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Static and Dynamic Behavior of Autocatalytic Replicators in Reactor Networks
Abstract
The static and dynamic behavior of the autocatalytic reaction R + 2P → 3P with decay P → D is studied in networks of coupled continuous stirred tank reactors (CSTRs). Numerical bifurcation studies of the system are performed, resulting in rich steady-state bifurcation structures with multiple steady states and isolas. The heterogeneity of the networks is influenced by the number of reactors as well as the network topology. It is shown that the number of steady states of the network increases with heterogeneity, thereby allowing those autocatalytic species to exist in the network that would normally not exist in the homogeneous environment of a single CSTR. Spatial patterns of stable steady states are evident in reactor networks. Dynamic simulation studies are performed to illustrate the transition from one stable state configuration to another or from stable steady states to periodic regimes.
... We use autocatalytic reactions in the network to formulate surrogates for predator-prey, virus propagation in a distributed population, multiple species of animals that rely on the same resources, or chemical manufacturing problems. Reactor networks exhibit highly complex behavior, with multiple steady state operating regimes, and have a large pool of candidates for manipulated variables [2]. Recent work on multiple reactor configurations with cubic autocatalytic reactions has demonstrated a rich spectrum of static and dynamic behavior [2]. ...
... Reactor networks exhibit highly complex behavior, with multiple steady state operating regimes, and have a large pool of candidates for manipulated variables [2]. Recent work on multiple reactor configurations with cubic autocatalytic reactions has demonstrated a rich spectrum of static and dynamic behavior [2]. The topography of interconnected CSTR networks has been shown to drastically affect the steady state bifurcation structure of the system [2], [3]. ...
... Recent work on multiple reactor configurations with cubic autocatalytic reactions has demonstrated a rich spectrum of static and dynamic behavior [2]. The topography of interconnected CSTR networks has been shown to drastically affect the steady state bifurcation structure of the system [2], [3]. Spatial inhomogeneity of the network can be increased by increasing the number of reactors in the network as well as manipulating the interconnection flow rates of the network. ...
Controlling reactors in distributed manufacturing processes producing different grades of a product requires intelligent reconfiguration strategies. Agent-based approaches are ideal for such cases, since they can provide flexible, robust, and emergent solutions during dynamically changing process conditions. A hierarchical, agent-based system with local and global control agents is developed to control networks of interconnected chemical reactors. This paper proposes a multi- layered, multiagent framework based on a decentralized approach for the supervision of grade transitions in autocatalytic reactor networks. The values for the manipulated variables are chosen to give the least disturbance to the system. The case studies show that the approach is successful in controlling the reactor network and being able to keep the desired grade even after the need of shutting down some of the reactors.
... In addition, the control structure for these systems tend to be distributed themselves, in the form of complex arrays of sensors, controllers, and actuators. Networks of interconnected continuous stirred tank reactors (CSTRs), for example, exhibit highly complex behavior, with multiple steady state operating regimes, and have a large pool of candidates for manipulated variables [20]. ...
... Recent work on multiple reactor configurations with cubic autocatalytic reactions has demonstrated a rich spectrum of static and dynamic behavior [2]. The topography of interconnected CSTR networks has been shown to drastically affect the steady state bifurcation structure of the system [20,22]. Spatial inhomogeneity of the network can be increased by increasing the number of reactors in the network as well as manipulating the interconnection flow rates of the network. ...
... In a single reactor configuration, only one autocatalytic species is able to survive at a time in a stable manner [2]. Furthermore, numerical experiments suggest that individual CSTRs in networks are capable of hosting only a single dominant species, while other competing species may be present only in trace quantities [20]. Consequently, if the control objective calls for one species to be replaced with another, a nonlinear control scheme must be used. ...
Control of spatially distributed systems is a challenging problem because of their complex nature, nonlinearity, and generally high order. The lack of accurate and computationally efficient model-based techniques for large, spatially distributed systems leads to challenges in controlling the system. Agent-based control structures provide a powerful tool to manage distributed systems by utilizing (organizing) local and global information obtained from the system. A hierarchical, agent-based system with local and global controller agents is developed to control networks of interconnected chemical reactors (CSTRs). The global controller agent dynamically updates local controller agent’s objectives as the reactor network conditions change. One challenge posed is control of the spatial distribution of autocatalytic species in a network of reactors hosting multiple species. The multi-agent control system is able to intelligently manipulate the network flow rates such that the desired spatial distribution of species is achieved. Furthermore, the robustness and flexibility of the agent-based control system is illustrated through examples of disturbance rejection and scalability with respect to the size of the network.
... In addition, the control structure for these systems tend to be distributed themselves, in the form of complex arrays of sensors and actuators. Networks of interconnected continuous stirred tank reactors (CSTRs), for example, exhibit highly complex behavior, with multiple steady state operating regimes, and have a large pool of candidates for manipulated variables (Tatara, Birol, Teymour, & Cinar, 2004). Most techniques for the control of nonlinear distributed processes focus on distributed parameter systems and involve mathematically complex model reduction and controller synthesis methodologies (Christofides, 2001). ...
... Recent work on multiple reactor configurations with cubic autocatalytic reactions has demonstrated that spatial heterogeneity enlarges the boundaries of species survival (Birol, Parulekar, & Teymour, 2002). The topography of interconnected CSTR networks has been shown to drastically affect the steady state bifurcation structure of the system (Tatara et al., 2004). Spatial inhomogeneity of the network can be increased by increasing the number of reactors in the network as well as manipulating the interconnection flow rates of the network. ...
... where j represents the neighboring reactor, thus j = 2 if i = 1 and j = I − 1 if i = I. Since the complexity of the system grows geometrically with the number of species and the number of reactors in the system (Tatara et al., 2004), for a network size of I > 3 with two or more species, analytical solutions become practically intractable, although it should be noted that a single trivial steady state (r i = 1, p in = 0) exists for all i for every combination of model parameter values. This trivial steady state is always stable and will always pose a threat to control efforts as it represents total extinction of the autocatalytic species in the system. ...
