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Title
CERTIFICATE 1
CERTIFICATE-2
ACKNOWLEDGEMENT
Preface
List of Figures
CONTENTS
1. Introduction
1.1 Chaos and Nonlinear Dynamics
1.2 Dynamical Systems
1.2.1 Autonomous Dynamical Systems
1.2.2 Non autonomous Dynamical Systems
1.2.3 Discrete-time Dynamical Systems
1.2.4 Conservative and Dissipative Systems
1.3 The Poincare Maps and invariant Manifolds
1.4 Lyapunov Exponents
1.5 Attractors
1.5.1 Equilibrium points
1.1 Basins of attraction of a simple damped pendulum along with the equilibrium points
1.5.2 Periodic Solutions and Limit Cycles
1.5.3 Quasiperiodic attractor
1.5.4 Chaos
1.6 Dimension
1.6.1 Capacity Dimension
1.6.2 Information dimension
1.6.3 Correlation Dimension
1.7 Stability and bifurcatiorn
1.7.1 Bifurcations
1.7.2 Bifurcations at Non Hyperbolic Equilibrium Points. . . .
1.2 Bifurcations from equilibrium point: (a) SN bifurcation; (b) trans critical bifurcation; (c) and (d) are pitch fork bifurcations
1.7.3 Hopf Bifurcations and Bifurcations of Limit Cycles from a multiple Focus
1.3 Hopf bifurcation from an equilibrium point: (a) shows supercritical and (b) the subcritical cases. In (b), a stable equilibrium coexists with stable and unstable periodic orbits
1.7.4 Bifurcations at Non Hyperbolic Periodic orbits
1.4 Neimark bifurcation or secondary Hopf bifurcation
1.5 Period doubling bifurcations. (a) is the supercritical and (b) the subcritical form
2. Analytical Techniques
2.1 Introduction
2.2 Transition from periodicity to chaos
2.2.1 Period Doubling
2.1 Poincare section of a trajectory undergoing period doubling bifur cation. On the left is the original periodic trajectory, giving the intersection point q*. On the right this is changed to two points..
2.2 Geometrical meaning of P (q)
2.3 Schematic representation of the period doubling route, using eigen values. The different figures correspond to (a) a stable cycle, (b) an unstable cycle, (c) a nonstable cycle and (d) an unstable cycle with all als outside the unit circle
2.4 Bifurcation to double period
2.5 Bifurcation of the simple damped pendulum [En. (2.7) 1 obtained numerically, Here H values are plotted corresponding to with l () = 0, agaist A. Parameters are w = 1.0, q = 0.2
2.2.2 Quasiperiodicity
2.6 Schematic representation of the quasiperiodic route. Fixed point in the Poincare map is a limit cycle. As the control parameter is changed a second frequency appears. Quasiperiodic behaviour follows when the frequencies are incommensurate
2.7 Arnold tongues: frequency locking occurs in the shaded regions..
2.8 The Complete Devils Staircase. The winding number given by Eq. (2.14), plotted against Il for sine-circle map
2.2.3 Intermittency
2.9 Intermittency shown by the logistic map. Fig. (a) is a plot of a period - 5 behaviour for p = 0.935; in (b), for y = 0.9342 intermittent behaviour is seen
2.10 The orbits of the surface of section map is shown as a function of the parameter p. Stable orbits are represented by solid lines and dashed curves represent unstable orbits
2.2.4 Crises
Boundary Crisis
Interior Crisis and Attractor Merging Crisis
2.11 Heteroclinic tangency crisis illustrated in (a) The homoclinic version is shown in (b)
Noise Induced Crisis
2.3 Analysis of the transition techniques
2.3.1 Melnikovs Method
2.12 Phase portraits showing (a) homoclinic orbit r, which is also the separatrix and (b) heteroclinic orbitsT1 ?and T2 forming separatrix, along with other phase trajectories
