Non-smooth feedback control for Belousov- Zhabotinskii reaction-diffusion equations: semianalytical

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1 University o Wollongong Research Online Faculty o Engineering and Inormation Sciences - Papers: Part A Faculty o Engineering and Inormation Sciences 6 Non-smooth eedback control or Belousov- Zhabotinskii reaction-diusion equations: semianalytical solutions Hassan Alii The University o Dammam, hyja973@uowmail.edu.au Timothy R. Marchant University o Wollongong, tim@uow.edu.au Mark Nelson University o Wollongong, mnelson@uow.edu.au Publication Details Alii, H. Y., Marchant, T. R. & Nelson, M. I. (6). Non-smooth eedback control or Belousov-Zhabotinskii reaction-diusion equations: semi-analytical solutions. Journal o Mathematical Chemistry, 54 (8), Research Online is the open access institutional repository or the University o Wollongong. For urther inormation contact the UOW Library: research-pubs@uow.edu.au

2 Non-smooth eedback control or Belousov-Zhabotinskii reactiondiusion equations: semi-analytical solutions Abstract The Belousov-Zhabotinskii reaction is considered in one and two-dimensional reaction-diusion cells. Feedback control is examined where the eedback mechanism involves varying the concentrations in the boundary reservoir, in response to the concentrations in the centre o the cell. Semi-analytical solutions are developed, via the Galerkin method, which assumes a spatial structure or the solution, and is used to approximate the governing delay partial dierential equations by a system o delay ordinary dierential equations. The orm o eedback control considered, whilst physically realistic, is non-smooth as it has discontinuous derivatives. A stability analysis o the sets o smooth delay ordinary dierential equations, which make up the ull non-smooth system, allows a band o Hop biurcation parameter space to be obtained. It is ound that Hop biurcations or the ull non-smooth system all within this band o parameter space. In the case o eedback with no delay a precise semi-analytical estimate or the stability o the ull nonsmooth system can be obtained, which corresponds well with numerical estimates. Examples o limit cycles and the transient evolution o solutions are also considered in detail. Disciplines Engineering Science and Technology Studies Publication Details Alii, H. Y., Marchant, T. R. & Nelson, M. I. (6). Non-smooth eedback control or Belousov-Zhabotinskii reaction-diusion equations: semi-analytical solutions. Journal o Mathematical Chemistry, 54 (8), This journal article is available at Research Online:

3 Non-smooth eedback control or Belousov-Zhabotinskii reaction-diusion equations: semi-analytical solutions H.Y. Alii T. R. Marchant M. I. Nelson August 5, 6 Abstract The Belousov-Zhabotinskii reaction is considered in one and two-dimensional reactiondiusion cells. Feedback control is examined where the eedback mechanism involves varying the concentrations in the boundary reservoir, in response to the concentrations in the centre o the cell. Semi-analytical solutions are developed, via the Galerkin method, which assumes a spatial structure or the solution, and is used to approximate the governing delay partial dierential equations by a system o delay ordinary dierential equations. The orm o eedback control considered, whilst physically realistic, is non-smooth as it has discontinuous derivatives. A stability analysis o the sets o smooth delay ordinary dierential equations, which make up the ull non-smooth system, allows a band o Hop biurcation parameter space to be obtained. It is ound that Hop biurcations or the ull non-smooth system all within this band o parameter space. In the case o eedback with no delay a precise semi-analytical estimate or the stability o the ull non-smooth system can be obtained, which corresponds well with numerical estimates. Examples o limit cycles and the transient evolution o solutions are also considered in detail. Keywords mathematical modelling, reaction-diusion-delay equations, Belousov-Zhabotinskii, Hop biurcations, non-smooth eedback control Mathematics Subject Classiication () 35,37,4 Introduction Oscillatory phenomena in chemical systems have been studied, by both theoreticians and experimentalists, or many decades. The Belousov-Zhabotinskii (BZ) reaction, Bray-Liebhasky and Briggs-Rauscher systems undergo periodic concentration variations and have the added interest that these oscillations can be visualized via colour changes, see []. The BZ reaction, discovered by Belousov [] in 95, is a classical one used or understanding periodic chemical and biological relaxation oscillations. The BZ reaction has a rich history o experimental, theoretical and numerical study. The range o phenomena or which the BZ reaction, and other chemical oscillator systems, prove a useul test-bed include multi-stability, chaos, bursting, reaction-diusion patterns and waves and eedback control, see Sagues and Epstein [3] or a comprehensive review o these phenomena in the context o chemical systems. H.Y. Alii School o Mathematics and Statistics, The University o Dammam, Dammam, Eastern, 344, Saudi Arabia T. R. Marchant M. I. Nelson School o Mathematics and Applied Statistics, The University o Wollongong, Wollongong, 5, N.S.W., Australia. tim marchant@uow.edu.au

