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TITLE
CERTIFICATE
DECLARATION
ACKNOWLEDGEMENT
PREFACE
List of publications
CONTENTS
1. An Approach to the Study of Chaos
1.1 Introduction
1.1.1 Control of chaos
1.1.2 Historical developments
1.2 Dynamical systems
1.2.1 Discrete-time dynamical system
1.2.2 Continuous-time dynamical systems
1.2.3 Poincare map
1.2.4 Conservative and dissipative dynamical systems.
1.3 Attractors
1.3.1 Point attractor
1.3.2 Periodic attractor
1.3.3 Quasiperiodic attractor
1.3.4 Strange attractor
1.4 Sensitive dependence on Initial Conditions [SIC-ness]
1.5 Fractal nature
2. Chaos in One Dimensional Systems
2.1 Introduction
2.1.1 Logistic map
2.2 Period-doubling Phenomena
2.2.1 Metric universality: Feigenbaum constants (α & δ)
2.3 Feigenbaum attractor as a fractal
2.4 Tangent bifurcation
2.5 Intermittency
2.6 Crisis
3. Characterisation of Chaos in Maps
3.1 Introduction
3.2 Time series, Power Spectrum and Correlation Function.
3.3 Lyapunov Exponents (LE)
3.4 Multifractal nature of the Feigenbaum attractor
3.5 Multifractal and (f - α) Spectrum
3.6 Self-similarity of the Feigenbaum attractor
4. Bimodal Maps
4.1 Introduction
4.1.1 The dynamics of bubbling and bistability
4.2 Graphical analysis
4.3 Parameter-space plot
4.4 Case studies of- type I maps
4.5 Exponential maps of type I and type II
4.6 Maps of type III
4.7 Conclusion
5. Critical Exponents in the Transition to Chaos in One Dimensional Discrete Systems
5.1 Introduction
5.2 Unimodal & bimodal maps
5.3 Scaling behaviour in fractal dimensions
5.4 Critical exponents in the (f - a) spectrum
5.5 Scaling of Lyapunov Exponents
5.6 Conclusion
6. Stochastic Resonance and Chaotic Resonance in Bimodal Maps
6.1 Introduction
6.2 Double-well potential model
6.3 Results of numerical simulation for SR and CR
6.4 Effect of coupling
6.5 Conclusion
7. Conclusion
BIBILIOGRAPHY