The experimentally known phenomenon of oscillatory instability in convective burning ofporous explosives is discussed. A simple phenomenological model accounting for theejection of unburned particles from the consolidated charge is formulated and analyzed. Itis shown that the post-front hydraulic resistance induced by the ejected particlesprovides a mechanism for the oscillatory burning.

A computer-aided method for accurately carrying out the Chapman-Enskog expansion of the Boltzmann equation, including its inelastic variant, is presented and employed to derive a hydrodynamic description of a dilute binary mixture of smooth inelastic spheres. Constitutive relations, formally valid for all physical values of the coefficients of restitution, are calculated by carrying out the pertinent Chapman-Enskog expansion to sufficient high orders in the Sonine polynomials to ensure numerical convergence. The resulting hydrodynamic description is applied to the analysis of a vertically vibrated binary mixture of particles (under gravity) differing only in their respective coefficients of restitution. It is shown that even with this “minor”difference the mixture partly segregates, its steady state exhibiting a sandwich-like configuration.

We construct interfacial solitary structures (spots) generated by a bistable chemicalreaction or a non-equilibrium phase transition in a surfactant film. The structures arestabilized by Marangoni flow that prevents the spread of a state with a higher surfacetension when it is dynamically favorable. In a system without surfactant massconservation, a unique radius of a solitary spot exists within a certain range of valuesof the Marangoni number and of the deviation of chemical potential from the Maxvellconstruction, but multiple spots attract and coalesce. In a conservative system, there isa range of stable spot sizes, but solitary spots may exist only in a limited parametricrange, beyond which multiple spots nucleate. Repeated coalescence and nucleation leads tochaotic dynamics of spots observed computationally in Ref. [1].

We determine the steady-state structures that result from liquid-liquid demixing in afree surface film of binary liquid on a solid substrate. The considered model correspondsto the static limit of the diffuse interface theory describing the phase separationprocess for a binary liquid (model-H), when supplemented by boundary conditions at thefree surface and taking the influence of the solid substrate into account. The resultingvariational problem is numerically solved employing a Finite Element Method on an adaptivegrid. The developed numerical scheme allows us to obtain the coupled steady-state filmthickness profile and the concentration profile inside the film. As an example wedetermine steady state profiles for a reflection-symmetric two-dimensional droplet forvarious surface tensions of the film and various preferential attraction strength of onecomponent to the substrate. We discuss the relation of the results of the present diffuseinterface theory to the sharp interface limit and determine the effective interfacetension of the diffuse interface by several means.

Spatiotemporal patterns near a codimension-2 Turing-Hopf point of the one-dimensionalsuperdiffusive Brusselator model are analyzed. The superdiffusive Brusselator modeldiffers from its regular counterpart in that the Laplacian operator of the regular modelis replaced by ∂ α /∂|ξ|α , 1 < α< 2, an integro-differential operator that reflects the nonlocal behavior ofsuperdiffusion. The order of the operator, α, is a measure of the rate ofsuperdiffusion, which, in general, can be different for each of the two components. Aweakly nonlinear analysis is used to derive two coupled amplitude equations describing theslow time evolution of the Turing and Hopf modes. We seek special solutions of theamplitude equations, namely a pure Turing solution, a pure Hopf solution, and a mixed modesolution, and analyze their stability to long-wave perturbations. We find that thestability criteria of all three solutions depend greatly on the rates of superdiffusion ofthe two components. In addition, the stability properties of the solutions to theanomalous diffusion model are different from those of the regular diffusion model.Numerical computations in a large spatial domain, using Fourier spectral methods in spaceand second order Runge-Kutta in time are used to confirm the analysis and also to findsolutions not predicted by the weakly nonlinear analysis, in the fully nonlinear regime.Specifically, we find a large number of steady state patterns consisting of a localizedregion or regions of stationary stripes in a background of time periodic cellular motion,as well as patterns with a localized region or regions of time periodic cells in abackground of stationary stripes. Each such pattern lies on a branch of such solutions, isstable and corresponds to a different initial condition. The patterns correspond to thephenomenon of pinning of the front between the stripes and the time periodic cellularmotion. While in some cases it is difficult to isolate the effect of the diffusionexponents, we find characteristics in the spatiotemporal patterns for anomalous diffusionthat we have not found for regular (Fickian) diffusion.