4 H.Y. Alii et al. Field et al. [4] described the chemistry o the BZ reaction and presented experimental data illustrating the sustained oscillations, rate constants or the reactions and stated the ten component reactions undamental to the system. There have been many subsequent approaches used to simpliy the original system o BZ equations. Field and Noyes [5] developed the Oregonator model, which consists o three coupled ODEs, to model the ive most important BZ reactions, while [6,7] proposed the Oregonator model ɛ du = qv uv + u( u), dt δ dv dw = qv uv + w, dt where ɛ = 4, δ = 4, q = 8 4, dt = u w, () where u, v and w are the reactant concentrations. Marchant [8] considered the Gray & Scott cubic autocatalytic model in a reaction-diusion cell. The Galerkin method was used to obtain a lower-order ODE model, as an approximation to the governing PDE system. Singularity and biurcation theory theory was then used to obtain semianalytical steady-state solutions and biurcation diagrams, together with the region o parameter space, where Hop biurcations occur. The comparison between the semi-analytical and numerical solutions o the governing PDEs was ound to be excellent. Marchant [9] extended these ideas to the Gray Scott model with Michaelis Menten decay. The Fourier Galerkin series solution method has also been used to obtain numerical solutions to a steady-state diusive BZ equation by Forbes [, ]. He ound a stripy pattern, corresponding to a standing wave and that the spatial pattern is not necessarily unique. It was also shown that small amplitude patterns are not stable but that large-amplitude patterns may be quasi-stable. Experimental studies o spatial phenomena or oscillatory chemical systems have relied on the development o new types o reactors, which allow the inlux o resh reactants without stirring the reactor contents. This has been achieved by the use o gel illed reactors (which prevents advective motion) coupled to well-stirred reactant reservoirs at the boundaries. Early experiments with gel reactors were perormed by Noszticzius et al. [] and Tam et al. [3] who reported results or the BZ reaction while later studies by Bagyan et al. [4] and Lavrova et al. [5] considered glycolytic reactions. Feedback control can be applied to chemical systems to achieve the stabilization o limit cycles and unstable steady states and also to generate chaotic behaviour. Sriram [6] studied numerical and experimental simulations o electrical eedback or the BZ reaction in a CSTR reactor. The amplitude and period o the limit cycle oscillations were increased by the eedback and it was shown that the experimental observations were accurately modelled by the Oregonator model, with a eedback term added to one o the dynamic variables. Biurcation diagrams were drawn and the eect o positive eedback on the Hop biurcation parameter region was investigated numerically. Vanag et al. [7] perormed numerical simulations o the BZ reaction. They used a three-variable model o the BZ reaction, proposed by [8], and obtained good comparisons between their simulations and experimental data. The control parameter was the inlow rate or the CSTR. They showed that there are two simple ways to modiy the transition rom chaotic behaviour, by varying the strength o the eedback and the delay response. Lipták et al. [9] considered a general open CSTR system obeying the mass action law. They proposed a class o polynomial eedback that stabilizes the system, which can then be described by a generalized Hamiltonian orm. Vanag and Epstein [] reviewed the design and control o patterns in both batch oscillators and gel reactors. Some o the varied control methods discussed include the use o initial conditions, photochemical control, periodic orcing and temperature. di Bernardo et al. [] considered biurcation theory or non-smooth piecewise continuous ODE systems. Many important applications, such as control and switching problems, impact oscillators and riction systems, are governed by such systems. They reviewed biurcation theory or steadystate solutions, which lie on discontinuity boundaries, and described the new types o instabilities which can occur in the non-smooth system. Camlibel et al. [] considered the stability o a plane piecewise smooth linear system with two dependent variables and discontinuous derivatives at the steady-state solution. They derived the conditions or the overall stability o the nonsmooth system, which relate the complex eigenvalues o the two smooth systems. Csikja et al. [3]

5 Non-smooth eedback control or BZ reaction-diusion equations 3 considered a linear piecewise aine ODE model o hysteresis. They constructed piecewise smooth limit cycle solutions and considered their stability. In this paper, we study the Oregonator model () in a reaction-diusion cell with eedback control. The eedback consists o varying the concentrations in the boundary reservoirs in response to the concentrations in the centre o the cell. A delay in the response time is also modelled. Also, to be physically realistic, the concentrations at the boundary must be positive, which requires the introduction o a modulus term in the eedback response, leading to a non-smooth system. This study has a number o aims; to illustrate that the Galerkin averaging method is useul in accurately approximating the ull reaction-diusion BZ system, to explore the eect o eedback on the BZ system, and to illustrate how approximations can be ound that predict Hop biurcations or nonsmooth systems. Moreover the Oregonator model represents a classical prototype or analyzing chemical oscillations; analytical techniques ound to be useul here can potentially be applied to the vast range o other chemical oscillators known to exist (see, or example, igure 3 o Sagues and Epstein [3]) or which the number o chemical species is very large and the corresponding sets o model equations do not enjoy the simplicity o the classical Oregonator. In Section the semi-analytical model, consisting o a set o non-smooth delay ODEs, is derived by using the Galerkin method. In Section 3 steady-state solutions are ound. In Section 4 the prediction o regions o parameters space, in which Hop biurcations occur, is considered in detail. The non-smooth system consist o eight dierent sets o smooth delay ODEs. The stability o each smooth ODE system is ound and combined. This approach gives three regions, a region in which all smooth parts o the system are stable, a region in which all are unstable, and an intermediate region where some smooth systems are stable and some are unstable. Hop biurcations or the ull non-smooth system always occur in this intermediate band o parameter space. In the case o no delay a precise prediction o the Hop biurcation parameter space, o the ull non-smooth system, is ound by considering the dominant eigenvalues and the ideas o [, ]. A good comparison between the semi-analytical model and numerical results is also obtained, or steady-state solutions, the transient evolution to the steady-state and or limit cycles. The semi-analytical model. Model equations The BZ equations in a -D reaction-diusion cell have the orm du dt = k u + dv (qv uv + u( u)), ɛ dt = k v + (w qv uv), δ () dw dt = k w + u w, u = v = w = at x, y = ±, u = u a, v = v a, w = w a, t =, (3) The system () is the BZ model proposed by [6], with Dirichlet boundary conditions (3). In - D we consider the natural simpliication o () where the concentrations in the y-direction are uniorm. The equations are in non-dimensional orm with the scaled reactant concentrations, u, v and w. It is an open system; the reactor has a permeable boundary at x, y = ±, joined to a reservoir in which the reactants have zero concentrations, see [, 3] or experimental scenarios. The parameters k,, ɛ, δ, q, deined in (), are all positive where is termed the stoichiometric actor and k the diusion coeicient. Jahnke and Winree [6] report that the range [, 4] or chemical systems. We let (u s, v s, w s ) be the steady-state concentrations at the centre o the reactor x = y =. The initial concentrations are (u a, v a, w a ). We are interested in examining the eect o eedback on the reaction-diusion cell () so consider the ollowing eedback algorithm u = H u s u(,, t τ), v = H v s v(,, t τ) (4) w = H w s w(,, t τ), at x, y = ±,

