CHEM 515: Chemical Kinetics and Dynamics
|
|
- Bertina Malone
- 5 years ago
- Views:
Transcription
1 Alejandro J. Garza S Department of Chemistry, Rice University, Houston, TX ext Submitted December 12, 2011 Abstract Spontaneous antispiral wave formation was observed during the simulation of typical reaction-diffusion systems using differential equations models. The equations for the FitzHugh-Nagumo (FHN), complex Ginzburg-Landau (CGL) and the two-variable Oregonator models were integrated using Runge-Kutta methods. It was found that antispiral waves can occur only near the Hopf bifurcation, i.e. the point at which the system starts to oscillate. The specific conditions required for antispiral wave formation were established through theoretical analysis and numerical simulations. The results explain a number of phenomena observed in oscillatory reactions such as the Belousov-Zhabotinsky (BZ) reaction. Introduction This paper is intended to describe and explain the findings of Gong and Christini in Antispiral Waves in Reaction-Diffusion systems [1]. With this purpose, this section will give an introduction on reaction diffusion-systems and their connection to chemistry. Excitable Media An excitable medium is a nonlinear dynamical system which has the capacity to propagate a wave, and which cannot support the passing of another wave until a so-called refractory amount of time has passed [2]. Arguably, excitable media are constitute one of the foremost important, and fascinating, nonlinear dynamical systems; excitable media can show selforganization, a requirement for the existence of life. Excitable media can be analyzed/simulated by means of three models; (1) stochastic models, (2) cellular automata models, and (3) partial differential equations. Theses models have been able to describe the most notable qualitative features observed in experiments (e.g formation of waves and self-organization). Gong and Christini have utilized the method of partial differential equations in their study (see the methods section). Reaction-diffusion systems have been show to act as excitable media under certain conditions. The most famous example of such a class of systems in chemistry is the unstirred Belousov-Zhbotinsky (BZ) reaction. The BZ-Reaction The BZ-reaction was discovered by Belousov in 1950 while trying to find and inorganic analogous to the Krebs cycle [3]. However, his findings were dismissed by the scientific community until Zhabotinsky confirmed his results [4]. Today, the BZ-reaction is the most studied, and arguably most important, nonbiological oscillatory reaction. The overall BZ-reaction is the oxidation of malonic acid by BrO 3 to form CO 2 and H 2 O 3 CH 2 (CO 2 H) BrO 3 4 Br + 9 CO H 2 O (1) However, this equation is deceptively simple. The mechanism for this reaction is extremely complicated and many models of varying complexity have been proposed [5]. The simplest model that is able to retain the qualitative features of the BZ-reaction is the Oregonator model [6]. This model can be explained as follows; consider a reaction described by the mechanism
2 Concentration X Y Z P impossibility of an oscillatory overall reaction (i.e. the reaction will be moving away from equilibrium). When coupled with diffusion, which is governed by the diffusion equation c t D c 2 c (6) the Oregonator forms a model for an excitable medium Time Figure 1: Time series for the solutions to the Field and Noyes equations. The parameters used where ɛ , ɛ , q , f 1, x(t 0) 0, y(t 0) and z(t 0) 0. A + Y k1 X + P X + Y k2 2 P A + X k3 2 X + 2 Z X + X k4 A + P B + Z k5 1 2 fy (O1) (O2) (O3) (O4) (O5) (i.e. the Oregonator mechanism). Assuming steadystate for A and B dx qy xy + x(1 x) dτ ɛ (2) dy qy xy + fz dτ ɛ (3) dz dτ x z (4) where x 2k 4 X/(k 3 A), y k 2 Y/(k 3 A), z k c k 4 BZ/(k 3 A) 2, τ k c Bt, ɛ k c B/(k 3 A) y ɛ 2k c k 4 B/(k 2 k 3 A), q 2k 1 k 4 /(k 2 k 3 ). These equations are known as the Field and Noyes equations. The solutions to these equations are oscillatory for certain sets of values of ɛ, ɛ, q and f. Figure 1 shows an example of these solutions for a set of parameters that produce oscillations. Note that, in this model, the concentration of products is always increasing in time (see Figure 1); only the concentration of intermediaries oscillate. Since the overall reaction is spontaneous, we conclude that S univ > 0 (5) for all time, t. This is very important because, when first discovered, oscillatory reaction where thought to be errors of measurement, due to the thermodynamic The FHN and CGL Model Apart from the oregonator model, two other models for excitable media were examined by Gong and Christini; the FitzHugh-Nagumo (FHN) and