We study molecular motor-induced microtubule self-organization in dilute and semi-dilutefilament solutions. In the dilute case, we use a probabilistic model of microtubuleinteraction via molecular motors to investigate microtubule bundle dynamics. Microtubulesare modeled as polar rods interacting through fully inelastic, binary collisions. Ourmodel indicates that initially disordered systems of interacting rods exhibit anorientational instability resulting in spontaneous ordering. We study the existence anddynamic interaction of microtubule bundles analytically and numerically. Our resultsreveal a long term attraction and coalescing of bundles indicating a clear coarsening inthe system; microtubule bundles concentrate into fewer orientations on a slow logarithmictime scale. In semi-dilute filament solutions, multiple motors can bind a filament toseveral others and, for a critical motor density, induce a transition to an ordered phasewith a nonzero mean orientation. Motors attach to a pair of filaments and walk along thepair bringing them into closer alignment. We develop a spatially homogenous, mean-fieldtheory that explicitly accounts for a force-dependent detachment rate of motors, which inturn affects the mean and the fluctuations of the net force acting on a filament. We showthat the transition to the oriented state can be both continuous and discontinuous whenthe force-dependent detachment of motors is important.

We study pattern-forming instabilities in reaction-advection-diffusion systems. Wedevelop an approach based on Lyapunov-Bloch exponents to figure out the impact of aspatially periodic mixing flow on the stability of a spatially homogeneous state. We dealwith the flows periodic in space that may have arbitrary time dependence. We propose adiscrete in time model, where reaction, advection, and diffusion act as successiveoperators, and show that a mixing advection can lead to a pattern-forming instability in atwo-component system where only one of the species is advected. Physically, this can beexplained as crossing a threshold of Turing instability due to effective increase of oneof the diffusion constants.

In a simple FitzHugh-Nagumo neuronal model with one fast and two slow variables, asequence of period-doubling bifurcations for small-scale oscillations precedes thetransition into the spiking regime. For a wide range of values of the timescale separationparameter, this scenario is recovered numerically. Its relation to the singularlyperturbed integrable system is discussed.

The discovery of nearly periodic vegetation patterns in arid and semi-arid regionsmotivated numerous model studies in the past decade. Most studies have focused onvegetation pattern formation, and on the response of vegetation patterns to gradients ofthe limiting water resource. The reciprocal question, what resource modifications areinduced by vegetation pattern formation, which is essential to the understanding ofdryland landscapes, has hardly been addressed. This paper is a synthetic review of modelstudies that address this question and the consequent implications for inter-specificplant interactions and species diversity. It focuses both on patch and landscape scales,highlighting bottom-up processes, where plant interactions at the patch scale give rise tospatial patterns at the landscape scale, and top-down processes, where pattern transitionsat the landscape scale affect inter-specific interactions at the patch scale.

A global feedback control of a system that exhibits a subcritical monotonic instabilityat a non-zero wavenumber (short-wave, or Turing instability) in the presence of a zeromode is investigated using a Ginzburg-Landau equation coupled to an equation for the zeromode. The method based on a variational principle is applied for the derivation of alow-dimensional evolution model. In the framework of this model the investigation of thesystem’s dynamics and the linear and nonlinear stability analysis are carried out. Theobtained results are compared with the results of direct numerical simulations of theoriginal problem.

We consider the stabilization of a rotating temperature pulse traveling in a continuousasymptotic model of many connected chemical reactors organized in a loop with continuouslyswitching the feed point synchronously with the motion of the pulse solution. We use theswitch velocity as control parameter and design it to follow the pulse: the switchvelocity is updated at every step on-line using the discrepancy between the temperature atthe front of the pulse and a set point. The resulting feedback controller, which can beregarded as a dynamic sampled-data controller, is designed using root-locus technique.Convergence conditions of the control law are obtained in terms of the zero structure(finite zeros, infinite zeros) of the related lumped model.

We investigate the origin of deterministic chaos in the Belousov–Zhabotinsky (BZ) reaction carried out in closed and unstirred reactors (CURs). In detail, we develop a model on the idea that hydrodynamic instabilities play a driving role in the transition to chaotic dynamics. A set of partial differential equations were derived by coupling the two variable Oregonator–diffusion system to the Navier–Stokes equations. This approach allows us to shed light on the correlation between chemical oscillations and spatial–temporal dynamics. In particular, numerical solutions to the corresponding reaction-diffusion-convection (RDC) problem show that natural convection can change the evolution of the concentration distribution as well as oscillation patterns. The results suggest a new way of perceiving the BZ reaction when it is conducted in CURs. In conflict with the common experience, chemical oscillations are no longer a mere chemical process. Within this framework the evolution of all dynamical observables are demonstrated to converge to the regime imposed by the RDC coupling: chemical and spatial–temporal chaos are genuine manifestations of the same phenomenon.