6 4 H.Y. Alii et al. where the reservoir concentrations are altered, in response to the concentrations in the cell at the centre o the cell (located at x = y = ). Many studies have been undertaken o eedback control or CSTRs, where the low rate is altered in response to the concentrations in the reactor; (4) represents an analogous orm o eedback control or a reaction-diusion cell, see [3, 4, 5]. Experimentally this can be achieved using a diusive gel coupled to a CSTR, which represents the reservoir; [3] considered a BZ reaction while [4, 5] considered a glycolytic reaction. The inert gel medium prevents convective motion but allows diusion o the chemical species. By using high low rates the concentrations in the CSTR remain close to the input values and the chemical reactions in the CSTR can be neglected. The eedback is proportional to the dierence between the transient concentrations and the steady-state values at the centre o the cell, while H is the strength o the eedback and τ is the delay response. The eedback response is zero at the steady-state so (4) does not alter the steady-state solutions o () and we investigate the eect o this eedback control on the stability o the reaction-diusion cell. Note that the concentrations must always be positive in the reservoir, hence the modulus signs on the eedback terms. The modulus terms result in a continuous eedback system but with non-smooth derivatives. A Crank Nicolson inite-dierence scheme is used to ind the numerical solutions o the governing PDE system. This implicit scheme is unconditionally stable. A ourth-order Runge Kutta scheme is used to solve the semi-analytical ODE models. Other numerical methods, such as operator splitting, exist or this class o reaction diusions equations, see or example, Ropp and Shadid [7].. The Galerkin method The semi-analytical models or () in the -D and -D geometries are ound using a Galerkin method. This method assumes a spatial structure o the concentration proiles, see [8, 8]. The Galerkin method allows the governing delay PDE to be approximated by a system o delay ODEs. In -D we use the expansion ( π ) u(x, t) = (u (t) H u s + u s u d u d ) cos x ( ) 3π + u (t) cos x + H u s + u s u d u d, ( π ) v(x, t) = (v (t) H v s + v s v d v d ) cos x ( ) 3π + v (t) cos x + H v s + v s v d v d, (5) ( π ) w(x, t) = (w (t) H w s + w s w d w d ) cos x ( ) 3π + w (t) cos x + H w s + w s w d w d. The subscript d implies a delay, that is u d = u (t τ). The trial unctions are chosen so that u + u, v + v and w + w are the concentrations at the centre o the reaction-diusion cell and so the boundary conditions at x = ± are satisied. Note that the orm o (5) is not unique; a more symmetric orm or could be chosen or quadratic expressions or the spatial proiles could be used, but the level o accuracy o the method is usually independent o the orms o the basis unctions used. The PDEs () are not satisied exactly, but the ree parameters in this expansion are obtained by evaluating averaged versions o the governing equations, weighted by the basis unctions cos ( πx) and cos ( 3 πx). Then, the ODEs du dt = π [ kπ Hπ 4H + π 4 u ( 8 ɛπ 5 u v u u + 7 ) 35 u v

7 Non-smooth eedback control or BZ reaction-diusion equations 5 ( 8 ɛπ 5 u v u u + 8 ) ] 3 u v πqv πu + HM, dv dt = π [ kπ Hπ 4H + π 4 v ( 8 δπ 3 u v u v + 7 ) 35 u v 8 5δπ u v + δ w ] δ qv + HM, ] dw π = [ kπ dt Hπ 4H + π 4 w + u w + HM 3, (6) du 3π = [ 9kπ dt 4H + 3π 4 u ( 8 ɛπ 5 u 8 9 u u v + 7 ) 35 u v + ( 8 ɛπ 9 u v 7 35 u v 44 ) ] 35 u u + πqv + πu + HM 4, dv dt = 3π [ 9kπ 4H + 3π 4 v ( 8 δπ 5 u v u v + 7 ) 35 u v + 8 9δπ u v + δ w ] δ qv + HM 5, ] dw 3π = [ 9kπ dt 4H + 3π 4 w + u w + HM 6, are obtained, where the M i are given in the Appendix. The series in (5) has been truncated ater two terms. The number o terms that are used in the truncated series represents a trade-o between the accuracy and complexity o the semi-analytical solution. It is ound that a two-term method gives suicient accuracy without excessive expression swell. A one-term solution is ound by letting each o u, v and w equal to zero. As the M i includes modulus terms, the ODEs (6) represent a non-smooth system (as there are discontinuous derivatives). Moreover, as there are three dierent modulus terms the non-smooth system (6) is composed o eight dierent smooth ODE systems. For the -D geometry, the expansion ( ) ( ) u(x, y, t) = (u (t) H u s + u s u d u d ) cos πx cos πy ( ) ( ) ( ) ( ) u (t) cos πx cos πy + u (t) cos πx cos πy + H u s + u s u d u d, ( ) ( ) v(x, y, t) = (v (t) H v s + v s v d v d ) cos πx cos πy ( ) ( ) ( ) ( ) v (t) cos πx cos πy + v (t) cos πx cos πy + H v s + v s v d v d, (7) ( ) ( ) w(x, y, t) = (w (t) H w s + w s w d w d ) cos πx cos πy ( ) ( ) ( ) ( ) w (t) cos πx cos πy + w (t) cos πx cos πy + H w s + w s w d w d. is used, which also satisies the relevant -D geometry boundary conditions in (). Symmetry implies that two o the terms have the same coeicient. Averaging using the weights cos ( πx) cos ( πy) and cos ( πx) cos ( 3 πy), gives the ollowing ODE model du π = [ kπ dt Hπ 6H + π u ( 64 ɛπ 9 u v + 8 ) 45 u v π qv

