Period doubling and chaos in partial differential equations for thermosolutal convection

Transition to turbulence in a five-mode Galerkin truncation of two-dimensional magnetohydrodynamics

The chaotic dynamics of a low-order Galerkin truncation of the two-dimensional magnetohydrodynamic system, which reproduces the dynamics of fluctuations described by nearly incompressible magnetohydrodynamic in the plane perpendicular to a background magnetic field, is investigated by increasing the external forcing terms. Although this is the case closest to two-dimensional hydrodynamics, which shares some aspects with the classical Feigenbaum scenario of transition to chaos, the presence of magnetic fluctuations yields a very complex interesting route to chaos, characterized by the splitting into multiharmonic structures of the field amplitudes, and a mixing of phase-locking and free phase precession acting intermittently. When the background magnetic field lies in the plane, the system supports the presence of Alfvén waves thus lowering the nonlinear interactions. Interestingly enough, the dynamics critically depends on the angle between the direction of the magnetic field and the reference system of the wave vectors. Above a certain critical angle, independently from the external forcing, a breakdown of the phase locking appears, accompanied with a suppression of the chaotic dynamics, replaced by a simple periodic motion.

A physically extended Lorenz system

The Lorenz system is a simplified model of Rayleigh-Bénard convection, a thermally driven fluid convection between two parallel plates. Two additional physical ingredients are considered in the governing equations, namely, rotation of the model frame and the presence of a density-affecting scalar in the fluid, in order to derive a six-dimensional nonlinear ordinary differential equation system. Since the new system is an extension of the original three-dimensional Lorenz system, the behavior of the new system is compared with that of the old system. Clear shifts of notable bifurcation points in the thermal Rayleigh parameter space are seen in association with the extension of the Lorenz system, and the range of thermal Rayleigh parameters within which chaotic, periodic, and intermittent solutions appear gets elongated under a greater influence of the newly introduced parameters. When considered separately, the effects of scalar and rotation manifest differently in the numerical solutions; while an increase in the rotational parameter sharply neutralizes chaos and instability, an increase in a scalar-related parameter leads to the rise of a new type of chaotic attractor. The new six-dimensional system is found to self-synchronize, and surprisingly, the transfer of solutions to only one of the variables is needed for self-synchronization to occur.

Time-periodic solitons in a damped-driven nonlinear Schrödinger equation

Time-periodic solitons of the parametrically driven damped nonlinear Schrödinger equation are obtained as solutions of the boundary-value problem on a two-dimensional spatiotemporal domain. We follow the transformation of the periodic solitons as the strength of the driver is varied. The resulting bifurcation diagrams provide a natural explanation for the overall form and details of the attractor chart compiled previously via direct numerical simulations. In particular, the diagrams confirm the occurrence of the period-doubling transition to temporal chaos for small values of dissipation and the absence of such transitions for larger dampings. This difference in the soliton's response to the increasing driving strength can be traced to the difference in the radiation frequencies in the two cases. Finally, we relate the soliton's temporal chaos to the homoclinic bifurcation.

Roll and square convection in binary liquids: a few-mode Galerkin model

We present a few-mode Galerkin model for convection in binary fluid layers subject to an approximation to realistic horizontal boundary conditions at positive separation ratios. The model exhibits convection patterns in form of rolls and squares. The stable squares at onset develop into stable rolls at higher thermal driving. In between, a regime of a so-called cross roll structure is found. The results of our few-mode model are in good agreement with both experiments and numerical multimode simulations.

Minimal model for complex dynamics in cellular processes

Cellular functions are controlled and coordinated by the complex circuitry of biochemical pathways regulated by genetic and metabolic feedback processes. This paper aims to show, with the help of a minimal model of a regulated biochemical pathway, that the common nonlinearities and control structures present in biomolecular interactions are capable of eliciting a variety of functional dynamics, such as homeostasis, periodic, complex, and chaotic oscillations, including transients, that are observed in various cellular processes.

Chaotic dynamics in an ionic model of the propagated cardiac action potential

We simulate the effect of periodic stimulation on a strand of ventricular muscle by numerically integrating the one-dimensional cable equation using the Beeler-Reuter model to represent the transmembrane currents. As stimulation frequency is increased, the rhythms of synchronization [1:1----2:2----2:1----4:2---- irregular----3:1----6:2----irregular----4:1----8:2----...----1:0] are successively encountered. We show that this sequence of rhythms can be accounted for by considering the response of the strand to premature stimulation. This involves deriving a one-dimensional finite-difference equation or "map" from the response to premature stimulation, and then iterating this map to predict the response to periodic stimulation. There is good quantitative agreement between the results of iteration of the map and the results of the numerical integration of the cable equation. Calculation of the Lyapunov exponent of the map yields a positive value, indicating sensitive dependence on initial conditions ("chaos"), at stimulation frequencies where irregular rhythms are seen in the corresponding numerical cable simulations. The chaotic dynamics occurs via a previously undescribed route, following two period-doubling bifurcations. Bistability (the presence of two different synchronization rhythms at a fixed stimulation frequency) is present both in the simulations and the map. Thus, we have been able to directly reduce consideration of the dynamics of a partial differential equation (which is of infinite dimension) to that of a one-dimensional map, incidentally demonstrating that concepts from the field of non-linear dynamics--such as period-doubling bifurcations, bistability, and chaotic dynamics--can account for the phenomena seen in numerical simulations of the cable equation. Finally, we sketch out how the one-dimensional description can be extended, and point out some implications of our work for the generation of malignant ventricular arrhythmias.