Spatially distributed systems such as reactor networks hosting multiple autocatalytic species demonstrate a rich spectrum of complex behavior. From a control systems perspective, spatially distributed systems offer a difficult control challenge because of their distributed nature, nonlinearity, and high order. Furthermore, manipulation of the network states may require simultaneous control actions in different parts of the system and may need transients through several operating regimes to achieve the desired operation. The lack of accurate and computationally efficient model-based techniques for large, spatially distributed systems results in complications in controlling the system, either in disturbance rejection or changing the operational regimes of the system. A hierarchical, agent-based control structure is presented whereby local control objectives may be changed in order to achieve the global control objective. The performance of the control system is demonstrated for several global objectives. The challenge posed is to control the spatial distribution of autocatalytic species in a network of five reactors hosting two species using the interaction flow rates as the manipulated variables. The multi-agent control system is able to effectively explore the parameter space of the network and intelligently manipulate the network flow rates such that the desired spatial distribution of species is achieved.
... An alternative approach is based on a hierarchical agent-based system with local and global control structures (Tatara et al., 2005c;Tatara et al., 2005b) that has been demonstrated on a network of interconnected continuous stirred tank reactors (CSTRs). Reactor networks exhibit highly complex behavior with multiple steady state operating regimes and have a large pool of candidates for manipulated variables (Tatara et al., 2004). ...
... The performance of the agent-based control architecture is demonstrated in a case study to control the distribution of autocatalytic species in a network model (Tatara et al., 2004) of 49 (7x7 grid) reactors hosting three autocatalytic species using the interaction flow rates as the manipulated variables. The species that populate the reactor network are characterized by identical growth and death rates, such that one species does not have an unfair advantage over the others. ...
Large-scale spatially distributed systems provide control challenges because of their nonlinearity, spatial distribution and generally high order. The control structure for these systems tend to be both discrete and distributed. A layered control structure interfaced with complex arrays of sensors and actuators provides a flexible supervision and control system that can deal with local and global challenges. An adaptive agent-based control structure is presented whereby local control objectives may be changed in order to achieve the global control objective. Information is shared through a global knowledge environment that promotes the distribution of ideas through reinforcement. The performance of the agent-based control approach is illustrated in a case study where the interaction front between two competing autocatalytic species is moved from one spatial configuration to another. The multi-agent control system is able to effectively explore the parameter space of the network and intelligently manipulate the network flow rates such that the desired spatial distribution of species is achieved.
... The steady-state bifurcation diagram of the resource concentration for a system of four reactors with species growth rate k r ) 25, species death rate k d ) 0.1, and feed flow rate f ) 0.002 is shown in Figure 2 for both the bidirectional and unidirectional cases with the interconnection flow rate as the bifurcation parameter. Bifurcation diagrams for the network average concentrations are shown in Figure 3. Extensive bifurcation studies on reactor networks of various sizes and configurations are available 52 and, as such, will not be reproduced here. The effects of the manipulating interconnection flow rate of the network serve to vary the spatial heterogeneity of the network. ...
... That is, for a two-CSTR bidirectional network with interconnection flow rate g 2 , there exists an equivalent two-CSTR unidirectional network with interconnection flow rate g 1 , when g 2 ) 2g 1 . 52 Instead, the steady states for the unidirectional case terminate at closely packed limit points with respect to g, disconnected from the single-CSTR states. The resulting effect on the dimension measure for unidirectional networks at high interaction flow rates can be observed in two ways: (1) the dimension for the unidirectional case (Figure 8b,d) is greater than that of the bidirectional case (Figure 8a,c), and (2) the point at which the dimension (and complexity) drops to 0 for the unidirectional case is independent of the network size for large networks. ...
Systems with high steady-state multiplicity and rich dynamic behavior are difficult to investigate using conventional reductionist methods. A network of more than five reactors hosting cubic autocatalytic reactions may potentially have more than 102 steady states and many distinct dynamic regimes, all for the same parameter set. This paper discusses how the static complexity of such systems can be measured to give a holistic picture. To achieve this, stochastic simulations were performed to statistically determine the bifurcation structure of the system, and the gathered information is summarized using a measure akin to fractal dimension. With this measure, the growth of static complexity is investigated as a function of the network size.
... Once the final configuration is found, the second sub problem is to decide on the actions to be taken in the correct order to move the system from its current state to the final state. Earlier works demonstrated autocatalytic reactions in CSTR networks to simulate population dynamics, multiple species of organisms that compete on same resources, or chemical manufacturing problems (Tatara et al., 2004;Tatara et al., 2005a;Tatara et al., 2005b). Controlling the spatial distribution of autocatalytic species that compete for the same resources in a network of reactors can be achieved by simultaneous manipulation of interconnection flow rates within the system. ...
Agent-based control structures provide flexible and emergent solutions to complex nonlinear problems benefiting from properties such as modularity, adaptability, scalability and robustness. One such problem is product grade transitions in distributed process. The framework proposed earlier (Tetiker, 2006a) is extended by adding several layers of agents to control species percentage distribution in autocatalytic reactor networks. A deadlock detection layer is implemented to detect and solve conflicting cases between local controller agents. An auctioning mechanism is employed to promote competition between local controller agents leading to emergent solutions satisfying global constraints. The proposed architecture performed successfully to change the species percentage distributions without specifying the final configuration.
... In a single reactor configuration, only one autocatalytic species is able to survive at a time in a stable manner (Birol and Teymour, 2000). Furthermore, numerical experiments suggest that individual CSTRs in networks are capable of hosting only a single dominant species, while other competing species may be present only in trace quantities Tatara et al., 2003b). Consequently, if the control objective calls for one species to be replaced with another, a nonlinear control scheme must be used. ...
