A model of the thermochemical reactor with well mixing is considered. An impact of random noise on the attractors of this model is studied in zones of saddle-node bifurcations with transitions from mono- to bimodal dynamics. Phenomena of the stochastic excitability and noise-induced transitions are demonstrated. Critical values of the noise intensities corresponding to the onset of generation of the large-amplitude stochastic oscillations are estimated

Bashkirtseva, I.

Pisarchik, A. N.

Ryashko, L.

IOP Conference Series Materials Science and Engineering

Bashkirtseva, Irina

Pisarchik, Alexander

Ryashko, Lev

Archivo Digital UPM

Multistability and stochastic phenomena in a randomly forced thermochemical system

I Bashkirtseva

A N Pisarchik

L Ryashko

Crossref

Institutional repository of Ural Federal University named after the first President of Russia B.N.Yeltsin

1Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd1234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012011 doi:10.1088/1757-899X/192/1/012011Multistability and stochastic phenomena in arandomly forced thermochemical systemI Bashkirtseva1, A N Pisarchik2 and L Ryashko11 Ural Federal University, Lenina, 51, Ekaterinburg, 620000, Russia2 Center for Biomedical Technology, Technical University of Madrid, 28223 Pozuelo deAlarcon, Madrid, SpainE-mail: lev.ryashko@urfu.ruAbstract. A model of the thermochemical reactor with well mixing is considered. An impactof random noise on the attractors of this model is studied in zones of saddle-node bifurcationswith transitions from mono- to bimodal dynamics. Phenomena of the stochastic excitability andnoise-induced transitions are demonstrated. Critical values of the noise intensities correspondingto the onset of generation of the large-amplitude stochastic oscillations are estimated.1. IntroductionIt is well known that in nonlinear systems even a weak noise can generate the wide varietyof stochastic phenomena [1-4]. Such effects as the stochastic resonance, noise-induced chaos,stochastic bifurcations are observed in various nonlinear models in different domains of science[5-9]. In particular, the mathematical models of chemical kinetics [10, 11] demonstrate thecomplex dynamic regimes extremely sensitive to noise. A nonlinear model of thermochemicalreactor [12-14] is one of such examples with excitable stochastic dynamics.In the present paper, we study this model in zones of saddle-node bifurcations where thetransitions from mono- to bimodal behavior are observed. An impact of random noise on theattractors of this model is investigated. We focus on two variants of the saddle-node bifurcationwhen the additional attractor (equilibrium or cycle) appears.In Section 2, the dynamic regimes of the deterministic model of the thermochemical reactorwith well mixing are shortly discussed.In Section 3, an influence of random noise on attractors of this model is studied both inthe mono- and bistable zones. In zones of stable equilibria near the saddle-node bifurcation,a phenomenon of the stochastic excitability is shown. We demonstrate how increasing noisetransforms the system to the regime of large-amplitude stochastic oscillations. The noise-inducedtransitions between coexisting attractors (equilibrium–equilibrium and equilibrium–cycle) arediscussed.2. Deterministic thermochemical modelConsider a deterministic systemx˙ =√y(−x exp(− δy)+ p (1− x))y˙ = 23q√y(x exp(− δy)+ r (1− y)) , (1)21234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012011 doi:10.1088/1757-899X/192/1/012011where x and y are the dimensionless concentration and temperature, respectively; parametersp, q, r and δ are positive. The system (1) models the dynamic processes in the thermochemicalreactor with well mixing [12].1 1.5 2 2.5−0.010  p=0.4p=0.45438p=0.5zyFigure 1. Plots of the function z(y).a)0.7 0.8 0.91234xy0 10 20 30 400.811.2txb)0.7 0.8 0.9123xy0 10 20 30 400.70.80.9txFigure 2. Phase trajectories and time series of deterministic system (1) with δ = 5, r =0.07, q = 40 and a) p = 0.4, b) p = 0.5.In the present paper, we study a behavior of this system in zones of the transition frommono- to bistability. Here, we consider a case when an appearance of the additional