8 6 H.Y. Alii et al. ( 876 ɛπ dv dt = π Hπ 6H + π dw dt du dt 575 u v u v u u + 64 ) ] 9 u π u + HN, [ kπ v ( 64 δπ 9 u v + 8 ) 45 u v + π qv δπ u v 8 45δπ u v + δ w + HN π = [ kπ Hπ 6H + π 3π = 6H + 3π ɛπ + qv ɛ dv dt = 3π 6H + 3π dw dt [ 5kπ u ɛπ ], ] w + u w + HN 3, (8) ( u v u v ) 575 u v ) ( u u u u v u π u ] + HN 4, ( 64 [ 5kπ v δπ 45 u v u v ) 575 u v δπ u v δ qv + ] δ w + HN 5, 3π ] = [ 5kπ 6H + 3π w + u w + HN 6. where N i are given in the Appendix. 3 Steady-state solutions In this section we study the steady-state solutions o the semi-analytical model or the -D and -D geometries. In order to ind steady-state solutions, we let u(t) = u s, v(t) = v s and w(t) = w s in the ODE models, which reduces them to sets o transcendental equations. At the steady state, the eedback terms, involving H, are all zero. The steady-state solutions or the -D and -D geometries are ound by solving the transcendental equations using a root-inding routine rom the Maple sotware package. For all igures in this section the diusion coeicient k = and ɛ, δ, q are given in (). Figure shows the steady-state reactant concentrations u, v and w versus stoichiometric actor, or the -D geometry. Shown are the one and two-term semi-analytical and numerical solutions at the centre o the domain, x =. There is a unique steady-state solution or the reactant concentrations. The igures show u and w decrease as increases, beore approaching a minimum at large. However, the curve or v increases as increases. Hence, or large, u and w are near zero while v increases linearly. There is an excellent comparison between the two-term semianalytical and numerical solutions, with less than.5% error or all values o stoichiometric actors up to =. The concentration versus response curve or the non-diusive BZ system (the classical Oregonator model ()), is qualitatively similar to igure with a unique steady-state solution (see Field and Noyes [5]). Figure shows the steady-state reactant concentrations u, v and w versus x, or the -D geometry. Shown are the one-term and two-term semi-analytical and numerical solutions o the governing PDE. The stoichiometric actor = 8. The solution or the reactant concentration u has two humps, or the two-term semi-analytical and numerical solutions, while the other reactants have a single central peak. The two-term solution can model the non-central peak accurately while the one-term solution cannot. The comparison between the two-term semi-analytical and numerical solutions is excellent, while the one-term solution is reasonably accurate at the centre o the domain, x =. These behaviours are qualitatively similar to the concentration proiles in cubic auto-catalytic reactions, see [8].

9 Non-smooth eedback control or BZ reaction-diusion equations 7 u.5 v w Fig. (color online) The steady-state reactant concentrations u, v and w versus stoichiometric actor, at x =, or the -D geometry. Shown are the one-term (dashed blue line) and two-term (black solid line) semi-analytical solutions and numerical (red dotted line) solutions o the governing PDEs. 3 u x -3 v 8 w x Fig. (color online) The steady-state reactant concentrations u, v and w versus x, or the -D geometry. The parameter is = 8. Shown are the one-term (dashed blue line) and two-term (black solid line) semi-analytical solutions and numerical (red dotted line) solutions o the governing PDEs. x

10 8 H.Y. Alii et al. u.5 v. w Fig. 3 (color online) The steady-state reactant concentrations u, v and w at x = y = versus stoichiometric actor, or the -D geometry. Shown are the one-term (dashed blue line) and two-term (black solid line) semi-analytical solutions and numerical (red dotted line) solutions o the governing PDEs (). u x - v x w x Fig. 4 (color online) The steady-state reactant concentrations u, v and w versus x, or the -D geometry. The parameter space is = 4. Shown are the one-term (dashed blue line) and two-term (black solid line) semi-analytical solutions and numerical (red dotted line) solutions o the governing PDEs. x