From a control system perspective, spatially distributed systems offer challenges because of their distributed nature, nonlinearity, and high order. In addition, the control structure for these spatially distributed networks combine discrete and distributed components, in the form of complex arrays of sensors and actuators. Manipulation of the network states may require simultaneous control actions in different parts of the system and may need transients through several operating regimes to achieve the desired operation. A hierarchical, agent-based control structure is presented whereby local control objectives may be changed in order to achieve the global control objective. The performance of the hierarchical agent-based control approach is illustrated in a case study where the interaction front between two competing autocatalytic species is moved from one spatial configuration to another.The multi-agent control system is able to effectively explore the parameter space of the network and intelligently manipulate the network flow rates such that the desired spatial distribution of species is achieved.
... Recent work on multiple reactor configurations with cubic autocatalytic reactions has demonstrated that spatial heterogeneity enlarges the boundaries of species survival (Birol et al., 2002). Furthermore, detailed analysis has shown that networks of reactors with autocatalytic replicators produce highly complex bifurcation structures and that the number of steady states increases exponentially with size of the system (Tatara et al., 2003). With the autocatalytic reaction scheme, larger networks permit more steady states and spatial combinations thereof than smaller networks. ...
Several recent studies on autocatalytic reactions in single and coupled continuous stirred-tank reactor (CSTR) networks have demonstrated a rich spec-trum of complex behavior. From a control systems perspective, the operating regime of a CSTR network can be manipulated by changing the flow rates between the reactors. Systems of more than one CSTR require multiple controllers and may need transients through several operating regimes to achieve the desired operation. This may require a hierarchical control structure whereby local control objectives can change dynamically in order to achieve the global control objective of the system. An agent-based control system is used to observe and control various aspects of a CSTR network. The primary focus of analyzing complex emergent behavior is demonstrating methods or combinations of methods that influence the behavior and capabilities of the agent-based control system. Simulation studies illustrate scenarios of interest, especially conditions that lead to emergent behavior.
... In a single reactor configuration, only one autocatalytic species is able to survive at a time in a stable manner [8]. Furthermore, numerical experiments suggest that individual CSTRs in networks are capable of hosting only a single dominant species, while other competing species may be present only in trace quantities [6,9]. Consequently, if the control objective calls for one species to be replaced with another, a nonlinear control scheme must be used. ...
Large-scale spatially distributed systems provide a unique and difficult control challenge because of their nonlinearity,
spatialdistribution and generally high order. The control structure for these systems tend to be both discrete and distributed
as well and contain discrete and continuous elements. A layered control structure interfaced with complex arrays of sensors
and actuators provides a flexible supervision and control system that can deal with local and global challenges. An adaptive
agent-based control structure is presented whereby local control objectives may be changed in order to achieve the global
control objective. Information is shared through a global knowledge environment that promotes the distribution of ideas through
reinforcement. The performance of the agent-based control approach is illustrated in a case study where the interaction front
between two competing autocatalytic species is moved from one spatial configuration to another. The multi-agent control system
is able to effectively explore the parameter space of the network and intelligently manipulate the network flow rates such
that the desired spatial distribution of species is achieved.
Autocatalytic phenomena in chemical kinetics originate from various reasons, such as acid-catalytic effects of reaction products or change of the physical properties of the reaction mixture. The impact of autocatalytic kinetics on the chemical reactor design is crucially important leading to deviations from standard design rules. The impact of autocatalytic effects on first, second and nth order kinetics was investigated for batch, plug flow, backmixed, axial dispersion and recycled plug flow reactors was analyzed by classical approach and numerical simulations. Efficient numerical strategies were developed for the different reactor models. The results showed how an optimal degree of backmixing (Péclet number) and optimal recycle ratio can be determined for tubular reactors and how the reactor volume can be minimized for specific cases. Generic examples as well as hydrolysis of alkyl formate were considered as case studies.
There is a trend in the process industries towards the use of multiple small reconfigurable production units rather than a single large unit to increase production, product range and operation flexibility. Monitoring, supervision and control of these processes is difficult because traditional methods typically do not handle well the increased heterogeneity, nonlinearity and complexity. In this work, statistical multiblock process monitoring techniques and an agent- based diagnosis and control methodology are integrated. The performance of this combined monitoring-diagnosis-control system is illustrated with a case study using a network of spatially distributed reactors.
From a control system perspective, spatially distributed systems offer challenges because of their distributed nature, nonlinearity, and high order. in addition, the control structure for these spatially distributed networks combine discrete and distributed components, in the form of complex arrays of sensors and actuators. Manipulation of the network states may require simultaneous control actions in different parts of the system and may need transients through several operating regimes to achieve the desired operation. A hierarchical, agent-based control structure is presented whereby local control objectives may be changed in order to achieve the global control objective. The performance of the hierarchical agent-based control approach is illustrated in a case study where the interaction front between two competing autocatalytic species is moved from one spatial configuration to another. The multi-agent control system is able to effectively explore the parameter space of the network and intelligently manipulate the network flow rates such that the desired spatial distribution of species is achieved.
In this paper, we show that heterogeneous networks can be configured and reconfigured to operate in flexible spatial patterns. The concept is demonstrated by using a square grid of continuous stirred tank reactors with nearest neighbor interactions and hosting multiple cubic autocatalytic species. Asymmetry is introduced in the system by allowing for different feed flow rates in the reactors. A heterogeneous system offers expanded phase space, improved stability, and coexistence characteristics. The advantages of asymmetry are studied using bifurcation diagrams, continuation studies, and dynamics.
