Investigations on the Dynamics of Combination Maps and the Analysis of Bifurcation in a Discontinuous Logistic Map

This thesis is a study of discrete nonlinear systems represented by one dimensional mappings.As one dimensional interative maps represent Poincarre sections of higher dimensional flows,they offer a convenient means to understand the dynamical evolution of many physical systems.It highlighting the basic ideas of deterministic chaos.Qualitative and quantitative measures for the detection and characterization of chaos in nonlinear systems are discussed.Some simple mathematical models exhibiting chaos are presented.The bifurcation scenario and the possible routes to chaos are explained.It present the results of the numerical computational of the Lyapunov exponents (λ) of one dimensional maps.This thesis focuses on the results obtained by our investigations on combinations maps,scaling behaviour of the Lyapunov characteristic exponents of one dimensional maps and the nature of bifurcations in a discontinous logistic map.It gives a review of the major routes to chaos in dissipative systems,namely, Period-doubling ,Intermittency and Crises.This study gives a theoretical understanding of the route to chaos in discontinous systems.A detailed analysis of the dynamics of a discontinous logistic map is carried out, both analytically and numerically ,to understand the route it follows to chaos.The present analysis deals only with the case of the discontinuity parameter applied to the right half of the interval of mapping.A detailed analysis for the n –furcations of various periodicities can be made and a more general theory for the map with discontinuities applied at different positions can be on a similar footing

Studies on the Effect of Randomness on the Synchronization of Coupled Systems and on the Dynamics of Intermittently driven systems

Nonlinear dynamics has emerged into a prominent area of research in the past few Decades.Turbulence, Pattern formation,Multistability etc are some of the important areas of research in nonlinear dynamics apart from the study of chaos.Chaos refers to the complex evolution of a deterministic system, which is highly sensitive to initial conditions. The study of chaos theory started in the modern sense with the investigations of Edward Lorentz in mid 60's. Later developments in this subject provided systematic development of chaos theory as a science of deterministic but complex and unpredictable dynamical systems. This thesis deals with the effect of random fluctuations with its associated characteristic timescales on chaos and synchronization. Here we introduce the concept of noise, and two familiar types of noise are discussed. The classifications and representation of white and colored noise are introduced. Based on this we introduce the concept of randomness that we deal with as a variant of the familiar concept of noise. The dynamical systems introduced are the Rossler system, directly modulated semiconductor lasers and the Harmonic oscillator. The directly modulated semiconductor laser being not a much familiar dynamical system, we have included a detailed introduction to its relevance in Chaotic encryption based cryptography in communication. We show that the effect of a fluctuating parameter mismatch on synchronization is to destroy the synchronization. Further we show that the relation between synchronization error and timescales can be found empirically but there are also cases where this is not possible. Studies show that under the variation of the parameters, the system becomes chaotic, which appears to be the period doubling route to chaos.

Nonlinear Dynamics of Nd:YAG lasers: Hopf bifurcation, Multistability and Chaotic Synchronization

In this thesis we have presented some aspects of the nonlinear dynamics of Nd:YAG lasers including synchronization, Hopf bifurcation, chaos control and delay induced multistability.We have chosen diode pumped Nd:YAG laser with intracavity KTP crystal operating with two mode and three mode output as our model system.Different types of orientation for the laser cavity modes were considered to carry out the studies. For laser operating with two mode output we have chosen the modes as having parallel polarization and perpendicular polarization. For laser having three mode output, we have chosen them as two modes polarized parallel to each other while the third mode polarized orthogonal to them.

Studies on function projective synchronization in chaotic and hyperchaotic systems

FPS is a more general form of synchronization. Hyperchaotic systems possessing more than one positive Lypaunov exponent exhibit highly complex behaviour and are more suitable for some applications like secure communications. In this thesis we report studies of FPS and MFPS of a few chaotic and hyperchaotic systems. When all the parameters of the system are known we show that active nonlinear control method can be efectively used to obtain FPS. Adaptive nonlinear control and OPCL control method are employed for obtaining FPS and MFPS when some or all parameters of the system are uncertain. A secure communication scheme based on MFPS is also proposed in theory. All our theoretical calculations are verified by numerical simulations.