the Complex Ginzburg-Landau (CGL) models [7, 8]. The FHN model is a generic model for excitable media described by the system of partial differential equations u/ t (u u 3 /3 v)/ɛ + D u 2 u v/ t u γv + δ + D v 2 v (7) The CGL equation arises in many physical systems, including Rayleigh-Bénard convection, and contains diffusion, non-linear dispersion and amplitudedependent frequency. The CGL equation can be written as W/ t W (1 + iα)w W 2 + (1 + iβ) 2 W (8) Note that the equation has a real and a complex part. Antispiral Waves Soon after the confirmation of the discovery of oscillatory reactions, the formation of patterns in unstirred reaction-diffusion systems was also discovered. Among a wide variety of patterns observed, spiral waves seem to be of particular interest and importance due to their occurrence in biological and inorganic processes. Nevertheless, little attention was paid to the direction of rotation of the spirals (normally observed to rotate outwardly) until the recent observation of inwardly rotating spirals, termed antispirals [9]. Gong and Christini utilized numerical simulations to qualitatively reproduce these observations. Methods The FHN, CGL and oregonator models were analyzed numerically using partial differential equations. For Garza 2
3 the FHN and CGL models, Equation 7 and Equation 8 were integrated; for the Oregonator, the twovariable Oregonator model was used ( u u 2 fv u q u + q ) + D u 2 u du dt 1 ɛ dv dt u v + D v 2 v (9) This model can be derived by noting that ɛ is considerable larger than ɛ, so that it is reasonable to assume steady state for y y ss fz q + x (10) Making x u and z v, and considering the corresponding diffusion terms, one obtains Equation 9. All the systems were solved in a grid with no-flux boundaries and from random initial conditions. The differential equations were integrated using the explicit Euler method. This method can be described by the equations u(t) t f(t, u(t)) (11) u(t) f(t, u(t)) t (12) u(t + t) u(t) + f(t, u(t)) t (13) which is basically a discretization of the differential equation. To calculate the Laplacian, the standard five-point approximation was utilized 2 u 2 u x u y 2 2 u u(x + h, y) + u(x h, y) 2u(x, y) x2 h 2 (14) 2 u u(x, y + h) + u(x, y h) 2u(x, y) y2 h 2 (15) where h is the size of the step. Here, h 0.5 while t for CGL and t 0.01 for the FHN and Oregonator models. The total number of integration steps was The results were also verified to remain the same using the fourth-order Runge-Kutta integrating method. Results Figure 2 summarizes and illustrates some of the most important results for the FHN and CGL models. Figure 2a shows the time series for the u in the FHN Figure 2: Spiral and antispiral waves in typical excitable (FHN with γ 0.5, δ 0.7, D u 1.0, D v 0.0) and oscillatory (CGL) media. (a) Time history (arbitrary units) of the fast activator variable u of the FHN model. (b) Snapshot of spirals in the FHN model; the arrow indicates outward propagation of waves. (c) The parameter plane (α, β) shows where spirals ( ) and antispirals ( ) exist in the CGL system. (d) Snapshot of a well-developed single antispiral in the CGL system. (e) (f) Snapshots of antispirals in the CGL system. model; the small amplitude oscillations occur for a relatively large ɛ (ɛ 1.39) near the Hopf bifurcation (i.e. the point at which oscillations start occuring as ɛ changes). It is near this point at which antispirals occur; in contrast, spirals are formed when the oscillations have large amplitudes, for relatively small values of ɛ (ɛ0.09). Figure 2b shows a typical spiral in the FHN system. Figure 2c shows the parameter plane (α, β) shows where spirals ( ) and antispirals ( ) exist in the CGL system; that is, the points at which spirals or antispirals are formed as a function of α and β. Spirals occur when β > α 0 or 0 α > β; antispirals when α > β 0 or 0 β > α. For α β the medium may either support phase waves or remain static, depending on the parameter choice. Nevertheless, Figure 2c does not shows the expected behavior when αβ < 0; Figure 2c was extended improved in a comment to the paper of Gong and Christini by L. Brusch et al [10]. The improved verison of Figure 2x is shown in Figure 3, where RD denotes a general reaction diffusion system of the form i ũ f(ũ, µ) + D 2 ũ (16) Figures 2d, 2e, and 2f show typical spirals and antispirals observed in the CGL system. Garza 3