Control of spatially distributed systems is a challenging problem because of their complex nature, nonlinearity, and generally high order. Agent-based control structures provide a powerful tool for managing distributed systems by utilizing local and global information obtained from the system. A hierarchical, agent-based system with local and global control agents is developed to control networks of interconnected chemical reactors hosting multiple autocatalytic species. The global controller agent dynamically updates the objective of local control agents as the reactor network conditions change. The case illustrated in this paper is to change the dominant species in one CSTR by modifying feed and interconnection flow rates with the constraint of shortest path possible, which causes the least amount of changes in the whole network. The agent-based system and the reactor networks are implemented using the agent-based system development framework RePast
Supervision of distributed manufacturing processes producing different grades of a product requires intelligent reconfiguration strategies during grade transition phases to minimize off-spec production. Agent-based approaches are ideal for such problems and they provide flexible, robust, and emergent solutions during dynamically changing process conditions. Three different multi-layered, multi-agent frameworks are proposed for the supervision of grade transitions in autocatalytic reactor networks. The first framework is the centralized framework and it is useful for small-scale grade transitions where only a small region of the network needs to be reconfigured. Alternatively, the other two frameworks use a decentralized approach. The first decentralized framework implements genetic algorithms and the second one uses self-organizing heuristics and auctions for large-scale grade transitions. The case studies demonstrate that as the complexity of the reconfiguration problem increases, decentralized solutions perform more efficiently.
In this paper, the design and implementation of an agent-based framework to safeguard an auto-catalytic reactor network against external disturbances is discussed. The autocatalytic reactor network producing a robust, host species is attacked by an unmeasured, un-sustained disturbance of a non-robust invading species. The invading species adversely affects host species and manifests different behavioural responses depending on its magnitude and location of injection. The agent framework is employed to detect and counteract the invasion. The agent architecture consists of the physical layer and multiple invasion detection and local operator agents. Based on the operation state of the process, the detection agent raise a variety of flags (for example, “normal operation”, “severe invasion” etc.), whereas the operator agents take control actions based on the flags raised by the detection agent. The agent knowledge in the form of rule-based heuristics designed based on process knowledge gained through simulation under open-loop. The agent knowledge base consists of several parameters which affect their behaviours. Here, we propose a methodology to optimize the parameters critical to agent design. With this method, the agent system evolves in its structure, which facilitates more accurate responses.
Controlling the individual reactors of a chemical reactor network producing different grades of a product requires intelligent reconfiguration strategies. Agent-based approaches are ideal for such distributed manufacturing processes, since they provide flexible, robust, and emergent solutions under dynamically changing process conditions. This paper proposes a multi-layered, multi-agent framework based on a decentralized online learning approach for the supervision of grade transitions in autocatalytic reactor networks. The values for the manipulated variables and the path to the target reactor are determined to give the least disturbance to the system. Case studies illustrate the performance of the approach in managing grade transition and disturbance rejection in a reactor network.
A master equation approach is used to study the influence of internal fluctuations on the dynamics of three excitable thermochemical systems exhibiting continuous as well as discrete changes of temperature. The systems differ by the types of excitability. The dependences of the relative deviations from mean values of the interspike intervals and escape times from the stable stationary state on the size of the systems calculated from simulations of stochastic trajectories exhibit minima, which testify to the appearance of resonance phenomena. An explanation for the appearance of the stochastic resonances is presented.
It is well known that the properties of polymeric materials depend strongly upon their chemical structure. Other more specific factors that may be related to the chemical structure also determine the macroscopic behaviour of such materials, namely the relative position of the different segments of the polymeric chain, the molecular architecture (molecular weight distribution, branching, copolymer organisation, cross-linking extent, etc.), the crystalline environment and the pressure/temperature conditions. All these factors have a common impact in the material: they are strongly correlated to the mobility on the molecular level. That is why a huge amount of work has been devoted to the study of translational/rotational mobility that occurs within the polymeric chains. This review is intended to provide a brief survey on such kinds of mobilities, how they can be studied and what are their main characteristics. Examples on systems studied in our groups will be provided, obtained by dielectric and mechanical spectroscopies and differential scanning calorimetry. It will be mainly focused on molecular motions that occur in the solid phase (i.e., to temperatures up to the rubbery plateau). The dynamics in blends or copolymers will be avoided here, as they would deserve a special discussion in their own context. Special attention will be paid to the glass transition and the mobility that occurs below and above it. The dynamics that are observed in peculiar systems, such as semi-crystalline or liquid crystalline polymers, will be addressed.
Intricate patterns of wave propagation are exhibited in a chemical reaction-diffusion system with spatiotemporal feedback. Wave behavior is controlled by feedback-regulated excitability gradients that guide propagation in specified directions. Waves interacting with boundaries and with other waves are observed when interaction terms are incorporated into the control algorithm. Spatiotemporal feedback offers wide flexibility for designing and controlling wave behavior in excitable media.
The paradigm of cubic autocatalytic replicators with decay in coupled isothermal continuous stirred tank reactors is selected as a model to study complex behavior in population dynamics of sexually reproducing species in a heterogenous environment. It is shown that, even a setup with single species in two coupled environments may have regions in parameter space that result in chaotic behavior, hence segregation in the environment causes complexity in the system dynamics. Furthermore, partitioning is found to lead to emergence phenomena exemplified by steady states not obtainable in the equivalent homogeneous system. These phenomena are illustrated through case studies involving single or multiple species. Results show that the coupled environments can host species, that would not survive should the coupling be removed.
The dynamic behavior of the continuous stirred tank reactor is analysed and classified for a variable reactor residence time. Although earlier work, treating the bifurcation to limit cycles and steady states with changing Damköhler number, yields a complete description of the problem, the evolution of multiple steady states and limit cycles is much more bizarre as the reactor residence time varies. In addition, it is the reactor residence time which is most easily varied experimentally so that the present results are more readily compared with experiment. Plots are given to show the influence of system parameters on the reactor behavior.
Multiplicity, stability and dynamics for autocatalytic reactions of the type A⋯ → rR + ⋯ with overall rate expression rA = kCAmCRn in a continuous stirred tank reactor (CSTR) are studied. Exact multiplicity criteria are defined in the parameter space. Stability analysis shows that no periodic oscillation is possible for the system. When multiplicity occurs, some minimum of R present initially in the reactor is required in order for the high steady state to be achieved. Loci of transient extreme for product are also investigated. Multiplicity and uniqueness criteria are further compared with reported experimental data in literature.
