Parity-breaking Bifurcation Research Articles

Parity-Breaking Bifurcation and Global Oscillation in Patterns of Ion Channels

Stationary spatiotemporal pattern formation emerging from the electric activity of biological membranes is widespread in cells and tissues. A known key instability comes from the self-aggregation of membrane channels. In a two-dimensional geometry, we show that the primary pattern undergoes four secondary instabilities: Eckhaus-like, period-halving, drift instabilities, and a global oscillation. The stability diagram is determined. The parity-breaking (drift) bifurcation of channel density is characterized analytically and numerically.

Wave-number selection and parity-breaking bifurcation in directional viscous fingering

We present a mechanism of limitation for the possible wave numbers above an instability threshold. This mechanism is experimentally investigated in the interfacial instability of directional viscous fingering in a finite system. It is shown experimentally to be controlled by the divergence of a phase-diffusion constant. Theoretically, this limitation on the low value of the accessible wave numbers is a consequence of the interaction between the fundamental and the first harmonic modes. The analysis of coupled amplitude equations demonstrates theoretically the existence of a divergence of a phase-diffusion constant when approaching the threshold

Parity-breaking bifurcation in inhomogeneous systems

The steady-state reflection-breaking bifurcation from a circle of nontrivial equilibria in O(2)-equivariant systems results in a pair of travelling waves. When the continuous part of the group O(2) is weakly broken, the corresponding instability may lead to nonsymmetric but steady states. The transition from this state to the travelling wave state with increasing bifurcation parameter is complex, and typically involves sequences of global bifurcations. The possible scenarios are described in detail and the results are related to the dynamics associated with parity-breaking instabilities of spatially periodic patterns in inhomogeneous systems.

Order parameter equations for front transitions: Planar and circular fronts

Near a parity breaking front bifurcation, small perturbations may reverse the propagation direction of fronts. Often this results in nonsteady asymptotic motion such as breathing and domain breakup. Exploiting the time scale differences of an activator-inhibitor model and the proximity to the front bifurcation, we derive equations of motion for planar and circular fronts. The equations involve a translational degree of freedom and an order parameter describing transitions between left and right propagating fronts. Perturbations, such as a space dependent advective field or uniform curvature (axisymmetric spots), couple these two degrees of freedom. In both cases this leads to a transition from stationary to oscillating fronts as the parity breaking bifurcation is approached. For axisymmetric spots, two additional dynamic behaviors are found: rebound and collapse.

Phase dynamics near a parity-breaking instability.

In a directional viscous fingering experiment, a phase diffusion behavior of the regular array of cells is demonstrated. At constant wave number, the evolution of the phase diffusion coefficient D with the parameter of the instability \ensuremath{\epsilon} is reported. This coefficient D is found to decrease at low \ensuremath{\epsilon} values in agreement with the presence of an Eckhaus instability, but to increase strongly at large \ensuremath{\epsilon} values. This unusual increase of D is also present in the analytical and numerical study of two coupled amplitude equations for the modes k and 2k. The phase diffusion coefficient is found to diverge at the threshold of the parity-breaking bifurcation.

Parity breaking and solitary waves in axisymmetric Taylor vortex flow.

We report, for the first time, the experimental observation of solitary waves in an axisymmetric Taylor-Couette flow. The solitary waves result from a parity-breaking bifurcation which occurs near the onset of Eckhaus instability for Taylor vortex flow in a wide-gap, counterrotating system (radius ratio \ensuremath{\eta}=0.5 and speed ratio \ensuremath{\mu}=-0.2). Our results agree quantitatively with a recent theoretical calculation by Riecke and Paap.

Analytical treatment for parity breaking in eutectic growth.

Using an ansatz for an asymmetric profile of the solidification front in lamellar eutectic growth, the von Neuman problem is solved. Imposing the Gibbs-Thomson condition, we derive a general expression for the tilt angle as a function of the control parameters. We find that the front undergoes a supercritical parity-breaking bifurcation at a critical value of \ensuremath{\sigma}=${\mathit{d}}_{0}$l/${\ensuremath{\lambda}}^{2}$ (${\mathit{d}}_{0}$ is the capillary length, l the diffusion length, and \ensuremath{\lambda} the wavelength of the pattern). We further find that parity-breaking causes a reduction of the average front undercooling. All these features agree with previous numerical results.

Parity-breaking and Hopf bifurcations in axisymmetric Taylor vortex flow.

In a linear stability analysis of axisymmetic Taylor vortex flow between counterrotating cylinders we find a parity-breaking bifurcation and a Hopf bifurcation. They are due to an internal resonance of modes with wave number q and 2q and do not break the azimuthal symmetry of the flow. They may lead to drift waves and standing waves, respectively. The standing waves resemble an optical mode, in which adjacent vortex pairs have opposite phase. A temporally periodic driving of the system is expected to excite and stabilize such standing waves. An analysis of the amplitude-phase equations for parity-breaking bifurcations shows that the drift waves can be localized although the parity-breaking bifurcation itself is supercritical. This result applies also to the other systems in which localized drift waves are observed as, for instance, in directional solidification.

Resonant interactions and traveling-solidification cells.

Recently a set of new phenomena has been observed in the directional solidification of liquid crystals, which include solitary inclusions of traveling waves and spatial period doubling. We propose that these patterns can be naturally explained as arising from the resonant interaction of modes with wave vectors q and 2q. This picture is substantiated by numerical calculations employing the material parameters of the liquid-crystal system of Simon, Bechhoefer, and Libchaber [Phys. Rev. Lett. 61, 2557 (1988)]. Our approach correctly predicts period-doubled (mixed) modes and the parity-breaking bifurcation to the traveling-wave state. Possible causes for the disagreement of some stability assignments as compared with the experiments are sketched.

Parity-breaking transitions of modulated patterns in hydrodynamic systems.

A model for subcritical parity-breaking bifurcation of a periodic pattern is introduced. It is found that nucleated regions of the new asymmetric state propagate with a well-defined velocity in a direction determined by their parity, and leave in their wake a pattern with an altered wavelength. Successive passage of these ``parity bubbles'' enables the system to relax to a selected wavelength. The possible relevance of these findings to recent observations of ``solitary modes'' in directional solidification and hydrodynamic experiments is discussed.