M. Lakshmanan

Chaos in nonlinear oscillators : controlling and synchronization

Duffing oscillator - bifurcation and chaos, analytic approaches chaotic dynamics of Bonhoeffer-Van der Pol (BVP) and Duffing-Van der POL (DVP) oscillators chaotic oscillators with Chuas diode controlling of chaos synchronization and secure communications.

Geometrical and gauge equivalence of the generalized hirota, Heisenberg and wkis equations with linear inhomogeneities

Integrable evolution equations can take several equivalent forms in a geometrical sense. Here we consider the equivalence of generalized versions involving linear inhomogeneities of three important nonlinear evolution equations, namely the Hirota, Heisenberg ferromagnetic spin and Wadati-Konno-Ichikawa-Shimizu (WKIS) equation through a moving helical space curve formalism and stereographic representation. From the geometrical consideration, we also construct suitable (2 × 2)-matrix linear eigenvalue equations, involving however non-isospectral flow: the eigenvalues evolve in time. However, these systems are also gauge equivalent. We briefly analyse the scattering problem and show that infinite number of constants of motion can exist for these systems.

Perturbation of solitons in the classical continuum isotropic Heisenberg spin system

The dynamics of a one-dimensional classical continuum isotropic Heisenberg ferromagnetic spin system in the presence of a weak relativistic interaction, which causes damping of the spin motion, is considered. The corresponding evolution equation is identified with a damped nonlinear Schrodinger equation in terms of the energy and current densities of the unperturbed system. A direct perturbation method, along the lines of Kodama and Ablowitz, is developed for the envelope soliton solution of the nonlinear Schrodinger equation and the explicit perturbed solution obtained. This solution is found to be valid in a finite domain of the propagation space. To cover the entire region, a uniform solution is constructed using the matched asymptotic expansion technique. Finally, the spin vectors are constructed using the known procedures in differential geometry and the consequences of damping analysed briefly.

Chaotic planar states of the discrete dynamical anisotropic Heisenberg spin chain

Planar states of the semi-infinite classical, discrete dynamical anisotropic Heisenberg spin chain are shown to be completely chaotic in the sense that for any degree of anisotropy the states are totally aperiodic for almost all values of the first spin S1 in the chain, while on a dense set of S1 one obtains, with minor exceptions, periodic states of all orders exceeding a certain number which depends on the degree of anisotropy. Such states are not present in the isotropic chain.

Bifurcation and chaotic motion of a soliton in a long Josephson junction oscillator

The fluxon dynamics of a long Josephson junction (LJJ) is investigated numerically. The effect of the surface resistance of the superconductor on the bifurcation phenomenon is examined in detail. We show the occurrence of a period doubling route to chaos and the reappearance of low order periodic motions in the LJJ by varying the magnitude of the external force. The chaotic attractors of the LJJ are characterized using maximal Lyapunov exponents and a fractal (correlation) dimension. We show the existence of noise induced oscillations in a force free LJJ system. Finally the influence of noise on the periodic and chaotic dynamics is also presented.

Spatio-Temporal Patterns

Patterns abound in nature. If we look around us, we can see a myriad of interesting patterns ranging from uniform to very complex types. They occur in varied phenomena encompassing physics, chemistry, biology, social dynamics, economics and so on. Essentially they are distinct structures on a space-time scale which arise as a collective and cooperative phenomenon due to the underlying large number of constituent subsystems. The latter could be aggregates of particles, atoms, molecules, circuits, cells, bacteria, defects, dislocations and so on. When these aggregates can move and/or interact, they give rise to various patterns. A small select set of patterns occurring in nature and in physical and chemical systems is shown in (Fig. 15.1) and (Fig. 15.2), respectively. Such patterns tell us much about the dynamics, both at the macroscopic as well as at the microscopic levels of the underlying systems. Naturally, when the interactions among the constituents are nonlinear, one might expect novel and unexpected patterns.