The types of dynamic behavior possible for a single first order reaction carried out in a stirred tank reaction are classified according to values of the parameters and plots in parameter space used to define the various possibilites. Analytic criteria are developed which predict the existence and stability character of limit cycles as a function of the system parameters. The types of dynamic behavior predicted are illustrated by numerical computation of the temperature and concentration trajectories. Several kinds of new jump phenomena for this problem are observed and explained in terms of the parameter space plots. The theory and analysis methods can be easily generalized to other types of reaction systems.
In the present work we report the measurement of stochastic resonance in the minimal-bromate (MB) reaction. The chemical reaction is placed in an excitable focal steady state in a CSTR. When a weak periodic signal together with noise is superimposed on the focus, the Hopf bifurcation is crossed and the system responds with irregular spikes. The periodic signal alone is not sufficient to cross the threshold. An optimal noise level exists at which the periodic signal is optimally detected via the spiking behavior of the reaction. This is done through the signal-to-noise ratio or the interspike histogram that pass through a maximum at stochastic resonance. For numerical simulations we use the Noyes-Field-Thomson (NFT) model which is known to agree with the kinetics of the MB reaction.
In this project, parallel general-purpose software for two classes of mathematical problems has been developed. PVODE is a portable solver for ordinary differential equation systems, based on robustmathematical algorithms, and targeted at large systems on parallel machines. It is the parallel extension of the earlier sequential solver CVODE. A related solver called KINSOL has been developed for systems of nonlinear algebraic equations. KINSOL was first developed as a sequential solver, on a design that permitted extending it to a parallel version with fairly minimal additions. Both PVODE and KINSOL are being used within a parallel version of the tokamak edge plasma model UEDGE. KINSOL is also being applied in the ParFlow groundwater flow model to solve a nonlinear pressure equation.
Electrical coupling of chemical oscillators is shown to be a versatile and rapid method to study the propagation of action potentials (spikes, oscillations) using three, four, and six electrically coupled continuous flow stirred tank reactors (CSTR). Starting with a prepared excitable steady state close to a SNIPER bifurcation in the Belousov−Zhabotinsky (BZ) reaction, conditions are specified for the initiation and propagation of these action potentials in a linear array of reactors and in a cyclic array. In the cyclic array, traveling waves of action potentials were measured for the first time. The coherence and incoherence of two, three, and four chemical BZ oscillators which were mutually electrically coupled was also studied. Numerical simulations with the seven-variables Montanator are in fair agreement with the experimental results.
Dynamical regimes arising due to mutual interactions of oscillatory and excitatory modes of the Belousov−Zhabotinskii (BZ) reaction in a two-array and linear and circular three-arrays (with different arrangements of intrinsic connections) of identical continuous stirred tank reactors (CSTRs) coupled via symmetric passive diffusion/convection mass exchange were studied both experimentally and by numerical simulations. The coupling strength among individual CSTRs and the threshold of excitability of the BZ reaction mixture were varied systematically. Firing numbers (vectors) were used for classification of observed oscillatory−excitatory modes. Full spectra of firing numbers ranging from 0 to 1 were detected in all CSTR arrays investigated in experiments. The numbers of oscillators and excitators, threshold of excitability, and the way of coupling and coupling strengths within the array are principal factors affecting firing patterns of the array. Numerical simulations with the dimensionless three-variable Oregonator based model of the BZ reaction predict qualitatively well dynamical regimes encountered in experiments. Noisy coupling among the individual CSTRs due to hydrodynamical fluctuations is suggested to explain some of the observed differences.
We present experiments and simulations of two mutually mass-coupled biochemical oscillators represented by the nonlinear peroxidase-oxidase reaction. The uncoupled oscillators show simple period-1 (P1) oscillations of different frequencies for different values of the oxygen concentration in the gas stream. A phase diagram is established where the ratio of the natural frequencies is plotted versus the mass exchange rate. For each frequency ratio four regimes of behavior have been observed for an increasing mass exchange rate: (I) two independent and uncorrelated P1 oscillations; (II) quasiperiodicity in one and P1 oscillations in the other reactor; (III) quasiperiodicity in both reactors; (IV) periodic in-phase synchronization of both reactors. We use the correlation coefficient, the frequencies of the system, and the phase difference between the two cells to characterize the four regimes. A new feature of this work is the experimental observation of an “infinite” modulation period at the transition from the quasiperiodic region to the limit cycle that can be explained by a collision of a torus with a saddle-node bifurcation of a limit cycle (SNIPER bifurcation of codimension two). Simulations using two coupled four-variable DOP models give very good agreement with the experimental results and support our interpretation of the dynamical behavior and the observed bifurcations.
Synchronizations of oscillatory regimes of the Belousov−Zhabotinskii (BZ) reaction in a circular array of three identical CSTRs coupled via symmetric passive diffusion/convection mass transfer were studied experimentally. Stability of symmetric and asymmetric phase-shifted oscillatory regimes with respect to variations of the coupling strength among the reaction cells was examined. The all-in-phase regime was found to be the only regime stable over the entire range of coupling strength values. Phase-shifted oscillatory regimes were found to be stable only within a narrow interval of very low coupling strength values. Spontaneous transitions of the phase-shifted regimes to the synchronized mode due to stochastic fluctuations of the coupling strength were observed. Numerical simulations with the four-variable Oregonator based model of the BZ reaction qualitatively confirmed the experimental findings. Propagation of an excitable response to periodic pulsed stimulations in a linear three-array of coupled chemical excitators (Belousov−Zhabotinskii reaction) was studied in dependence on the coupling strength, on the excitability level of the reaction mixture, and on the period and amplitude of pulse stimulation. Regimes of complete and partial propagation of the excitable response and the regimes of partial and complete propagation failure were observed. Numerical simulations predict qualitatively well excitatory regimes observed in experiments.