2.13 The stable and unstable manifolds of an unperturbed system join ing smoothly
2.14 Appearance of Homoclinic tangle
2.15 Variation of R° (w) as a function of w for the Duffing oscillator in Eq. (2.47)
2.3.2 Harmonic Balance Method
2.3.3 Stability and Mathieu Equation
2.16 The stability regions of Mathieu equation Liz + (a - 2q cos 21) y = 0.
2.3.4 Lyapunov Exponent Computation
2.17 The evolution of the principal axes of an n-sphere is shown in (a) The application of the CSR is illustrated in (b)
3. The Froude Pendulum and its Dynamics
3.1 Introduction
3.2 The Froude Pendulum
3.1 Schematic diagram of the driven Frode pendulum
3.3 Attractors of the Pendulum
3.2 Basin of attraction of the undiven Frode pendulum. Parameters are gl=0.1, g2=0.7andw=0.2
3.3 Alimit cycle of the undriven FP is shown in (a), and (b) shows a chaotic trajectory of the driven system (3.6)
3.4 Bifurcation diagrams for the driven FP. θ is plotted as a function of, f in (a) and w in (b)
3.5 Plot of maximal LE for the driven FP. The parameters are q, = 0- 1, q2 = 0.7and w = 0.2
3.6 Blow up of figure 3.5, for f < 0.34 shown
3.7 A Poincare attractor for f = 0.1 is shown in (a); (b) is the winding number plot. Parameters are q1 = 0. 1, q2 = 0. 7, w = 0.2
3.8 The period doubling cascades shown for same parameters as in previous figures
3.9 The crisis route to chaos. Figure above shows a boundary crisis while interios crisis can be seen below. Parameters are as in Fig. 3.8.
3.4 Melnikov analysis of the Pendulum
3.10 The Melnikov threshold fth plotted as a function of w for the FP. Parameters are q1 = 0.3 and Q2 = 0.5
3.11 Phase space trajectoris of the FP (a) below and (b) above the Melnikov threshold in Fig. 3.10
3.5 Harmonic Balancing Analysis
3.12 Chaotic attractor of the RP for f=1.7; other parameters are as in the previous figure
3.13 The limit cycle of the system (3.6) obtained by numerical integra. tion, compared to the one got by harmonic balance method J
3.6 Discussion
4. Stability Analysis for the Froude pendulum
4.1 Introduction
4.2 Analysis using harmonic balance method
4.1 Response curves of the FP: (a) The response curves for f =0.08, 0.1 (b) . Curves for f = 0.094, 0.114, 0.134 (f), 0.154 7
4.2 The values of β, ƒ and ω for different q2 values are plotted in figure l
4.3 Stability analysis of the solutions
4.3.1 Case A
4.3.2 Case B
4.3.3 Case C
4.3 The regions of stability identified for all the three cases above. Different shades represent different no. of coexisting solutions 1
4.4 Discussion
5. Suppression of chaos via secondary perturbations
5.1 General Formalism
5.2 Application to Froude pendulum
5.2.1 Parametric modulation of the drive term
5.1 (a) The bifurcation diagram, b) the plot of maximal LE and (c) chaotic trajectories; parameters are ql = 0.3, q2 = 0.5 and w = 0.7. l
5.2 ∆n for modulation of drive term plotted as a function of f. (b) is bifurcation diagram. (c) is LE plot and (d) trajectories of modulated pendulum
5.2.2 Parametric modulation of the damping term
5.2.3 Parametric modulation of the restoring term
5.3 Modulation amplitude ∆n for modulation of damping term plotted against f. Parameters and numbers as in previous figure l
5.2.4 Addition of secondary forcing term
5.4 Modulation amplitude ∆n for modulation of restoring term plotted against f. Parameters and numbers as in previous figure l
5.3 Discussion
5.5 ∆n on adding a secondary forcing, plotted against original amplitude f; details are as in figure. 5.2.a
5.4 Conclusions
BIBILIOGRAPHY
List of Research Papers