11 Non-smooth eedback control or BZ reaction-diusion equations 9 Figure 3 shows the steady-state reactant concentrations u, v and w versus the stoichiometric actor, at the centre o the domain, x = y =, or the -D geometry. As in the -D case, the one and two-term semi-analytical and numerical solutions are shown. The curves in this case are qualitatively similar to igure but the dierence between the semi-analytical and numerical solutions is slightly larger, or large values. There is a dierence o 8% between the two-term semi-analytical and numerical solutions, up to = 5. Semi-analytical solutions or a -D geometry generally have slightly larger errors than those in a -D geometry, see [8,9,8]. Figure 4 shows the steady-state reactant concentrations u, v and w versus x, or the -D case. Shown are the one-term and two-term semi-analytical solutions and numerical solutions o the governing PDE (). The stoichiometric actor = 4 and a slice or y = are shown. The numerical solutions or v show a latter concentration proile, which is more challenging to model with a series o trial unctions. The two-term semi-analytical solution is superior to the one-term solution, but extra terms in the trial unctions would be needed to urther improve the comparison. 4 Stability analysis and Hop biurcations In this section we discuss the stability o the BZ model in order to determine the parameter regions in which Hop biurcations points occur. Standard texts in biurcation theory and dynamic systems describe the theory o Hop biurcations or delay systems, see [3, 3]. Stability theory is well understood or systems o smooth ODEs but is less well developed or the non-smooth ODE system considered here, see [] or a review o current theories. We use two approaches or the analysis o our semi-analytical non-smooth ODE system, which is comprised o eight sets o smooth ODEs. The irst approach leads to the prediction o a band o parameter space, in which Hop biurcations or the ull non-smooth system occur. The second approach is to consider the dominant eigenvalues o the system and the ideas o [, ] who constructed a hybrid stability condition or a non-smooth system. This approach leads to a precise prediction o the region in which Hop biurcations occur, or the case o eedback with no delay. 4. Theoretical considerations The -D model (6) and -D model (8) consist o ODEs or u i, v i and w i. We write the modulus terms, o the orm p, as p sgn(p) in the ODEs, and separately consider the stability o each smooth part o the system. As there are three dierent sign unctions in the equations we get eight dierent sets o smooth ODEs. The smooth ODEs are expanded in a Taylor series about the steady-state solution. Let u i = u is + ɛce µt, v i = v is + ɛge µt, w i = w is + ɛme µt, i =, ɛ, (9) and substitute (9) into the systems (6) and (8), and then linearize around the steady state. The eigenvalues o the Jacobian matrix describes the growth o small perturbations in the system. This gives the characteristic equation F (µ) = m + im = or the decay rate µ = w + iw. Hop biurcation points may occur at points where µ is purely imaginary. Here, the Hop biurcation points or the -D and -D cases are ound by solving the system o equations i =, i =,,... 6 and w = m = m =, () where the i are the steady-state versions o the -D (6) and -D (8) models. For our system, this gives eight dierent regions in which Hop biurcations occur. Combining these results gives three regions: a region in which all sets o smooth ODEs o the system are stable; a region in which all smooth parts are unstable; and an intermediate region in which some smooth parts are stable and some are unstable. Hop biurcations or the ull non-smooth ODE system occur in this intermediate region o the parameter space.

12 H.Y. Alii et al. To help resolve the exact parameter space in which Hop biurcations occur the system considered in [] provides some useul insights. They considered a linear system ẋ = { A x i c T x, A + x i c T x, () where the eigenvalues o A ±, λ = ς ± ± iω ±, (ω ± > ), are complex, the steady state solution x = and A ± R, c = R. Then the discontinuous system () is stable i S < where S = ς+ ω + + ς ω. () An insight into this hybrid stability condition can be seen by considering the solutions x = de ς±t cos(ω ± t), (3) o the two ODEs ẋ = A ± x. The solution stays in its portion o the phase plane space or time t = π ω, beore the components o x change sign. Hence, the growth or decay during this time is ± ς ± π ω, which leads to the hybrid stability condition S <. ± For our system without delay, the one-term model consists o three ODEs and the two-term model consists o six ODEs. In the parameter region where the Hop biurcation occurs the long time behaviour is dominated by a complex conjugate pair o eigenvalues. The other eigenvalues have real parts that are more negative so can be ignored as they only aect the dynamics o the system at short times. Hence the dynamics o each smooth part o our system without delay is governed by a complex conjugate pair o eigenvalues, as in the linear system (). We now examine the perturbation (3) in more detail. I the components o the initial perturbation d are all positive (termed the ppp case), then the evolution o the perturbation (3) can only switch the system between the ppp and nnn (all negative components) cases. Hence in this case the perturbation to the non-smooth system will be stable i the dominant eigenvalues or the smooth ODE systems nnn and ppp satisy S <. There are three other pairs o ODE systems (the npn and pnp, ppn and nnp, pnn and npp cases) or which similar switching can occur, so or each pair, the eigenvalues must also satisy S <. The neutrally stable points are given by S = or each pair o eigenvalues and points in the (, k) plane which satisy S = are ound using a bisection method. However, or the case with delay the dynamics is no longer dominated by a single set o complex conjugate eigenvalues and a precise analytical description o the parameter space is not available. The reason or this is that the eigenvalue equation is not a cubic equation but a transcendental equation with an ininite number o solutions. 4. Hop biurcation regions and limit cycles or the -D geometry In this section, the Hop biurcation parameter regions are obtained or -D geometry in two cases: eedback with no delay and eedback with delay. In each case, the Hop biurcation curves o the eight dierent smooth systems are ound and curves plotted which divide the parameter space into three regions: one where no Hop points occur; one where all the smooth systems predict Hop points; and one region that has mixed stability. For the case with no delay precise semi-analytical estimates o the Hop biurcation parameter region are also obtained. Figure 5 represents the region o Hop biurcations in the k plane or the -D geometry with no delay. The parameters are H =. and τ =. Figure 5(a) shows the two-term semianalytical and numerical solutions. The semi-analytical predictions are o the stability o the sets o smooth ODEs, which divides the parameter space into three regions: one in which all smooth ODEs are stable, one in which they are all unstable; and one o mixed stability. The numerical Hop biurcation points lie inside the band o mixed stability. The Hop biurcation curves corresponding to the eight sets o smooth ODE systems do not intersect each other so the curves, which separate the parameter space into the three regions are smooth. Figure 5(b) shows the numerical Hop