attractor31234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012011 doi:10.1088/1757-899X/192/1/012011occurs as a result of the saddle-node bifurcation. In system (1), two variants of the saddle-nodebifurcation can be observed. One of them (variant A) is connected with the birth of the pair ofthe stable and unstable equilibria. The second one (variant B) is a result of the appearance ofstable and unstable cycles.At first, consider the variant A of the saddle-node bifurcation for the fixed δ = 5, r =0.07, q = 40 and varying p.The coordinates of equilibria of model (1) satisfy the following systemz(y) = 0, x =pp+ exp(− δy) ,wherez(y) =pp exp(δy)+ 1+ r(1− y).In Fig. 1, plots of the function z(y) are shown for three values of parameter p. As can be seen,system (1) has a single equilibrium for p = 0.4, and has three equilibria for p = 0.5. The valuep∗ = 0.45438 where the plot of z(y) is tangent to the y-axis corresponds to the saddle-nodebifurcation. In Fig. 2, the phase portraits and time series of system (1) for p = 0.4 and p = 0.5are presented.The system (1) with p = 0.4 is monostable: all trajectories tend to the single stableequilibrium x¯ = 0.9561, y¯ = 1.2508 (see Fig. 2a).For p = 0.5, system (1) is bistable with two stable equilibria M1(0.7544, 2.7544) andM2(0.9634, 1.2612), separated by the saddle equilibrium M3(0.8719, 1.9153) (see Fig. 2b). Thebasins of attraction of equilibria M1 and M2 are separated by the stable manifold of the saddleM3.0.5 11234xy0 100 200 300 4001234tyFigure 3. Phase trajectories and time series of stochastic system (2) with δ = 5, r = 0.07, q =40, p = 0.4543 for ε = 0.01 (blue) and ε = 0.1 (red).3. Stochastic modelTo study an impact of random disturbances on the dynamics of the thermochemical system, wewill use the following stochastic model:x˙ =√y(−x exp(− δy)+ p (1− x))+ ε1ξ1(t),y˙ = 23q√y(x exp(− δy)+ r (1− y))+ ε2ξ2(t).(2)41234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012011 doi:10.1088/1757-899X/192/1/012011a)0.7 0.8 0.9 1123yx 0 100 2000.51txb)0.5 11234yx 0 100 2000.51txFigure 4. Phase trajectories and time series of the stochastic system (2) with δ = 5, r =0.07, q = 40, p = 0.5 for a) ε = 0.01 (blue), ε = 0.03 (red); b) ε = 0.08 (green).10−2 10−1012345εyFigure 5. Random states of the stochastic system (2) δ = 5, r = 0.07, q = 40, p = 0.5.Here, ξ1,2(t) are the standard Gaussian uncorrelated processes, and ε1,2 are the noise intensities.In what follows, we put ε1 = ε2 = ε.3.1. Stochastic excitabilityConsider an influence of noise on the monostable system with p = 0.4543 that is close to thesaddle-node bifurcation point p∗. In Fig. 3, for two values of the noise intensity, the randomtrajectories of system (2) starting from the stable equilibrium are shown.For a weak noise (ε = 0.01), the random trajectories are localized in a small vicinity of the51234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012011 doi:10.1088/1757-899X/192/1/012011equilibrium, and system (2) exhibits the small-amplitude stochastic oscillations (SASO). Whenthe noise intensity increases and exceeds some threshold value, along with SASO, the large-amplitude stochastic oscillations (LASO) appear (see Fig. 3 for ε = 0.1). So, an alternation ofSASO and LASO specifies a stochastic excitability of the stochastic system (2) close to p∗.3.2. Bistability and noise-induced transitionsConsider now an impact of noise on the system in the bimodal zone where two stable equilibriaM1 and M2 coexist.a)0.7 0.8 0.91.52xy0 1000 20000.70.80.9xtb)0.7 0.8 0.91.52xy0 1000 20000.70.80.9xtc)0.7 0.8 0.91.522.5xy0 1000 20000.70.80.9xtFigure 6. Phase trajectories and time series of the deterministic system (2) with δ = 5, r =0.0622, p = 0.2475 for a) q = 39.6, b) q = 40, c) q = 43.5.In Fig. 4, for three values of the noise intensity, the random trajectories of system (2) startingfrom the stable equilibrium M1 are shown.For a weak noise (ε = 0.01), the random trajectories are still reside in a small vicinity of theequilibrium M1. When the noise intensity increases, a transition from the basin of attraction of61234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012011 doi:10.1088/1757-899X/192/1/012011a)0.7 0.8 0.91.52xy0 1000 20000.80.9xtb)0.7 0.8 0.91.52xy0 1000 20000.80.9xtFigure 