Spatially ‘‘chaotic’’ solutions in reaction-convection models and their bifurcations to moving waves Olga Nekhamkina and Moshe Sheintuch Department of Chemical Engineering, Technion—Israel Institute of Technology, Technion City, Haifa 32 000, Israel ~Received 14 June 2001; revised manuscript received 26 March 2002; published 12 July 2002! The emergence of stationary spatially multiperiodic or even spatially chaotic patterns is analyzed for a simple model of convection, reaction, and conduction in a cross-flow reactor. Spatial patterns emerge much like dynamic temporal patterns in a mixed system of the same kinetics. Moving waves are formed in an unbounded system but they are transformed into stationary spatially inhomogeneous patterns in a bounded system. The sequence of period doubling bifurcations is determined numerically. The incorporation of a slow nondiffusing inhibitor leads to chaotic spatiotemporal patterns. DOI: 10.1103/PhysRevE.66.016204 PACS number~s!: 05.45.2a, 82.40.Bj, 82.40.Ck I. INTRODUCTION The increasing interest in reaction-convection-diffusion systems was recently recognized by assigning it a new PACS number ~82.40.Ck! that distinguishes it from that of the well studied reaction-diffusion systems. Reaction-convection- diffusion systems are typically described by a system of the form: Lx t 1Vx z 2Dx zz 5f~ x! , ~1! where x is the vector of state variables, L5diag$ L i % , V 5diag$ V i % , D5diag$ D i % , and L i , V i , and D i are the capaci- ties, velocities, and diffusivities of the various state vari- ables. Reactants can be fed to the reactor either through one port or may be distributed along the reactor via many ports ~to which we refer as cross flow!. Cross-flow conditions can also be achieved by feeding through a membrane or through a preceding reaction. In the cross-flow reactor we can find a homogeneous solution @ f( x s ) 50 # . The technological advan- tages of such a reactor were argued in Ref. @1#. Stationary pattern formation mechanism in diffusive- reactive systems was suggested in the pioneering work of Turing @2#. The diffusive Turing instability applies to a two- variable system when the inhibitor diffuses sufficiently faster than the activator. This mechanism was able to account for certain patterns in chemistry and biology @3,4#, but largely was unable to induce patterns in liquid-phase oscillatory re- action where the reactant diffusivities are usually of similar magnitudes, or in catalytic systems, in which the diffusivity of the activator is typically larger than the diffusivity of the inhibitor. In the presence of convection a stationary pattern forma- tion mechanism has been recently suggested by Kuznetsov et al. @5#. The behavior of spatially distributed system cru- cially depends on whether the instability is convective or absolute. An instability is called convective if a small pertur- bation induces a local growth from the spatially uniform so- lution, but disturbances propagate as a wave packet and are advected out of the system. An instability is termed absolute if a localized initial perturbation gives rise to growing am- plitudes at all points in space. The distinction between abso- lute and convective instabilities in unbounded systems de- pends on the choice of the coordinate system and with appropriate transformations to the moving coordinate we can convert one instability to another. The problem becomes definite if we consider a bounded ~or semibounded! system with a boundary condition that is fixed at one end. Perturba- tions applied at the boundary can either penetrate the system, which then acts as a nonlinear filter and a spatial amplifier, or be damped. The pattern-formation mechanism suggested in Ref. @5# is based on the amplification of the stationary per- turbations in the convectively unstable systems. Such pertur- bations can be introduced by the stationary boundary condi- tions that differ from the steady state solution. This mechanism accounts for stationary patterns in several recent studies: ‘‘flow distributed oscillations’’ ~FDO! were exten- sively investigated in Ref. @6# for the Brusselator model, in Ref. @7# for a Gray-Scott kinetics, in Ref. @8# for the CDIMA reaction, and in Ref. @9# for the Oregonator models and in our previous studies of cross-flow reactors @10–12# with a single Arrhenius first order reaction. The mechanisms above can also be classified according to the activator/inhibitor parameter ratios, V 1 / V 2 , D 1 / D 2 , and L 1 / L 2 , which define the emergence of stationary patterns. In the FDO stationary patterns emerge even when D 1 5D 2 @6,8# and it is claimed therefore that these patterns are not due to the Turing mechanism. Diffusion is important for the station- arity of these patterns and the stationary solution breaks down with D 2 50. In recent works @10,11# we showed that stationary spatially periodic patterns emerge in a bounded system even when D 2 50 provided that the activator capac- ity L 1 is sufficiently large ~for catalytic nonisothermal sys- tems x 1 is typically the temperature and the heat capacity is large, L 1 @1). Other studies have focused on spatiotemporal patterns of Eq. ~1!. Most notably the well studied differential flow in- duced chemical instability mechanism @13,14# is connected with the separation of variables due to different convection rates ( V 1 ,V 2 ). The studies above were devoted to formation of stationary and moving spatially period-one patterns. In this work we present a general approach for designing stationary patterns of desired complexity. The steady state solutions of the sys- tem ~1! are governed by a system of ordinary differential equations ~ODEs! written in the dimensionless form as x z 2Px zz 5F( x), where P5diag$ Pe i 21 ,Pe i 5LV i / D i % , and L is the reactor length. In the limit case Pe i →‘ the spatially PHYSICAL REVIEW E 66, 016204 ~2002! 1063-651X/2002/66~1!/016204~5!/$20.00 ©2002 The American Physical Society 66 016204-1