13 Non-smooth eedback control or BZ reaction-diusion equations k (a) 4 3 k (b) Fig. 5 (color online) The regions o the -k plane in which Hop biurcations can occur or the -D geometry. The parameter values are H =. and τ =. Shown in (a) are: the two-term semi-analytical (blue and red line) and numerical solutions (black dots), Shown in (b) are: numerical points o Hop biurcation or PDEs (black dots) and the theoretical prediction o S = (red crosses). 7 6 Region C 5 Region B k 4 3 Region A Fig. 6 (color online) The regions o the -k plane in which Hop biurcations can occur or the -D geometry. Shown are the two-term semi-analytical (blue and red line) and numerical (black dots) solutions at parameter values H =. and τ =.

14 H.Y. Alii et al.. k.. 3 (a)- Feedback no delay (τ = ) k (b)- Feedback delay (τ = ) Fig. 7 (color online) The regions o the -k plane in which Hop biurcations can occur, or dierent values o H and small values o k. Shown are two-term semi-analytical solutions in the -D geometry or our dierent values o H: H = (black solid line), H = (red dotted line), H = (dashed blue line) and H = 4 (green dotted line). biurcation points and the theoretical predictions o S =, rom the two-term model. The neutral stability predictions, or each pair o ODEs, are the same, to graphical accuracy hence only one set o crosses are plotted on the igure. It can be seen that the theoretical estimates are very accurate with an error o less than 5% or all choices o. It is also worth noting that [, 4] or real chemical systems which corresponds to the lhs o the igures. For smaller the inner and outer stability regions lie close together, except or some variations at large k. Figure 6 represents the region o Hop biurcations in the k plane or the -D geometry with delay eedback control. The parameters are τ = and H =.. Shown are the two-term semi-analytical and numerical solutions. Again the numerical Hop biurcation points occur in the theoretical band o mixed stability. In the case with delay the smooth ODE Hop biurcation curves do intersect so the composite curves shown in Figure 6 are not smooth, unlike those in igure 5 or the no delay case. The stability ormula () does not generalise to our system with delay and a precise estimate o the Hop biurcation region is not available. However or smaller values o the diusion coeicient and the stoichiometric actor, k and 4 (which corresponds to chemically realistic values), the intermediate region o mixed stability is small and hence precise estimates o the region in which Hop biurcations occur, are available. For example, on the lhs o igure 6, at k =, the transition rom region C to region A occurs or =.65 and.7 or

15 Non-smooth eedback control or BZ reaction-diusion equations 3 u.5 v.5. w. 3 4 t Fig. 8 (color online) The reactant concentrations u, v and w at x = versus t or the -D geometry. The two-term semi-analytical solution (black solid line) and numerical solution (red dotted line) are shown. The parameters are τ =, H =., u a = v a = w a =., k = and =.6. u x - v x + w x Fig. 9 (color online) The reactant concentrations u, v and w at x = versus t or the -D geometry. The two-term semi-analytical solution (black solid line) and numerical solution (red dotted line) are shown. The parameters are τ =, H =., u a = v a = w a =., k = and = 3.

16 4 H.Y. Alii et al. v w..4 u Fig. (color online) The limit cycle when = 3, H =., k = and τ =. The two-term (black solid line) semi-analytical and numerical solutions (red dotted line) are shown. the numerical, and two-term semi-analytical solutions respectively. The two predictions are close, with only a 3% dierence. Figure 7 represents the Hop biurcation region in the k plane, or the -D geometry, or small values o the diusion coeicient. Shown is the two-term semi-analytical solution or H =,, and 4. (a) is the no delay τ = case and (b) is or τ =. For small values o the diusion coeicient the stability predictions rom the eight smooth systems are the same to graphical accuracy, so there is no band o mixed stability and a precise Hop prediction is obtained. We can see that appropriately chosen values o H can stabilize or destabilize regions o parameter space. For igure 7(a), the case with no delay, we can see that as the eedback parameter H increases, the region o instability is decreasing. Figure 7(b) shows that as H increases, the region o instability increases. The eect o increasing H is stabilizing or small τ and destabilizing or large τ; the critical value o τ at which the behaviour changes is τ.5. Figures 8 and 9 show the reactant concentrations u, v and w at x =, versus t or the -D geometry. The parameters are u a = v a = w a =., k =, H =. and τ =, with =.6 (in region C o igure 6) or igure 8 and = 3 (in region A o igure 6) or igure 9. The two-term semi-analytical and numerical solutions are shown. For igure 8 =.6 and the solution evolves to a steady state, with u s.47, v s.49 and w s.5 as the time becomes large, ater some initial relaxation oscillations. The comparisons between the two-term semi-analytical and numerical solutions is excellent with only a maximum % error at the steady state. For igure 9 = 3 and a periodic solution occurs. It can be seen that the maximum concentration o v is two orders o magnitude greater than that o u and w. This example represents a challenging test case or the semi-analytical solution method but the results prove to be highly accurate. The numerical amplitudes o the limit cycle or the reactant concentrations u, v and w are.6, 85.9 and., respectively. These values are very close to the two-term semi-analytical results o.6, 85.6 and., respectively. The errors in the two-term semi-analytical values are less than.5%. Figure is a view o the 3-D phase space. The two-term semi-analytical and the numerical solutions in the -D geometry are shown. The parameters are = 3, k = and τ =. The numerical period o the limit cycle or the reactants is 3., while the two-term semi-analytical period o the limit cycle is.97, a dierence o only %. The two-term semi-analytical approximation is airly close to the numerical solution over the whole parameter space and the semi-analytical limit cycle has many quantitative similarities to the numerical solution. 4.3 Hop biurcation regions and limit cycles or the -D geometry Figure represents the region o Hop biurcations in the k plane or the -D geometry with no delay. The parameters are H =. and τ =. Figure (a) shows the two-term semi-analytical and numerical solutions. The stability o the smooth ODE systems give a band o parameter space in which Hop biurcations occur. As in the -D, there are three regions and the numerical Hop biurcations occur in the band region which has mixed stability. As or the -D case the composite

