Bifurcations, Chaos, Controlling and Synchronization of Certain Nonlinear Oscillators

In this set of lectures, we review briefly some of the recent developments in the study of the chaotic dynamics of nonlinear oscillators, particularly of damped and driven type. By taking a representative set of examples such as the Duffing, Bonhoeffer-van der Pol and MLC circuit oscillators, we briefly explain the various bifurcations and chaos phenomena associated with these systems. We use numerical and analytical as well as analogue simulation methods to study these systems. Then we point out how controlling of chaotic motions can be effected by algorithmic procedures requiring minimal perturbations. Finally we briefly discuss how synchronization of identically evolving chaotic systems can be achieved and how they can be used in secure communications.Comment: 31 pages (24 figures) LaTeX. To appear Springer Lecture Notes in Physics Please Lakshmanan for figures (e-mail: [email protected]

On some aspects of the dynamic behavior of the softening Duffing oscillator under harmonic excitation

The Duffing oscillator is probably the most popular example of a nonlinear oscillator in dynamics. Considering the case of softening Duffing oscillator with weak damping and harmonic excitation and performing standard methods like harmonic balance or perturbation analysis, zero mean solutions with large amplitudes are found for small excitation frequencies. These solutions produce a ”nose-like” curve in the amplitude–frequency diagram and merge with the inclining resonance curve for decreasing (but non-vanishing) damping. These results are presented without any additional discussion in several textbooks. The present paper discusses the accurateness of these solutions by introducing an error estimation in the harmonic balance method showing large errors. Performing a modified perturbation analysis leads to solutions with non-vanishing mean value, showing very small errors in the harmonic balance error analysis

Bifurcation and chaos in the double well Duffing-van der Pol oscillator: Numerical and analytical studies

The behaviour of a driven double well Duffing-van der Pol (DVP) oscillator for a specific parametric choice ($\mid \alpha \mid =\beta$) is studied. The existence of different attractors in the system parameters ($f-\omega$) domain is examined and a detailed account of various steady states for fixed damping is presented. Transition from quasiperiodic to periodic motion through chaotic oscillations is reported. The intervening chaotic regime is further shown to possess islands of phase-locked states and periodic windows (including period doubling regions), boundary crisis, all the three classes of intermittencies, and transient chaos. We also observe the existence of local-global bifurcation of intermittent catastrophe type and global bifurcation of blue-sky catastrophe type during transition from quasiperiodic to periodic solutions. Using a perturbative periodic solution, an investigation of the various forms of instablities allows one to predict Neimark instablity in the $(f-\omega)$ plane and eventually results in the approximate predictive criteria for the chaotic region.Comment: 15 pages (13 figures), RevTeX, please e-mail Lakshmanan for figures, to appear in Phys. Rev. E. (E-mail: [email protected]

Bifurcation structure of two Coupled Periodically driven double-well Duffing Oscillators

The bifurcation structure of coupled periodically driven double-well Duffing oscillators is investigated as a function of the strength of the driving force $f$ and its frequency $\Omega$. We first examine the stability of the steady state in linear response, and classify the different types of bifurcation likely to occur in this model. We then explore the complex behaviour associated with these bifurcations numerically. Our results show many striking departures from the behaviour of coupled driven Duffing Oscillators with single well-potentials, as characterised by Kozlowski et al \cite{k1}. In addition to the well known routes to chaos already encountered in a one-dimensional Duffing oscillator, our model exhibits imbricated period-doubling of both types, symmetry-breaking, sudden chaos and a great abundance of Hopf bifurcations, many of which occur more than once for a given driving frequency. We explore the chaotic behaviour of our model using two indicators, namely Lyapunov exponents and the power spectrum. Poincar\'e cross-sections and phase portraits are also plotted to show the manifestation of coexisting periodic and chaotic attractors including the destruction of $T^2$ tori doubling.Comment: 16 pages, 8 figure