Stability of coherent pattern formation through invasion in the FitzHugh–Nagumo system

In an earlier paper, we developed a general physical picture for the linear-marginal-stability mechanism governing the dynamics of front propagation into linearly unstable states. The main conclusion from this approach and the expressions for the resulting front velocity are similar to those obtained along different lines for the space-time evolution of instabilities in plasma physics and fluid dynamics with the so-called pinch-point analysis (a special type of saddle-point analysis). However, as stressed by Ben-Jacob et al. [Physica 14D, 348 (1985)], it is known from the work of Aronson and Weinberger [in Partial Differential Equations and Related Topics, edited by J. A. Goldstein (Springer, Heidelberg, 1975); Adv. Math. 30, 33 (1978)] on a class of simple model equations that exceptions can occur to the linear-marginal-stability velocity selection. In this paper, we generalize these observations and incorporate such exceptions into our general picture of front propagation into unstable states. We show that a breakdown of linear marginal stability occurs if the linear-marginal-stability front profile becomes unstable against a particular nonlinear ``invasion mode.''If this happens, a larger front speed is selected at a point at which the front profile is now marginally stable against this nonlinear invasion mode. We therefore refer to this as the nonlinear-marginal-stability mechanism. (Ben-Jacob et al. call it case-II marginal stability.) We present the results of detailed numerical studies that support our identification of the nonlinear-marginal-stability mechanism, and present the first examples of it for fronts in pattern-forming systems. In the neighborhood of a transition from linear to nonlinear marginal stability, the wavelength of the pattern generated by the front is only weakly dependent on the nonlinearities. We also analyze front propagation properties close to the threshold for instability at a pitchfork bifurcation. We conclude that linear marginal stability generally holds near a continuous transition (corresponding to a supercritical or forward bifurcation point), while front propagation close to a first-order transition (corresponding to a subcritical or inverted-bifurcation point) is generally governed by nonlinear marginal stability. These results are of importance for recent applications of the various approaches in fluid dynamics and other fields.Finally, we derive an expression for the rate of convergence of the front velocity to its asymptotic value. For the class of equations studied by Aronson and Weinberger, our expression reduces to a rigorous result by Bramson [Mem. Am. Math. Soc. 285, 1 (1983)], but it differs from the one often quoted in the pinch-point or saddle-point analysis. We argue that the latter one is only valid in a limited region of space, and show how to extend the usual analysis to arrive at our result. Several experimental systems to which our results are relevant are discussed.

In this paper the propagation of fronts into an unstable state are studied. Such fronts can occur e.g., in the form of domain walls in liquid crystals, or when the dynamics of a system which is suddenly quenched into an unstable state is dominated by domain walls moving in from the boundary. It was emphasized recently by Dee et al. that for sufficiently localized initial conditions the velocity of such fronts often approaches the velocity corresponding to the marginal stability point, the point at which the stability of a front profile moving with a constant speed changes. I show here when and why this happens, and advocate the marginal stability approach as a simple way to calculate the front velocity explicitly in the relevant cases. I sketch the physics underlying this dynamical mechanism with analogies and, building on recent work by Shraiman and Bensimon, show how an equation for the local ``wave number'' that may be viewed as a generalization of the Burgers equation, drives the front velocity to the marginal stability value. This happens provided the steady-state solutions lose stability because the group velocity for perturbations becomes larger than the envelope velocity of the front.For a given equation, our approach allows one to check explicitly that the marginal stability fixed point is attractive, and this is done for the amplitude equation and the Swift-Hohenberg equation. I also analyze an extension of the Fisher-Kolmogorov equation, obtained by adding a stabilizing fourth-order derivative -\ensuremath{\gamma} ${\ensuremath{\partial}}^{4}$\ensuremath{\varphi}/\ensuremath{\partial}${x}^{4}$ to it. I predict that for \ensuremath{\gamma}(1/12 the fronts in this equation are of the same type as those occurring in the Fisher-Kolmogorov equation, i.e., localized initial conditions develop into a uniformly translating front solution of the form \ensuremath{\varphi}(x-vt) that propagates with the marginal stability velocity. For \ensuremath{\gamma}>(1/12, localized initial conditions may develop into fronts propagating at the marginal stability velocity, but such front solutions cannot be uniformly translating. Differences between the propagation of uniformly translating fronts \ensuremath{\varphi}(x-vt) and envelope fronts are pointed out, and a number of open problems, some of which could be studied numerically, are also discussed.

