation of these observations based on the idea that stress due to long-lived orientational order enhances concentration fluctuations in DC regime. In the previous chapter we had studied the dynamics of wormlike micellar gels of pure CTAT 2wt% and found superdiffusive relaxation of concentration fluctuations due to a nonlinear coupling of long-lived stress and orientational fluctuations to the con- centration. In chapter 5 we present results from dynamic light scattering experiments to quantify the diffusive motion of polystyrene (PS) colloids in the same system. This chapter is split in two parts. In Part-I, we discuss dynamics of PS particles of radius 115 nm and 60 nm in CTAT 2wt%. The radius of the colloidal spheres is comparable to the mesh size ξ = 80 nm of the wormlike micellar network and hence we are probing the network dynamics. We find that ∆r2(t) is wavevector independent at small and large lag times. However at intermediate times, we find an anomalous wavevector dependence which we believe arises from the rapid restructuring of the gel network. This anomalous wavevector dependence of ∆r2(t) disappears as the temperature is increased. In Part-II we discuss the dynamics of PS particles of radius 25 nm and 10 nm, smaller than ξ, in CTAT 1wt% & 2wt%. We once again find an anomalous wavevector dependence of ∆r2(t) at intermediate times for the 2wt% sample. Surprisingly, at large times the particle motion is not diffusive, rather ∆r2(t) saturates. We do not have a clear understanding of this as yet. Also for the 10 nm particle, the motion at small lag times is superdiffusive. The motion of these particles is probably influenced by the superdiffusion of concentration fluctuations observed in pure CTAT 2wt% system (chapter 4). In chapter 6, we report the observation of an intermediate mesh phase with rhom- bohedral symmetry, corresponding to the space group R¯3m, in the ternary system consisting of CTAHN/NaBr/water. It occurs at lower temperatures between a random mesh phase (LDα ) and a lamellar phase (Lα) on increasing the surfactant concentration φs. The micellar aggregates, both in the intermediate and random mesh phases, are found to be made up of a two-dimensional network of rod-like segments, with three rods meeting at each node. SAXS studies also show the presence of small angle peaks corresponding to ad−spacing of 25 nm. Freeze fracture electron microscopy results shows that this peak may correspond to the presence of nodule like structures with no long-range correlations. The thesis concludes with a summary of main results and a brief discussion of the scope for future work in Chapter 7.

Ausgedehnte dissipative Systeme können fernab vom thermodynamischen Gleichgewicht instabil gegenüber Oszillationen bzw. Wellen oder raumzeitlichem Chaos werden. Die komplexe Ginzburg-Landau Gleichung (CGLE) stellt ein universelles Modell zur Beschreibung dieser raumzeitlichen Strukturen dar. Diese Arbeit ist der theoretischen Analyse komplexer Muster gewidmet. Mittels numerischer Bifurkations- und Stabilitätsanalyse werden Instabilitäten einfacher Muster identifiziert und neuartige Lösungen der CGLE bestimmt. Modulierte Amplitudenwellen (MAW) und Super-Spiralwellen sind Beispiele solcher komplexer Muster. MAWs können in hydrodynamischen Experimenten und Super-Spiralwellen in der Belousov-Zhabotinsky-Reaktion beobachtet werden. Der Grenzübergang von Phasen- zu Defektchaos wird durch den Existenzbereich der MAWs erklärt. Mittels der selben numerischen Methoden wird Bursting vom Fold-Hopf-Typ in einem Modell der Kalziumsignalübertragung in Zellen identifiziert.

In this thesis we investigate the formation of patterns in a simple activator-inhibitor model supplemented with an inhibitory nonlocal coupling term. This model exhibits a wave instability for slow inhibitor diffusion, while, for fast inhibitor diffusion, a Turing instability is found. For moderate values of the inhibitor diffusion these two instabilities occur simultaneously at a codimension-2 wave-Turing instability. We perform a weakly nonlinear analysis of the model in the neighbourhood of this codimension-2 instability. The resulting amplitude equations consist in a set of coupled Ginzburg-Landau equations. These equations predict that the model exhibits bistability between travelling waves and Turing patterns. We present a study of interfaces separating wave and Turing patterns arising from the codimension-2 instability. We study theoretically and numerically the dynamics of such interfaces in the framework of the amplitude equations and compare these results with numerical simulations of the model near and far away from the codimension-2 instability. Near the instability, the dynamics of interfaces separating small amplitude Turing patterns and travelling waves is well described by the amplitude equations, while, far from the codimension-2 instability, we observe a locking of the interface velocities. This locking mechanism is imposed by the absence of defects near the interfaces and is responsible for the formation of drifting pattern domains, i.e. moving localised patches of travelling waves embedded in a Turing pattern background and vice versa.

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