17 Non-smooth eedback control or BZ reaction-diusion equations k (a).5.5 k (b) Fig. (color online) The region o the -k plane in which Hop biurcations can occur or the -D geometry. The parameters are H =. and τ =. Shown in (a) are the two-term semi-analytical (blue and red line) and numerical solutions (black dots), Shown in (b) are the numerical points o Hop biurcation (black dots) and the theoretical prediction S = (red crosses). curves in igure (a) are smooth. Figure (b) shows numerical Hop biurcations and theoretical predictions o S = or the our pairs o ODE systems rom the two-term model (which are all the same to graphical accuracy). Also, as or the case o -D geometry the prediction is excellent, with an error o less than 9% between them at all choices o. As or the -D geometry the inner and outer stability regions lie close together or small (which corresponds to chemically realistic values), except or some variations at large k. Figure represents the region o Hop biurcations in the k plane, or the -D geometry with delay. The parameters are τ = and H =.. Shown are the two-term semi-analytical and numerical solutions. As in the -D geometry, this igure shows three dierent regions: one stable, one unstable, and one o mixed stability. It can be seen that the numerical Hop biurcation points occur in the region o mixed stability and are close to the border with the unstable region. Hence nearly all o smooth ODE systems need to be unstable or the ull non-smooth system to destabilize. As or the -D geometry case the intermediate region is small or smaller values o the diusion coeicient k. One the let hand side o the igure at k =, or the transition rom region C to region A, the Hop points are given by =.9 and 3., or the numerical and two-term semi-analytical solutions respectively, which is a 7% dierence.

18 6 H.Y. Alii et al. 4 k Region C Region B Region A Fig. (color online) The region o the -k plane in which Hop biurcations can occur or the -D geometry. Shown are the two-term semi-analytical (blue and red line) and numerical (black dotted) solutions at parameters H =. and τ =. Figure 3 represents the Hop biurcation region in the k plane or the -D geometry, or small values o the diusion coeicient. Shown is the two-term semi-analytical solution or H =,.5, and. (a) is the no delay τ = case and (b) is or τ =. As in the -D case, it can see that appropriately chosen values o H can stabilize or destabilize regions o parameter space. For igure 3(a) shows the case with eedback with no delay and we can seen that as the eedback parameter H increases, the region o instability decreases. The igure 3(b) shows the case with delay eedback at τ =. In this case as H increases the region o instability grows. As in -D, the eect o increasing H or small τ is stabilizing and destabilizing or large τ; the critical τ.6. Figures 4 and 5 show the reactant concentrations u, v and w at x = y = versus t or the -D geometry. The parameters are u a = v a = w a =. and τ = with =.7 (rom region C in igure ) or igure 4 and = 3.5 (rom region A in igure ) or igure 5. The two-term semianalytical and numerical solutions are shown. For igure 4 =.7 and the solution evolves to a steady state, with u s.47, v s.57 and w s.9 as the time becomes large. The comparison between the numerical and the two-term semi-analytical solutions shows a 8% dierence in the steady state. For igure 5 = 3.5, so periodic solutions occur. The numerical amplitudes o the limit cycle or u, v and w are.5, 34.4 and.7 respectively. These values are airly close to the two-term values o.58, 4.9 and.8 respectively. As in the -D case, the two-term semi-analytical method is again accurate, with errors o less than 8%. Figure 6 is a view o the 3-D phase space. The two-term semi-analytical and the numerical solutions or the PDEs in the -D geometry are shown. The parameter choices are = 3.5, k = and τ =. The numerical and two-term semi-analytical periods o the limit cycle o the reactant at concentrations u, v and w are.55 and.49 respectively. The errors in the two-term semi-analytical values are less than 4%. As or the -D case, the semi-analytical solution or the ODEs model is close to the numerical solution o the PDEs, in this 3-D parameter space. 5 Conclusion This paper has presented semi-analytical solutions or the BZ model in a reaction-diusion cell with eedback control or both the -D and -D geometries. The Galerkin method was used to approximate the governing delay PDEs by a system o delay ODEs. A key eature o the problem is the non-smooth nature o the eedback control and the challenges this presents or analytical investigation. For the no delay case the consideration o the dominant eigenvalues together with a hybrid stability condition allows an accurate semi-analytical prediction o the Hop biurcation region to be ound. For cases with eedback delay a band o parameter space is ound in which the