The random replacement interaction-by-exchange-with-the-mean (IEM) model, which was proposed previously as a numerically more preferable IEM scheme, had been employed for investigating the micromixing effects on the steady-state multiplicity and the relative stability of the oscillatory and stationary state of the Gray−Scott model (Gray, P.; Scott, S. K. Chem. Eng. Sci. 1983, 38, 29). In the bistable region, incomplete micromixing tends to largely reduce the attraction basin for the thermodynamic branch, while in the region where the stable limit cycle exists, it tends to weaken the relative stability of the oscillatory states. Owing to the random replacement nature of the present micromixing model, a stochastic response behavior is observed for the resulting limit cycle attractor and the attraction basin boundary on the parameter space.
We show that the phenomenon of stochastic resonance exists in the enzymatic peroxidase−oxidase reaction. The free-running system is placed in an excitable steady state close to a supercritical Hopf bifurcation. When a sinusoidal flow rate signal and increasing noise are imposed on this focal steady state and a threshold is crossed, the system responds with the generation of bursts. When stochastic resonance is achieved, the interspike histogram and the plot of the signal-to-noise ratio pass through a maximum at an optimal noise level. We compare the experimental results with calculations of the Hung−Schreiber−Ross model.
The recent contention that isothermal systems in a continuously stirred tank reactor can exhibit only extinction (ΔG is positive) as the flow rate is increased (the residence time reduced) and only ignition (ΔG is negative) as the flow rate is decreased is shown not to be correct. The simple, autocatalytic sequence A + 2B → 3B, rate = k1ab2, for which the catalyst B is not indefinitely stable, but undergoes a first-order removal of decay B → inert, rate = k2b, may also have ignition accompanying an increase in the flow rate and extinction accompanying a decrease. This gives rise to S-shaped curves, Z-shaped curves, and also to closed curves in the dependence of the extent of conversion on residence time or flow rate. Under these circumstances, the sign of the free energy change accompanying the observed behavior is determined by the kinetics: the behavior is not determined by the sign of ΔG.
We simulate a pattern recognition device constructed from mass-coupled chemical reactors. Each reactor is an open system containing a bistable (iodate-arsenous acid) reaction. We store arbitrary patterns of high and low iodide concentrations in the network using a Hebb type rule. When the network is initialized with a pattern which is similar to a stored pattern, errors in the initial pattern are corrected and the similar stored pattern is recalled. When the network is initialized with a pattern which is not similar to a stored pattern, the network evolves to a homogeneous state which signals nonrecognition. The network is a programmable parallel computational device.
Experiments on pattern recognition are: performed with a network of eight open, bistable, mass-coupled chemical reactors. A programming rule is used to determine the network connectivity in order to store sets of stationary patterns of reactors with low or high concentrations. Experiments show that these stored patterns can be recalled from similar initial patterns. To our knowledge, this is the first chemical implementation of a type of neural network computing device. The experiments on this small network agree with simulations and support the predictions of the performance-of large:networks.
Two types of nonlinear feedback control schemes are introduced and analyzed for their capability of recovering the original state of an isothermal continuous-flow stirred tank reactor with one robust cubic autocatalytic species, perturbed by a temporary disturbance of an invading cubic autocatalytic species in the inflow. The control objectives are to eliminate the invading species from the system and to restore the original state of the host species. The extent of applicability of the control design to different nonrobust invading species is studied, when the controller is tuned for a specific invader. Moreover, a time-delay feature is suggested in one of the control schemes developed to achieve the control objectives in systems with poor detection of invading species.
The competition of two species for a common resource is illustrated using the paradigm of autocatalytic replicators inhabiting a continuous stirred tank reactor (CSTR) environment that is continuously fed with the resource. In most cases presented, one species is robust (appears in reactor feed) while the other is not. The introduction of the second (invading) species into the CSTR via an unsustained disturbance has a strong effect on the steady-state and dynamic behavior of the first (host) species. New steady states are added to the bifurcation diagram that show that the invading species can coexist in the system with the host species when its growth and death characteristics are similar to those of the latter. The population levels of the host species are greatly reduced in these cases as a result of the considerable decrease in resource concentration at steady state. Open-loop strategies for the elimination of the invading species are developed and discussed. These strategies involve the manipulation of the reactor residence time to destabilize the states of coexistence.
The purpose of this paper is to consider the well agitated continuous reactor from the standpoint of stability of the steady state. It has been shown in the past that chemical-reaction systems may be unstable in the sense that on slight perturbation they tend to move to a more stable state or that they are stable in their steady states, small perturbations being self-correcting so that the system possesses autoregulation. In this paper methods of developing criteria for the quantitative determination of stability or instability or presented and applied to some simple problems. In order that the effect of large perturbations on the system may be determined, complete solutions of the rigorous equations are obtained on the analogue computer (R.E.A.C.). A complete plot of reaction paths in the concentration-temperature plane may be obtained in this manner. Because of the nonlinearity of the system one cannot predict with certainty what steady state will be approached after a given large perturbation, multiple steady states being assumed possible. From the phase plot of reaction paths the regions in the plane which lead to certain steady states are delineated. Also it is shown that the natural behavior of a reactor is not to approach an unstable state. So far as the reactor is concenrned, the unstable state does not exist. The stability of the system is important to the engineer, as control will be easy or difficult and product quality will be satisfactory or not depending upon the relative stability of the steady state. An unstable state would require more elaborate control than a stable state.
The question of the multiplicity of the steady states of a chemical reactor was one of the concerns in the pioneering work of Bilous and Amundson. Their diagrams showed quite clearly the geometry of the situation, and this kind of analysis sufficed for many years. It remained for Balakotaiah and Luss, using the methods of singularity theory, to give a comprehensive treatment of the question. After a brief survey, we take up the case of consecutive first-order reactions and show that up to seven steady states are possible.
