Razvan A. Satnoianu

Parameter space analysis, pattern sensitivity and model comparison for Turning and stationary flow-distributed waves (FDS)

A new type of instability in coupled reaction-diffusion-advection systems is analysed in a one-dimensional domain. This instability, arising due to the combined action of flow and diffusion, creates spatially periodic stationary waves termed flow and diffusion-distributed structures (FDS). Here we show, via linear stability analysis, that FDS are predicted in a considerably wider domain and are more robust (in the parameter domain) than the classical Turing instability patterns. FDS also represent a natural extension of the recently discovered flow-distributed oscillations (FDO). Nonlinear bifurcation analysis and numerical simulations in one-dimensional spatial domains show that FDS also have much richer solution behaviour than Turing structures. In the framework presented here Turing structures can be viewed as a particular instance of FDS. We conclude that FDS should be more easily obtainable in chemical systems than Turing (and FDO) structures and that they may play a potentially important role in biological pattern formation.

Spatio-temporal structures in a differential flow reactor with cubic autocatalator kinetics

Abstract A model of a differential-flow reactor with Gray-Scott kinetics is considered in which the reactant is immobilised while the autocatalyst is allowed to flow (with uniform velocity) and to diffuse. A linear stability analysis shows that there is a critical flow rate above which the spatially uniform stationary state is convectively unstable. A weakly nonlinear analysis (close to criticality) results in a complex Ginzburg-Landau equation (CGLE) which governs the envelope amplitude of the packet of differential-flow-induced waves being the first derivation of such a modulation equation for such a problem. The analysis of the CGLE shows that this bifurcation is supercritical with proparating wave packets developing for flow rates above the critical value and identifies a region of parameter space close to the Hopf bifurcation in the kinetic scheme where more complex propagating wave structures can arise. The theoretical predictions are confirmed by numerical integrations of the initial-value problem which also give results for flow rates well in excess of those amenable to analysis.

Pattern formation in a differential-flow reactor model

Abstract A model for a differential–flow reactor is considered, based on cubic autocatalator kinetics in which the substrate is immobilized and the autocatalyst flows through the reactor at a constant rate. Linear stability analysis shows that there is a critical flow rate above which the spatially uniform steady state becomes convectively unstable. Numerical simulations show that this convective instability leads to the formation of a wave packet propagating through the reactor. The nature of this wave packet depends on the flow rate and on the two cases that are identified for the kinetic parameter μ , namely a generic case for general values of μ and when μ is close to the value which gives a Hopf bifurcation in the kinetic system.

Controlled pattern formation in the CDIMA reaction with a moving boundary of illumination

A reaction–diffusion (RD) system that grows axially as one of its boundaries moves is equivalent to a boundary-forced open flow in which all species have identical flow coefficients. Depending on the flow or growth rate, ϕ, and on the intrinsic spreading velocity, c0, of the RD structure, such systems are either absolutely (ϕ   c0) unstable. We previously showed how periodic boundary forcing of an axially growing domain could be used to control the formation of space-periodic structures in biological morphogenesis. This paper proposes, as a chemical equivalent of an axially growing embryo, the design of a continuously fed unstirred flow reactor (CFUR), characterized by a photo-chemically controlled moving boundary. Using the Turing-unstable CDIMA system as an example, we illustrate by simulations the kinds of wave structures that are expected to arise in the absolutely and convectively unstable regimes when boundary forcing is either constant or time-periodic.

Forced convective structures in a differential-flow reactor

A model of an autocatalytic reaction in a differential-flow reactor is considered in which the reactant is immobilized and the autocatalyst allowed to flow (with uniform velocity) and to diffuse. Consideration is given to the influence that a periodic external signal (fluctuations in the inflow concentration of the autocatalyst) can have upon the convective structures which exist in this model. It is shown that, depending on the frequency of the signal, there are two parameter regions giving distinct spatio-temporal responses. There is a region of high excitation waves (convected forced waves) which appear for a finite range of frequencies. Outside this range there are two domains of natural response waves. Close to the boundary of these regions there are irregular responses suggestive of an excitable system: packets of travelling waves of complex form develop which appear at non-regular intervals surrounded by small amplitude periodic waves.