19 Non-smooth eedback control or BZ reaction-diusion equations 7. k (a)- Feedback no delay (τ = ) k. 4 6 (b)- Feedback delay (τ = ) Fig. 3 (color online) The region o the -k plane in which Hop biurcations can occur, or dierent values o H and small values o k. Shown are two-term semi-analytical solutions in the -D geometry or three dierent examples o H: H = (black solid line), H =.5 (red dotted line) and H = (dashed blue line). Hop points occur. The eect o eedback is stabilizing or small delay and destabilizing or large delay. Examples o stable and unstable limit cycles were obtained with a good comparison between semi-analytical and numerical solutions. This work illustrates the useulness o the Galerkin averaging technique, or reaction diusion equations and also contributes to the understanding o stability or non-smooth systems with multiple delay terms. Future work could involve extending the method to other classes o oscillatory chemical systems, or which the model equations are much more complex than the Oregonator model. Also the results illustrate a range o interesting behaviours that can occur in the reaction-diusion cell with boundary eedback control. Hopeully this study will motivate new experimental work using this type o eedback scenario. Acknowledgement: The authors would like to thank an anonymous reeree or their useul comments

20 8 H.Y. Alii et al..6 u.3.8 v.4. w 3 4 t Fig. 4 (color online) The reactant concentrations u, v and w at x = y = versus t or the -D geometry. The twoterm semi-analytical solution (black solid line) and numerical solution (red dotted line) are shown. The parameters are τ =, H =., u a = v a = w a =., k = and =.7. 6 u x - v x + w x Fig. 5 (color online) The reactant concentrations u, v and w at x = y = versus t or the -D geometry. The twoterm semi-analytical solution (black solid line) and numerical solution (red dotted line) are shown. The parameters are τ =, H =., u a = v a = w a =., k = and = 3.5. t

21 Non-smooth eedback control or BZ reaction-diusion equations 9 v 4.4 w.8.3 u Fig. 6 (color online) The limit cycle when = 3.5, H =., k = and τ =. The two-term (black solid line) semi-analytical and numerical solutions (red dotted line) are shown. Appendix: Expressions or the semi-analytical ODEs. This appendix presents relevant expression or or the semi-analytical models. The M i or the -D model (6) are M = H ɛ u dv d + H ɛ u d H 3ɛπ u dv d ɛ u d + 6u 3ɛπ u d + 6u 5ɛπ u d + 8u 3ɛπ v d + kπ 4 u d q ɛ v d + 8v 5ɛπ u d v ɛ u d H 3ɛπ u d u ɛ u d u ɛ v d, M = H δ u dv d + 8u 3δπ v d H 3δπ u dv d + 8v 5δπ u d u δ v d 4q δπ v d + 8v 3δπ u d + kπ 4 v d v δ u d δ w d + 8u 5δπ v d + 4 δπ w d + q δ v d, M 3 = 4 π u d u d + w d + kπ 4 w d 4 π w d, M 4 = 8v 5ɛπ u d + 8u 5ɛπ v d ɛπ u d 4 3ɛπ u d + 4H 5ɛπ u dv d + 4H 5ɛπ u d + 7u 35ɛπ v d u ɛ u d + 6u 5ɛπ u d + 7v 35ɛπ u d v ɛ u d u ɛ v d + 4q 3ɛπ v d, M 5 = 4q 3δπ v d 4 3δπ w d + 7v 35δπ u d + 7u 35δπ v d + 4 5δπ u dv d + 8u 5δπ v d + 8v 5δπ u d u δ v d v δ u d, M 6 = 4 3π w d 4 3π u d. The N i or the -D model (8) are N = 64u 9ɛπ v d + 8v 45ɛπ u d + 6q ɛπ v d + 8u 45ɛπ u d + H ɛ u d + 64v 9ɛπ u d q ɛ v d + kπ u d + 56u 45ɛπ u d + 6 ɛπ u d 8H 9ɛπ u dv d + H ɛ u dv d 8H 9ɛπ u d u ɛ u d + 8u 9ɛπ u d v ɛ u d u ɛ v d ɛ u d, N = 8H 9δπ v du d + H δ v du d + 64v 9δπ u d + kπ v d + 8v 45δπ u d + 64u 9δπ v d v δ u d 6q δπ v d + q δπ v d + 6 δπ w d δ w d + 8u 45δπ v d u δ v d, N 3 = kπ w d 6 π w d + 6 π u d + w d u d, N 4 = 6q 3ɛπ v d u ɛ u d + 988u 575ɛπ v d v ɛ u d + 76H 45ɛπ u d + 76H 45ɛπ u dv d + 64v 45ɛπ u d + 988v 575ɛπ u d + 64u 45ɛπ v d + 876u 575ɛπ u d + 8u 45ɛπ u d u ɛ v d 6 3ɛπ u d, N 5 = 6q 3δπ v d v δ u d + 64v 45δπ u d + 988u 575δπ v d + 76H 45δπ u dv d + 64u 45δπ v d 6 3δπ w d + 988v 575δπ u d u δ v d, N 6 = 6 3π w d 6 3π u d. The delay terms are deined as u d = u s + u s u (t τ) u (t τ), w d = w s + w s w (t τ) w (t τ). v d = v s + v s v (t τ) v (t τ),

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