switching system (original) (raw)

Physical Review Letters, 1988

We investigate the eA'ect of temporal modulation on the spatial patterns produced by a spatially ex- tended system undergoing a Hopf bifurcation. It is shown that physically this modulation can stabilize standing waves, which would otherwise be stable to traveling waves. Mathematically a codimension-3 point is naturally introduced which encompasses a Takens-Bogdanov codimension-2 point with vanishing Hopf frequency. Experimentally this setup is attractive since the three relevant parameters are easily accessible and the steady bifurcation can be made forward as well as backward near a codimension-2 point.

Drifting pattern domains in a reaction-diffusion system with nonlocal coupling

Physical review. E, Statistical, nonlinear, and soft matter physics, 2002

Drifting pattern domains (DPDs), i.e., moving localized patches of traveling waves embedded in a stationary (Turing) pattern background and vice versa, are observed in simulations of a reaction-diffusion model with nonlocal coupling. Within this model, a region of bistability between Turing patterns and traveling waves arises from a codimension-2 Turing-wave bifurcation (TWB). DPDs are found within that region in a substantial distance from the TWB. We investigated the dynamics of single interfaces between Turing and wave patterns. It is found that DPDs exist due to a locking of the interface velocities, which is imposed by the absence of space-time defects near these interfaces.

Bifurcation phenomena and cellular-pattern evolution in mixed-convection heat transfer

Journal of Fluid Mechanics, 1987

This investigation is concerned with the numerical calculation of multiple solutions for a mixed-convection flow problem in horizontal rectangular ducts. The numerical results are interpreted in terms of recent observations by Benjamin (1978a) on the bifurcation phenomena for a bounded incompressible fluid. The observed mutations of cellular flows are discussed in terms of dynamic interchange processes. Each cellular flow may be represented by a solution surface in the parametric space of Grashof number Gr and aspect ratio γ, which is delimited by stability boundaries. Such a stability map has been generated for each type of cellular flow by a series of numerical experiments. Once these boundaries are crossed one cellular flow mutates into another via a certain dynamical process. Although the nature of the singular points on this map have not been determined precisely, a plausible general structure of the cellular-flow exchange process emerges from this map with several features in ...

Spatio-temporal patterns in a reaction–diffusion system with wave instability

Chemical Engineering Science, 2000

We utilize a simple three-variable reaction}di!usion model to study patterns that emerge beyond the onset of the (short-)wave instability. We have found various wave patterns including standing waves, traveling waves, asymmetric standing}traveling waves and target patterns. We employ both periodic and zero #ux boundary conditions in the simulations, and we analyze the patterns using space}time two-dimensional Fourier spectra. A fascinating pattern of waves which periodically change their direction of propagation along a ring is found for very short systems. A related pattern of modulated standing waves is found for systems with zero #ux boundary conditions. In a two-dimensional system with small overcriticality we observe a wide variety of standing wave patterns. These include plain and modulated stripes, squares and rhombi. We also "nd standing waves consisting of periodic time sequences of stripes, rhombi and hexagons. The short-wave instability can lead to a much greater variety of spatio-temporal patterns than the aperiodic Turing and the long-wave oscillatory instabilities. For example, a single oscillatory cycle may display all the basic patterns related to the aperiodic Turing instability * stripes, hexagons and inverted hexagons (honeycomb) * as well as rhombi and modulated stripes. A rich plethora of patterns is seen in a system with cylindrical geometry * examples include rotating patterns of standing waves and counter-propagating waves.

Mode-locking in advection-reaction-diffusion systems: An invariant manifold perspective

Chaos (Woodbury, N.Y.), 2018

Fronts propagating in two-dimensional advection-reaction-diffusion systems exhibit a rich topological structure. When the underlying fluid flow is periodic in space and time, the reaction front can lock to the driving frequency. We explain this mode-locking phenomenon using the so-called burning invariant manifolds (BIMs). In fact, the mode-locked profile is delineated by a BIM attached to a relative periodic orbit (RPO) of the front element dynamics. Changes in the type (and loss) of mode-locking can be understood in terms of local and global bifurcations of the RPOs and their BIMs. We illustrate these concepts numerically using a chain of alternating vortices in a channel geometry.