Autocatalytic reactions are often complicated, and analyses of their behaviour in open systems can seem too particular to permit useful generalisation. We study here the simplest of circumstances (uniform temperatures and concentrations in the isothermal CSTR) and the simplest of reaction schemes: (i) quadratic autocatalysis (A + B→2B); and (ii) cubic autocatalysis (A + 2B→3B). The catalyst B may be stable or have a finite lifetime (B→ inert products). Allowing for this finite lifetime adds another dimension to the interest.The phenomena encountered include multistability, hysteresis, critical extinctions, critical ignitions, and anomalous relaxation times (though infinite values do not arise). Patterns of stationary states as function of residence time can show isolas and mushrooms. All these aspects yield to simple algebraic analysis. The presence of the catalyst B in the inflow can make qualitative differences of a kind parallelled by an additional, non-catalytic reaction of the same stoichiometry (e.g. A→B). Invoking the reversibility of the reactions neither increases nor diminishes their variety, and thermodynamic considerations have little to do with the many different patterns of reactivity displayed.The local stability of the various stationary states has also been characterized. Quadratic autocatalysis shows limited variety (stable node, stable focus); cubic autocatalysis generates all the kinds of stationary state possible in a two-variable system. Again all the algebra is straightforward if not always simple. Sustained oscillatory behavior (limit cycles) also occur.All these remarks relate to isothermal systems, but there are the most striking parallels between isothermal autocatalysis and the exothermic, first-order reaction in the CSTR. Behaviour with an autocatalyst of complete stability corresponds to perfect heat insulation (adiabatic operation) in the non-isothermal, exothermic system.
A study of the dynamic behavior of two coupled nonisothermal CSTRs carrying out an irreversible first-order reaction is presented. In particular, we study the case where the behavior of both—not necessarily identical—reactors is oscillatory when uncoupled. Emphasis is placed on the way the dynamics develop with increasing coupling strength from the asymptotic limit of weak coupling where the behavior can be deduced from that of the individual reactors. The detailed two-parameter study provides a coherent framework in which many previous experimental and computational studies can be consistently integrated. A scenario of common dynamic features emerges, which is independent of the order of the reaction, whether the reaction is isothermal or catalytic, the details of the mechanism, or the particular form of the coupling. A strong qualitative relation between this framework and the dynamic behavior of periodically forced chemical reactors is also noted.
The prototype, cubic autocatalytic reaction (A + 2B → 3B) forms the basis for the simplest homogeneous system to display “exotic” behaviour. Even under well-stirred, isothermal, open conditions (CSTR) we may find multistability, hysteresis, extinction, ignition and anomalous relaxation times. Allowing for the finite lifetime of the catalyst (B→inert products) adds another dimension. The dependence of the stationary-states on residence-time now yields isolas or mushrooms. Sustained oscillations (stable limit cycles) are also possible. The onset of oscillation corresponds to a change in the character of the stationary-state (from stable focus to unstable focus) and the conditions for this change can be evaluated analytically. The period of the oscillations and their amplitudes increase as the residence time is lengthened. A total of nine different phase-portraits in the a–b plane has been found.The isothermal system is less complex than the exothermic, first-order reaction in a CSTR, but there are strong analogies between the two.
Static and dynamic properties of multiple cubic autocatalytic reactions in an isothermal continuous stirred tank reactor are investigated. It is shown that in the bifurcation diagram of the system should an n-interaction isola exist, it is to reside in the intersection of all possible (n−1)-interaction isolas of the n species constituting the n-interaction. Next, it is discussed that once the ratio of reproduction to death rates for a species is fixed, then the size of its single species isola is fixed, in logarithmic residence time. Furthermore, it is found that any interaction isola in the bifurcation diagram is unstable, whereas single species isolas will have ranges of stable steady states. Finally, these findings and some dynamic properties of the system are demonstrated in case studies.
A family of parallel and consecutive autocatalytic reaction systems has been examined. It is shown that chaotic behavior is possible for several members of this family . In particular, systems of two parallel or consecutive reactions based on cubic, or mixed quadratic and cubic, autocatalysis give rise to chaos. Lyapunov exponents are used to delineate ranges of parameter values for which chaotic behavior is possible. It is found that chaos can occur over wide ranges of values of certain of the parameters, with very complex transitions between chaos and periodicity occuring as the parameters are varied. The transitions to chaos can occur either via a period-doubling route or via a process involving intermittency. A comparison is made between these autocatalytic systems and the analogous nonisothermal reaction systems for which chaotic behavior has previously been shown to be possible.
The interactions between three nonlinear oscillators have been studied for two different experimental systems: (1) the Briggs-Rauscher chemical oscillator induced by a catalytic oxidative reaction; and (2) the salt-water oscillator driven by a concentration difference between two solutions connected through a small pore. In both systems, it has been found that the oscillations are synchronized spontaneously in a tri-phasic mode. The stability of the tri-phasic mode is discussed using coupled nonlinear differential equations.
A model of a reaction system consisting of two parallel, isothermal autocatalytic reactions in a CSTR has been examined. It is shown that self-sustained chaotic behavior can occur in this system. The region of chaos is entered and exited according to period-doubling and halving sequences, with both sets of bifurcations giving rise to Feigenbaum's number. Power spectral density calculations show that the nature of the chaotic behavior depends quite strongly on the parameter values. From a calculation of the Lyapunov exponents it is found that the Lyapunov dimension of the strange attractor is only slightly greater than that of a two-periodic torus.