I review the differences and similarities between the marginal stability approach to front propagation into unstable states and the “pinch point” analysis for the space-time evolution of perturbations developed in plasma physics. I then briefly discuss the following developments and surprises: (i) the resolution of a discrepancy between the theory and experiments on Taylor vortex fronts; (ii) some new results for the regime where front propagation is dominated by nonlinear effects (nonlinear marginal stability regime); (iii) ongoing work on fronts and pulses in the complex Ginzburg-Landau equation.KeywordsUnstable StateMarginal StabilityConvective InstabilityFront VelocityFront PropagationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Upconverting nanoparticles (UCNPs) have recently shown great potential as contrast agents in biological applications. In developing different UCNPs, the characterization of their quantum yield (QY) is a crucial issue, as the typically drastic decrease in QY for low excitation power densities can either impose a severe limitation or provide an opportunity in many applications. The power density dependence of the QY is governed by the competition between the energy transfer upconversion (ETU) rate and the linear decay rate in the depopulation of the intermediate state of the involved activator in the upconversion process. Here we show that the QYs of Yb(3+) sensitized two-photon upconversion emissions can be well characterized by the balancing power density, at which the ETU rate and the linear decay rate have equal contributions, and its corresponding QY. The results in this paper provide a method to fully describe the QY of upconverting nanoparticles for arbitrary excitation power densities, and is a fast and simple approach for assessing the applicability of UCNPs from the perspective of energy conversion.

In this paper, we present the results of an experimental sensitivity analysis on a vertical electrically heated Rijke tube. We examine the shift in linear decay rates and frequencies of thermoacoustic oscillations, with and without control devices. To measure the decay rate, we wait for the system to reach a steady state and then excite it with an acoustic pulse from a loudspeaker. We identify the range of amplitudes over which the amplitude decays exponentially with time. In this range, the rate of change of the amplitude is linearly proportional to the amplitude, and we calculate the constant of proportionality, the linear decay rate, which can be compared with model predictions. The aim of this work is (i) to improve the experimental techniques implemented by Rigas et al. (J Fluid Mech 787, 2016), Jamieson et al. (Int J Spray Combust Dyn, 2016), using a technique inspired by Mejia et al. (Combust Flame 169:287–296, 2016), and (ii) to provide experimental data for future comparison with adjoint-based sensitivity analysis. Our experimental setup is automated and we can obtain thousands of decay rates in 1/12 the time of our previous method.

In this article, we report the results of an experimental sensitivity analysis on a vertical electrically heated Rijke tube. We examine the stability characteristics of the system due to the introduction of a secondary heat source. The experimental sensitivity analysis is quantified by measuring the shift in linear growth and decay rate as well as the shift in the linear frequency during periods of growth and decay of thermoacoustic oscillations. Linear growth and decay rate measurements agree qualitatively well with the theoretical predictions from adjoint-based methods. A discrepancy in the linear frequency measurements highlight deficiencies in the model used for those predictions and shows that the experimental measurement of sensitivities is a stringent test of any thermoacoustic model. The findings suggest that adjoint-based methods are, in principle, capable of providing industry with a cheap and efficient tool for developing optimal control strategies for more complex thermoacoustic systems.