7. Phase trajectories and time series of the stochastic system (2) with δ = 5, r =0.0622, p = 0.2475 for a) q = 39.6 and ε = 0.0001 (blue), ε = 0.001 (green); b) q = 40 andε = 0.0001 (blue), ε = 0.0003 (green).M1 to M2 occurs (see Fig. 4a for ε = 0.03). So, the noise induces a transition of the consideredthermochemical system to another alternative noisy equilibrium regime.A further increase of noise generates the repetitive transitions between basins of attraction ofM1 and M2. As a result, the large-amplitude oscillations are observed (see Fig. 4b for ε = 0.08).Some details of such changes in stochastic dynamics are well seen in Fig. 5 where the randomstates of the stochastic system (2) versus noise intensity ε are plotted. Here, the critical valuesof noise intensities corresponding to the onset of generation of the large-amplitude stochasticoscillations can be estimated.3.3. Stochastic dynamics near the bifurcation of the cycle birthConsider now the variant B of saddle-node bifurcation with the appearance of stable and unstablecycles.Following [14], we put δ = 5, r = 0.0622, p = 0.2475 and consider the dynamics dependingon parameter q.In Fig. 6, for three values of parameter q, the phase portraits and time series of deterministicsystem are shown. For q = 39.6, the system is monostable with a stable focus as a singleattractor, and all trajectories tend to it (see Fig. 6a). For q = 40, the system is bistable anddemonstrates a coexistence of the stable equilibrium and stable cycle. Their basins of attractionare separated by the unstable cycle (see Fig. 6b). For q = 43.5, the system is again monostablewith a stable cycle as a single attractor (see Fig. 6c).It is interesting that near the saddle-node bifurcation, in the zone of monostability where astable equilibrium is a single attractor, system (2) is excitable as well. Indeed, for a small noise,71234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012011 doi:10.1088/1757-899X/192/1/01201110−4 10−32εyFigure 8. Random states of the stochastic system (2) with δ = 5, r = 0.0622, p = 0.2475, q =39.6.system (2) exhibits SASO with random trajectories located near the stable equilibrium (seeFig. 7a) for ε = 0.0001). With increasing noise, an alternation of SASO and LASO is observed(see Fig. 7a) for ε = 0.001).Such alternation in the zone of bistability (see Fig. 7b) is also observed.Some details of such noise-induced transitions to LASO are shown in Fig. 8 where the randomstates of stochastic system (2) versus noise intensity ε are plotted for q = 39.6.As a conclusion, we have to note that the nonlinear model under consideration demonstratesa wide variety of noise-induced effects which must be taken into consideration in the engineeringproblems.4. References[1] Moss F and McClintock P V E 1989 Noise in Nonlinear Dynamical Systems (Cambridge: Cambridge UniversityPress)[2] Horsthemke W and Lefever R 1984 Noise-Induced Transitions (Berlin: Springer)[3] Fedotov S, Bashkirtseva I and Ryashko L 2002 Phys. Rev. E 66 066310[4] Bashkirtseva I, Chen G and Ryashko L 2010 Int. J. Bifurcat. Chaos 20 1439[5] Anishchenko V S, Astakhov V V, Neiman A B, Vadivasova T E and Schimansky-Geier L 2007 NonlinearDynamics of Chaotic and Stochastic Systems. Tutorial and Modern Development (Berlin/Heidelberg:Springer-Verlag)[6] Lindner B, Garcia-Ojalvo J, Neiman A and Schimansky-Geier L 2004 Phys. Rep. 392 321[7] Bashkirtseva I, Ryashko L and Schurz H 2009 Chaos Solitons Fractals 39 72[8] Ryashko L and Bashkirtseva I 2011 Phys. Rev. E 83 061109[9] Pisarchik A N and Feudel U 2014 Phys. Rep. 540 167[10] Field R J and Burger M 1985 Oscillations and Traveling Waves in Chemical Systems (New York: Wiley)[11] Jakobsen H A 2014 Chemical Reactor Modeling: Multiphase Reactive Flows (Berlin: Springer)[12] Nowakowski B and Kawczynski A L 2005 J. Phys. Chem. A 109 3134[13] Kawczynski A L and Nowakowski B 2008 Phys. Chem. Chem. Phys. 10 289[14] Kolbus A, Nowakowski B and Kawczynski A L 2013 Eur. Phys. J. B 86 262AcknowledgmentsThis work was supported by RFBR (16-08-00388).

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Multistability and stochastic phenomena in a randomly forced thermochemical system

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