Finite-dimensional dynamical system modeling thermal instabilities

Physica D: Nonlinear Phenomena, 2000

We describe a three-dimensional dynamical system, which is obtained as a pseudo-spectral approximation to a free boundary problem modeling solid combustion and rapid solidification, and is capable of generating its major dynamical patterns. These patterns include a Hopf bifurcation followed by a sequence of secondary period doubling and a transition to chaos, reverse sequences, and sequences followed by Shilnikov type trajectories. A computer-assisted bifurcation analysis uncovers some novel mechanisms of stability exchange. The most striking of them is an infinite period bifurcation which resembles the classical Shilnikov bifurcation, but instead of a funnel-shaped spiral along which the period is continually increasing, the continuation produced a series of isolas. Each isola is a closed branch of solutions of roughly the same period, and with the same number of oscillations. The isolas corresponding to consecutive numbers of low amplitude oscillations about the equilibrium are adjacent to each other, and appear to accumulate on a saddle-focus homoclinic connection of Shilnikov type.

A domain wall between single-mode and bimodal states and its transition to dynamical behavior in inhomogeneous systems

Physica D: Nonlinear Phenomena, 1997

We consider domain walls (DW's) between single-mode and bimodal states that occur in coupled nonlinear diffusion (NLD), real Ginzburg-Landau (RGL), and complex Ginzburg-Landau (CGL) equations with a spatially dependent coupling coefficient. Group-velocity terms are added to the NLD and RGL equations, which breaks the variational structure of these models. In the simplest case of two coupled NLD equations, we reduce the description of stationary configurations to a single second-order ordinary differential equation. We demonstrate analytically that a necessary condition for existence of a stationary DW is that the group-velocity must be below a certain threshold value. Above this threshold, dynamical behavior sets in, which we consider in detail. In the CGL equations, the DW may generate spatio-temporal chaos, depending on the nonlinear dispersion. A spatially dependent coupling coefficient as considered in this paper can be realized at least in two different convection systems: a rotating narrow annulus supporting two traveling-wave wall modes, and a large-aspect-ratio system with poor heat conductivity at the lateral boundaries, where the two phases separated by the DW are rolls and square cells.

Spatially periodic modulated Rayleigh-Bénard convection

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1996

Two-dimensional thermal convection in a fluid layer between two rigid walls at different mean temperatures is investigated. The top container boundary is undulated and the temperatures at the top and bottom boundaries are spatially periodic modulated, with modulation wavelengths large compared to the thickness of the fluid layer. The continuous translational invariance in the fluid layer is broken by these spatial modulations. Consequently phase differences between two periodic modulations give rise to an interesting drifting pattern, with the drift direction depending on the sign of the relative phase between the modulations. At distinguished ratios between the modulation wave numbers and relative phases the onset of convection changes as function of the modulation amplitudes from a stationary into an oscillatory one: We call this phenomenon Hopf bifurcation by frustrated drifts. Possible experiments are described in detail where this phenomenon can be expected. ͓S1063-651X͑96͒05805-5͔

Boundary effects on the dynamics of monostable reaction-diffusion systems

Physics Letters A, 1997

We study a monostable reaction-diffusion model in a bounded domain, subjected to partially reflecting boundary conditions. We analyze the stability of the arising patterns and detect a bifurcation of the uniform solution induced by changes in the reflectivity of the boundaries. We examine the critical slowing down of the system's dynamics in the neighborhood of the bifurcation point by analyzing its non-equilibrium potential. @ 1997 Elsevier Science B.V. PACS: 05.7O.Ln; 47.54.+r; 82.2O.Mj.