Multi-user bidirectional communication using isochronal synchronisation of array of chaotic directly modulated semiconductor lasers

Isochronal synchronisation between the elements of an array of three mutually coupled directly modulated semiconductor lasers is utilized for the purpose of simultaneous bidirectional secure communication. Chaotic synchronisation is achieved by adding the coupling signal to the self feedback signal provided to each element of the array. A symmetric coupling is effective in inducing synchronisation between the elements of the array. This coupling scheme provides a direct link between every pair of elements thus making the method suitable for simultaneous bidirectional communication between them. Both analog and digital messages are successfully encrypted and decrypted simultaneously by each element of the array.

Effect of a fluctuating parameter mismatch and the associated time-scales on coupled Rossler oscillators

We study the effect of parameter fluctuations and the resultant multiplicative noise on the synchronization of coupled chaotic systems. We introduce a new quantity, the fluctuation rate Ф as the number of perturbations occurring to the parameter in unit time. It is shown that ϕ is the most significant quantity that determines the quality of synchronization. It is found that parameter fluctuations with high fluctuation rates do not destroy synchronization, irrespective of the statistical features of the fluctuations. We also present a quasi-analytic explanation to the relation between ϕ and the error in synchrony.

Hopf bifurcation in parallel polarized Nd:YAG laser

Dynamics of Nd:YAG laser with intracavity KTP crystal operating in two parallel polarized modes is investigated analytically and numerically. System equilibrium points were found out and the stability of each of them was checked using Routh–Hurwitz criteria and also by calculating the eigen values of the Jacobian. It is found that the system possesses three equilibrium points for (Ij, Gj), where j = 1, 2. One of these equilibrium points undergoes Hopf bifurcation in output dynamics as the control parameter is increased. The other two remain unstable throughout the entire region of the parameter space. Our numerical analysis of the Hopf bifurcation phenomena is found to be in good agreement with the analytical results. Nature of energy transfer between the two modes is also studied numerically.

Dynamics of two coupled chaotic multimode Nd:YAG lasers with intracavity frequency doubling crystal

The effect of coupling two chaotic Nd:YAG lasers with intracavity KTP crystal for frequency doubling is numerically studied for the case of the laser operating in three longitudinal modes. It is seen that the system goes from chaotic to periodic and then to steady state as the coupling constant is increased. The intensity time series and phase diagrams are drawn and the Lyapunov characteristic exponent is calculated to characterize the chaotic and periodic regions.

Existence of multiple attractors and the nature of bifurcations in a discontinuous logistic map

We present the analytical investigations on a logistic map with a discontinuity at the centre. An explanation for the bifurcation phenomenon in discontinuous systems is presented. We establish that whenever the elements of an n-cycle (n > 1) approach the discontinuities of the nth iterate of the map, a bifurcation other than the usual period-doubling one takes place. The periods of the cycles decrease in an arithmetic progression, as the control parameter is varied. The system also shows the presence of multiple attractors. Our results are verified by numerical experiments as well.

Chaotic Dynamics of Semiconductor Laser with Current Modulation and optoelectronic feedback

Chaotic dynamics of directly modulated semiconductor lasers have been studied extensively over the last two decades because of their application in secure optical communication. However, chaos is generally suppressed in such systems when the nonlinear gain reduction factor is above 0.01 which is very much smaller than the reported values in semiconductors like InGaAsP. In this paper we show that by giving an optoelectronic feedback with appropriate delay one can increase the range of the values of the gain reduction factor for which chaos can be observed. Numerical studies show that negative feedback is more efficient in producing chaotic dynamics.

Scaling relations in the Lyapunov exponents of one dimensional maps

We establish numerically the validity of Huberman-Rudnick scaling relation for Lyapunov exponents during the period doubling route to chaos in one dimensional maps. We extend our studies to the context of a combination map. where the scaling index is found to be different.

Universal behaviour in a ``modulated'' logistic map

We study the period-doubling bifurcations to chaos in a logistic map with a nonlinearly modulated parameter and show that the bifurcation structure is modified significantly. Using the renormalisation method due to Derrida et al. we establish the universal behaviour of the system at the onset of chaos.

A study of hydrodynamic turbulence using laser transmission

Using laser transmission, the characteristics of hydrodynamic turbulence is studied following one of the recently developed technique in nonlinear dynamics. The existence of deterministic chaos in turbulence is proved by evaluating two invariants viz. dimension of attractor and Kolmogorov entropy. The behaviour of these invariants indicates that above a certain strength of turbulence the system tends to more ordered states.