Spatiotemporal chaos in a differential flow reactor

The spatiotemporal evolution of a chemical system close to a Hopf bifurcation in a differential flow reactor is studied. The interaction of the Hopf-differential flow induced instabilities for the cubic autocatalator model is determined through the appropriate form of the complex Ginzburg–Landau equation for the evolving amplitude. New behaviour, including spatiotemporal chaos, is observed from this equation. These predictions are shown also to be a feature of the initial-value problem for the original autocatalator equations.

Differential-flow-induced instability in a cubic autocatalator system

The formation of spatio-temporal stable patterns is considered for a reaction-diffusion-convection system based upon the cubic autocatalator, A + 2B → 3B, B → C, with the reactant A being replenished by the slow decay of some precursor P via the simple step P → A. The reaction is considered in a differential-flow reactor in the form of a ring. It is assumed that the reactant A is immobilised within the reactor and the autocatalyst B is allowed to flow through the reactor with a constant velocity as well as being able to diffuse. The linear stability of the spatially uniform steady state (a, b) = (µ−1, µ), where a and b are the dimensionless concentrations of the reactant A and autocatalyst B, and µ is a parameter reflecting the initial concentration of the precursor P, is discussed first. It is shown that a necessary condition for the bifurcation of this steady state to stable, spatially non-uniform, flow-generated patterns is that the flow parameter φ > φc(µ, λ) where φc(µ,λ) is a (strictly positive) critical value of φ and λ is the dimensionless diffusion coefficient of the species B and also reflects the size of the system. Values of φc at which these bifurcations occur are derived in terms of µ and λ. Further information about the nature of the bifurcating branches (close to their bifurcation points) is obtained from a weakly nonlinear analysis. This reveals that both supercritical and subcritical Hopf bifurcations are possible. The bifurcating branches are then followed numerically by means of a path-following method, with the parameter φ as a bifurcation parameter, for representatives values of µ and λ. It is found that multiple stable patterns can exist and that it is also possible that any of these can lose stability through secondary Hopf bifurcations. This typically gives rise to spatio-temporal quasiperiodic transients through which the system is ultimately attracted to one of the remaining available stable patterns.

Erdős-Mordell-Type Inequalities in a Triangle

with equality if and only if the triangle is equilateral and P is its center. This inequality was conjectured by Erdős [1] and proved by Mordell and Barrow [2]. Oppenheim [3] established a number of additional inequalities relating the six distances p, q, r , x , y, and z. Such an inequality will be referred to as an Erdős-Mordell-type inequality. A survey of some of these inequalities can be found in [4]. The aim of this note is to obtain a large family of Erdős-Mordell-type inequalities. To do so we first note that inequality (1) compares the elements f (p), f (q), f (r) with f (x), f (y), f (z) for the function f (t) = t . By applying one of the standard proofs for (1) (for example, as in [4]) and replacing f with a monotone multiplicatively convex function, a large class of Erdős-Mordell-type inequalities is obtained. Montel was among the first to consider multiplicative convexity [5]. A modern presentation is given by Niculescu [6]. We begin by recalling the definition of this concept.

A General Method for Establishing Geometric Inequalities in a Triangle

q can be any prime power. The material necessary to see this can be found in [3]. A different approach to finite field arithmetic can be found in [4]. The latter approach takes less space, but the table approach is faster, once the table is constructed. And it is very easy to implement. We have not addressed the question of existence and constructability of primitive polynomials. Existence can be shown rather easily using M6bius inversion in number theory [1, Theorem 16.9]. They can be constructed using Conway polynomials [2].

Travelling waves in a differential flow reactor with simple autocatalytic kinetics

A simple prototype model for a differential flow reactor in which the possible initiation and propagation of a reaction-diffusion-convection travelling-wave solution (TWS) in the simple isothermal autocatalytic system A+mB→ (m+1)B, rate kabm (m ≥ 1) is studied with special attention being paid to the most realistic cases (m=1,2). The physical problem considered is such that the reactant A (present initially at uniform concentration) is immobilised within the reactor. A reaction is then initiated by allowing the autocatalyst species to enter and to flow through the reaction region with a constant velocity. The structure of the permanent-form travelling waves supported by the system is considered and a solution obtained valid when the flow rate (of the autocatalyst) is very large. General properties of the corresponding initial-value problem (IVP) are derived and it is shown that the TWS are the only long-time solutions supported by the system. Finally, these results are complemented with numerical solutions of the IVP which confirm the analytical results and allow the influence of the parameters of the problem not accessible to the theoretical analysis to be determined.