Particle swarm optimization is employed here to evaluate the parametric regions where different dynamic phenomena (periodic oscillations, double-period oscillations, chaos) can be expected in dynamic models. The proposed algorithm comprises two fundamental steps: the rough evaluation of regions where the desired solutions can be found and solution refining. The refining step allows the search for unstable solutions that may coexist with the other stable attractors. No preliminary bifurcation analysis is required. Simulations performed for distinct dynamic models show that the proposed algorithm is indeed able to locate different dynamic phenomena in the parameter space and that the algorithm may be of help for those interested in increasing the speed of more traditional dynamic bifurcation analysis.
In a recent paper, Gray and Scott have considered the autocatalytic reactions: A → B; rate ∝ abn, n = 0, 1 or 2 where a and b are the concentrations of A and B respectively. Interest centred mainly on irreversible systems but for which the catalytic species was not indefinitely stable, decaying instead by a rate proportional to its concentration b. In practice all chemical reactions are to some extent reversible. The present work investigates the effect of reversibility for the cases in which B does not decay. Bounds are established on the ranges of residence-time tres for which multiple stationary-states are possible. The stability of the different solutions is assessed and the relaxation times t* characterizing the decay of perturbations are given. For deceleratory reactions reversibility enhances the decay rates (decreases t*): for autocatalytic systems the decay rates may be unaltered or even reduced by a finite reverse reaction-rate. Also treated is the influence of non-zero inlet concentrations b0 of the autocatalyst. This may lead to greater changes in the patterns of behaviour possible than those observed in the absence of reversibility. The algebraic analysis remains tractable throughout so the various effects can readily be interpreted in physical terms.
The interactions of two coupled oscillating systems of the Belousov-Zhabotinsky chemical reaction were investigated experimentally. A mapping diagram of coupling was experimentally obtained in the plane of omega2/omega1 and S1-2 where omega1 and omega2 were the natural frequencies of reactor 1 and 2, S1-2 was the area of the window that coupled the two reactors. The irregular synchronization region in this diagram may correspond to Tomita and Kai's region chi where a chaotic response has been predicted.
Phase shifts between four Belousov-Zhabotinsky (BZ) oscillators are applied to encode phase patterns in an experimental network consisting of four reactors. Oscillations are established in a focus of the BZ reaction, which is sinusoidally driven by an applied electrical current. In addition to the global electrical coupling by the sinusoidal function the four reactors are electrically coupled by an optimized feedback function including time delay. Two of three possible phase patterns can be encoded in this Hopfield-type network of four reactors. With a phase pattern in which all four reactors are in-phase, one of the two stored phase patterns is recalled with a 50% probability. This indicates that the two encoded patterns have the same dynamic stability. It is possible to reversibly switch between the two patterns by adding Ce4+ solution. Higher order phase patterns permit a higher phase resolution. The phase method provides for a large amount of information to be stored and recalled in a multiunit network. Numerical calculations with the seven-variable Gyrgyi-Field model of the BZ reaction are in good agreement with the experimental results. Generic similarities with the so-called binding problem in neurology are discussed.
We demonstrate the phenomenon of stochastic resonance in a nonlinear chemical reaction. The term stochastic resonance (SR) denotes the detection of a weak periodic signal in a noisy system displaying a threshold. If the sum of the periodic signal and the noise amplitude crosses the threshold, an output pulse is triggered. At an optimal noise amplitude the distribution of pulse intervals as well as the signal-to-noise ratio will pass through a maximum. In the continuously stirred tank reactor (CSTR) experiments we superimpose a periodic flow rate signal on an excitable focal steady state located near a Hopf bifurcation in the Belousov-Zhabotinsky (BZ) reaction. In the Showalter-Noyes-Bar-Eli model of this reaction we vary the perturbation frequency and amplitude as well as the pulse length of the applied noise to elaborate the optimal conditions for stochastic resonance to occur in the model.
Various modes in three coupled chemical oscillators in a triangular arrangement were observed. As a well-defined nonlinear oscillator, the Belousov-Zhabotinsky reaction was studied in a continuous-flow stirred tank reactor (CSTR). Coupling among CSTR’s was performed by mass exchange. The coupling strength was quantitatively controlled by changing the flow rate of reacting solutions among the three CSTR’s using peristaltic pumps between each pair of the reactors. As a key parameter to control the model of coupling, we changed the symmetry of the interaction between the oscillators. In the case of the symmetric coupling, a quasiperiodic state or a biperiodic mode, an all-death mode and two kinds of synchronized modes appeared, depending on the coupling strength. On the other hand, under the asymmetric coupling, a quasiperiodic state or a biperiodic mode, an all death mode and four kinds of synchronized modes appeared. Those modes have been discussed in relation to the idea of ‘‘frustration’’ in the Ising spin system, where the three-phase mode appears as a transition from the Ising spin system to the XY spin system.
The study of networks pervades all of science, from neurobiology to statistical physics. The most basic issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous network of interacting dynamical systems-be they neurons, power stations or lasers-will behave collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning to unravel the structure and dynamics of complex networks.
A numerical algorithm for continuation of stationary solutions to nonlinear evolution problems representable in the form u t = F (uxx ; ux ; u; x; ff); 0 ! x ! 1; f 0 (ux ; u; ff) = 0; x = 0; f 1 (ux ; u; ff) = 0; x = 1; is described as implemented in content. Here F : R n ThetaR n ThetaR n ThetaR m ! R n and f 0;1 : R n ThetaR n ThetaR m ! R n are sufficiently smooth nonlinear functions. The algorithm is based on the second-order finite-difference approximation with an adaptive non-uniform mesh selection. Special methods for efficient solution of linear systems appearing in the continuation are presented. Several examples are given. AMS Subject Classification (1991): 35B32, 35B60, 35G30, 65N06, 65N50, 68N99 Keywords & Phrases: Numerical continuation, partial differential equations, nonlinear boundary-value problems, finite-difference approximation, interactive software. Note: This work has been partially supported by the Visitor Program of the Neder...
Parallel software for differential and nonlinear systems; Technical Report UCRL-ID-129739; Lawrence Livermore National Laboratory: Liv-ermore, CA
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