Dee et al. have advanced the idea that the natural velocity of fronts propagating into an unstable state is related to the stability of these fronts through ``marginal stability.'' It is shown that this is indeed the case if front solutions lose stability through one particular mechanism. Marginal stability is derived for front propagation in the Swift-Hohenberg equation, but does not hold in a certain range of parameters of an extension of the Fisher-Kolmogorov equation.

In this paper, we study the minimax optimization problem in the smooth and strongly convex-strongly concave setting when we have access to noisy estimates of gradients. In particular, we first analyze the stochastic Gradient Descent Ascent (GDA) method with constant stepsize, and show that it converges to a neighborhood of the solution of the minimax problem. We further provide tight bounds on the convergence rate and the size of this neighborhood. Next, we propose a multistage variant of stochastic GDA (M-GDA) that runs in multiple stages with a particular learning rate decay schedule and converges to the exact solution of the minimax problem. We show M-GDA achieves the lower bounds in terms of noise dependence without any assumptions on the knowledge of noise characteristics. We also show that M-GDA obtains a linear decay rate with respect to the error's dependence on the initial error, although the dependence on condition number is suboptimal. In order to improve this dependence, we apply the multistage machinery to the stochastic Optimistic Gradient Descent Ascent (OGDA) algorithm and propose the M-OGDA algorithm which also achieves the optimal linear decay rate with respect to the initial error. To the best of our knowledge, this method is the first to simultaneously achieve the best dependence on noise characteristic as well as the initial error and condition number.

Front propagation into unstable states

The nonlinear evolution and stabilization of absolute whistler instabilities driven by electron pressure anisotropy and propagating obliquely to the external magnetic field in finite β plasmas are investigated without introducing the assumption of random phases. Both in the firehose and the mirror instability regimes the pressure anisotropy relaxes quasilinearly toward a more stable state. The nonlinear wave equation has also been obtained. By choosing the parameters in such a way that the plasma is sufficiently close to linear marginal stability, it has been possible to show both numerically and analytically that the (firehose) instability stabilizes nonlinearly at sufficiently low levels of excitation. The fluctuation level does not asymptotically approach a constant value, but oscillates in time, the maximum amplitude being proportional to the linear growth rate. Also, close to marginal stability, finite Larmor radius effects can play a significant role even in the limit kR ≪ 1. It is found that finite Larmor radius effects are stabilizing if tan2α > 4β‖/β⊥, where α is the angle between the direction of wave propagation and the direction of the background magnetic field and β‖(⊥) is the ratio of parallel (perpendicular) kinetic to magnetic pressure.

Hopf bifurcation for general network-organized reaction-diffusion systems and its application in a multi-patch predator-prey system

In this paper we investigate the diffusive zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form: W t (x,t) + 1 eA(x,D)W(x,t) = 1 e 2 B(x, W(x,t)) + 1 e D(W(x,t)) + E(W(x,t)). We analyze the singular convergence, as e ↓ 0, in the case which leads to a limit system of parabolic type. The analysis is carried out by using the following steps: (i) We single out algebraic structure conditions on the full system, motivated by formal asymptotics, by some examples of discrete velocity models in kinetic theories. (ii) We deduce energy estimates, uniformly in e, by assuming the existence of a symmetrizer having the so-called block structure and by assuming dissipativity conditions on B. (iii) We assume a Kawashima type condition and perform the convergence analysis by using generalizations of compensated compactness due to Tartar and Gerard. Finally, we include examples that show how to use our theory to approximate any quasilinear parabolic systems, satisfying the Petrowski parabolicity condition, or general reaction diffusion systems, including Chemotaxis and Brusselator type systems.

A new kind of integral transform and its application to periodic solutions

We summarize our results related with mathematical modeling of Aedes aegypti and its Lie symmetries. Moreover, some explicit, group-invariant solutions are also shown. Weak equivalence transformations of more general reaction diffusion systems are also considered. New classes of solutions are obtained.

Chaotic step bunching during crystal growth