Avijit Lahiri

Bifurcations of families of invariant curves in four-dimensional reversible mappings

Abstract The bifurcation of families of closed invariant curves near a fixed point of a four-dimensional reversible mapping as two pairs of its multipliers collide and move off the unit circle, is studied with reference to an analogous bifurcation in reversible vector fields. A description of the principal features of the bifurcation is presented by way of a conjecture, and for a particular class of maps an analytic expression is derived for distinguishing normal from inverted bifurcations. Evidence in support of the conjecture is indicated in terms of an order-by-order perturbation scheme as well as of numerical computations.

Stability of invariant curves in four-dimensional reversible mappings near 1 : 1 resonance

Recent work on reversible Neimark-Sacker bifurcation in 4D maps is summarized. Linear stability analysis of families of invariant curves appearing in this bifurcation is presented by (a) referring to the analogous stability problem in reversible Hopt bifurcation in vector fields and (b) perturbatively calculating a set of quantities, termed quasi-multipliers, for the invariant curves. In particular, the critical rotation numbers corresponding to transition from elliptic to hyperbolic invariant curves on the subthreshold side in the so-called inverted bifurcation are calculated. Results of numerical iterations corroborating the above analysis are presented. The question of exploring the structure of the phase space close to the invariant curves is briefly addressed in the conclusion.

The Hamiltonian Hopf bifurcation: an elementary perturbative approach

Abstract The Hamiltonian Hopf bifurcation is briefly introduced. Its occurrence in a problem involving a Rydberg electron in a rotating electric field is pointed out by way of illustration. Starting from a set of canonical variables in terms of which the quadratic part of the Hamiltonian assumes a simple standard form, we go over to polar co-ordinates that enable us in a natural way to identify ‘fast’ and ‘slow’ variables in the problem. The role of terms of higher degree is then analysed in a perturbative approach, employing the technique of averaging over the fast variables. The terms necessary to describe all the essential features of the bifurcation are identified, arriving at a simple separable Hamiltonian incorporating the bifurcation characteristics involving families of periodic orbits. It is shown that the bifurcation can essentially be of two types and that one of those involves an intersting ‘secondary bifurcation’ phenomenon, thereby confirming results of earlier analyses of the problem. The simple approach presented here additionally allows us to compute tori around the periodic orbits. We briefly illustrate our method by referring to the ‘spinning orthogonal double pendulum’. Remarks on the significance of the Hamiltonian Hopf bifurcation in the context of KAM theory and on the quantisation of the bifurcation are included.

The “Van Der Waals interaction” between loose structures

Abstract Defining a “loose” structure as that possessed by a molecule requiring for configurational transformation to a metastable state, an energy less than or comparable to the energy of interaction with another such structure, we construct a model to study the peculiarities of the interaction of such objects, when attendent non-linear effects are taken into account. It is shown that the interaction potential possesses some rather intriguing features whose possible relevance to enzymatic and other processes involving biomolecules is indicated. Our work is along the line of certain speculations of Frohlich.

Spatial patterns in a simple reaction-diffusion system

Abstract Bifurcations of the steady homogeneous solution of a simpel reaction-diffusion system, distributed over a one-dimensional discrete lattice, are examined, and the different types of steady spatially inhomogeneous solutions that can appear are indicated. Bifurcations in the infinite-dimensional system are related to branchings (see below) of a two-dimensional area-preserving map, and the result is applied to establish the appearance and stability of wavelength-two solutions. Looking into bifurcations of these wavelength-two solutions, we show that no futher wavelength doubling takes place. The possibility of appearance of spatially chaotic time-invariant structures, and of more complex spatio-temporal structures including temporal intermittency, is briefly speculated upon.

ELECTRIC FIELD DEPENDENCE OF PHASE TRANSITIONS IN BILAYER LIPID MEMBRANES AND POSSIBLE BIOLOGICAL IMPLICATIONS

Bilayer lipid membranes consist of an inner hydrocarbon tail region with the hydrophilic polar heads on either side. The order-disorder transition in the hydrocarbon tail reigon, from liquid crystalline (fluid) to gel state, is characterised in terms of a Landau-de Gennes description, in which the effect of an external electric field is incorporated through its description, in which the effect of an external electric field is incorporated through its interaction with the surface charges on the bilayer (placed as it is in an ionic medium) or with the polar heads. Biological implications of such a phase transition, for excitable membranes, resides in a model wherein ion channels (taken to be composed of protein bundles) are postulated to be surrounded by lipid molecules in the fluid phase when the membrane is in its resting state, while surface charges and/or the polar heads of the lipid molecules responding to an electric stimulus, if of adequate magnitude, induces a transition in the hydrocarbon tail region of the (boundary) lipid surrounding the ion channels from the liquid crystalline (fluid) to the crystalline (gel) phase which, in turn, through coupling with the relevant modes of the protein bundles, results in the opening of the ion channels, provinding thereby a mechanism for the desired response.

Effect of the intervening medium and of feeding of energy into polar modes on macro-molecular interactions

Abstract Macromolecules possess pliable structures and could be raised through interactions into metastable states with the excitation of giant dipolar modes (with zero wavenumber). This in turn would qualitatively alter the nature of the interaction between two such objects from the familiar Van der Waals form. The dispersive character of the intervening medium in the space between the interacting molecules gives The interaction between such “labile” molecules is amenable to profound influences due to the presence of external energy pumping into polar modes and leads to interesting possibilities of control at the molecular level and storage of energy in ordered states which may be of importance in biological processes.

Entropy production due to coupling to a heat bath in the kicked rotor problem

Considering a kicked rotor coupled to a model heat bath both the classical and quantum entropy productions are calculated exactly. Starting with an initial wave packet, the von Neumann entropy as a function of time is determined from the reduced density matrix while the Liouville evolution of the corresponding Husimi distribution provides us with the classical entropy. It is found that both these entropies agree reasonably satisfying the same asymptotic growth law and more importantly both are proportional to the classical Lyapunov exponent.

The functional approach in biology

The concepts of structural and functional approaches are analysed. The existence of a logical limit to the domain of applicability of the structural approach is indicated. Some sources of possible failure of the structural method in biology are pointed out. Two fundamental characteristics of biological systems, inductive development and inductive functioning, necessitating the functional approach are discussed.

Fourth order resonant collisions of multipliers in reversible maps: period-4 orbits and invariant curves

Resonant collision of multipliers at ±i of a symmetric fixed point for a 2-parameter family of 4-dimensional reversible maps is considered. Bifurcation of period-4 orbits from the fixed point and their linear stability characteristics are briefly reviewed. In one of the three possible types of bifurcation (see text), a small angle secondary collision of the Floquet multipliers of the bifurcating periodic orbit takes place, leading to the bifurcation of invariant curves from the orbit. The invariant curves are calculated in a perturbation scheme in the leading order of perturbation. The secondary bifurcation is found to be of superthreshold type. An interesting pattern in the vicinity of the resonant collision, involving families of invariant curves and 2-tori, emerges. Results of numerical iterations, corroborating the picture conjectured on the basis of perturbation calculations, are presented. Corresponding results on resonant collisions at −1 are briefly stated.