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A topological analysis of the attractor associated with the Moore–Spiegel nonlinear system is performed, following the basic idea laid down by Gilmore and Lefranc (2002, The Topology of Chaos, Wiley, Hoboken, NJ). Starting with the usual fixed point analysis and their stability, we proceed to study in detail the process of chaotic orbit extraction with the help of close return map. This is then used to construct the symbolic dynamics associated with it, which is helpful in understanding the sequential change taking place inside the attractor. In the next part, we show how to characterize the evolution of the attractor from its birth to the crisis by finding out the homoclinic orbit and the corresponding unstable manifold. In the concluding part of the paper, we show how all the pertinent information of the attractor can be encoded in the template, leading to the explicit realization of linking numbers and the relative rotation rates. In the concluding section, we have touched upon a new approach to chaotic dynamics, using the flow curvature manifold to display the relative positioning of the attractor in relation to the fixed points and the null lines.

Generalized synchronization between two different nonlinear systems under influence of noise is studied with the help of an electronic circuit and numerical experiment. In the present case, we have studied the phenomena of generalized synchronization between the Lorenz system and another nonlinear system (modified Lorenz) proposed in Ray et al. (2011, “On the Study of Chaotic Systems With Non-Horseshoe Template,” Frontier in the Study of Chaotic Dynamical Systems With Open Problems, Vol. 16, E. Zeraoulia and J. C. Sprott, eds., World Scientific, Singapore, pp. 85–103) from the perspective of electronic circuits and corresponding data collected digitally. Variations of the synchronization threshold with coupling (between driver and driven system) and noise intensity have been studied in detail. Later, experimental results are also proved numerically. It is shown that in certain cases, noise enhances generalized synchronization, and in another it destroys generalized synchronization. Numerical studies in the latter part have also proved results obtained experimentally.

State space linearization using the concept of Brunovsky form and Lie derivative is applied to the case of a Hyperchaotic Lorentz System. It is observed that the necessary and sufficient conditions can be satisfied, the analytic form of the controller ‘u’ and the final form of the linearized equations can be obtained. Numerical simulation is used to ascertain the feasibility of the procedure in practice. It may be added that the case of an ordinary Lorentz equation is distinctively different as the controller is to be added in a different manner. The most important aspect of the present analysis is that the controller can be determined and not chosen ad hoc.

Forced Lorenz system, important in modeling of monsoonlike phenomena, is analyzed for the existence of heteroclinic orbit. This is done in the light of the suggested new mechanism for the onset of chaos by Magnitskii and Sidorov (2006, “Finding Homoclinic and Heteroclinic Contours of Singular Points of Nonlinear Systems of Ordinary Differential Equations,” Diff. Eq., 39, pp. 1593–1602), where heteroclinic orbits plays important and dominant roles. The analysis is performed based on the theory laid down by Shilnikov. An analytic expression in the form of uniformly convergent series is obtained. The same orbit is also obtained numerically by a technique enunciated by Magnitskii and Sidorov, reproducing the necessary important features.