4 Figure 3: Extended parameter space (α, β) of the CGL equation and a general reaction-diffusion (RD) model. The parameters k s and ω s denote the wavenumber and frequency, repsectively, of the (spiral or antispiral) waves. The direction of wave-propagation is governed by the result of the competition between waves (whether spiral or antispiral) and their bulk oscillations [9]. Also, spiral and antispiral waves behave asymptotically as plane waves far from their cores. Because of this, Gong and Christine hypothesize that antispirals waves will be observed when the frequency of the bulk oscillation is larger than the asymptotic frequency of the wave (antispiral in this case). This hypothesis is in agreement with Figure 2c. Similarly, antispiral waves were observed in the two-variable Oregonator model. Figure 4 shows similar antispiral wave patterns in the FHN and Oregonato models. The case is similar to the FHN model; when ɛ is relatively small (e.g ɛ 0.01), the system becomes a typical relaxation oscillator with large amplitude. The small ɛ assumption was widely used in previous theoretical studies of the BZ reaction, which may account for the fact that antispirals waves were not observed. If, however, ɛ is near the Hopf bifurcation (ɛ for q and f 0.95), the motion will be sinusoidal with small amplitude oscillations, which in turn lead to the spontaneous formation of antispirals. In addition, in order to form antispirals, the diffusion coefficients in the Oregonator model must be considerably small ( 0.001) in comparison to the large amplitude case ( 0.01). Conclusions The spontaneous formation of antispiral waves in typical reaction-diffusion systems (the CGL, FHN and Oregonator models) was demonstrated by means of numerical simulations. In addition, the condi- Figure 4: Antispiral waves in the FHN (a-d) and the Oregonator (e,f) models, where the arrows indicate inward propagation of waves. For FHN, ɛ 1.95, γ 0.5, δ 0.01, D u D v For the Oregonator model, q 0.002, f 0.95, D u D v tions for formation of antispiral waves in reactiondiffusion systems were determined. The results qualitatively reproduce phenomena observed in Belousoz- Zhabotisnky-type reactions. The study of pattern formation in reactionsdiffusion is important for the understanding of several physical phenomena. Spiral waves form in cardiac tissue, and it has been speculated that antispiral waves might similarly occur. Reaction-diffusion systems have promising applications in the development of chemical computers. In addition, morphogenesis and self-organization in nature remain in considerable mystery among science and the study of reaction diffusion systems may shed light on these processes. References [1] Gong, Y.; Christini, D. J. Phys. Rev. Lett , 90, (4). [2] Karfunkel, H. R.; Seelig, F. F. Journal of Mathematical Biology, 1975, 2, 123. [3] Epstein, I. R.; Pojman, J. A. An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns and Chaos. Oxford University Press, New York, [4] Zaikin, A. N.; Zhabotinsky, A. M. Nature, 1970, 225, 535. Garza 4
5 [5] Kalishin, E. Yu.; Gonchareko M. M.; Khavrus, V. A.; Strizhak, P. E. Kinet. Catal , 43, 256. [6] Noyes, R. M.J. Chem Ed. 1989, 66, 190. [7] FitzHugh, R. A. Biophysics, 1980, 25, 906. [8] Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence, 1984 Springer-Verlag, Berlin. [9] Vanag, V. K.; Epstein, I. R. Science, 2001, 294, 835. [10] Brusch, L.; Nicola, E. M.; Bär, M. Phys. Rev. Lett. 2004, 92, (comment). Garza 5
7. Well-Stirred Reactors IV
7. Well-Stirred Reactors IV The Belousov-Zhabotinsky Reaction: Models and Experiments Oregonator [based on the FKN mechanism; Field, R. J. & Noyes, R. M.: Oscillations in chemical systems. IV. Limit cycle
More informationStabilization of Pattern in Complex Ginzburg Landau Equation with Spatial Perturbation Scheme
Commun. Theor. Phys. (Beijing, China) 49 (2008) pp. 1541 1546 c Chinese Physical Society Vol. 49, No. 6, June 15, 2008 Stabilization of Pattern in Complex Ginzburg Landau Equation with Spatial Perturbation
More informationChemical Wave Packet Propagation, Reflection, and Spreading
11676 J. Phys. Chem. A 2002, 106, 11676-11682 Chemical Wave Packet Propagation, Reflection, and Spreading Lingfa Yang and Irving R. Epstein* Department of Chemistry and Center for Complex Systems, MS 015,
More informationResonance in periodically inhibited reaction diffusion systems
Physica D 168 169 (2002) 1 9 Resonance in periodically inhibited reaction diffusion systems Karl Martinez, Anna L. Lin, Reza Kharrazian, Xaver Sailer, Harry L. Swinney Center for Nonlinear Dynamics and
More informationDrift velocity of rotating spiral waves in the weak deformation approximation
JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 8 22 AUGUST 2003 Drift velocity of rotating spiral waves in the weak deformation approximation Hong Zhang a) Bambi Hu and Department of Physics, University
More informationMulti-mode Spiral Wave in a Coupled Oscillatory Medium
Commun. Theor. Phys. (Beijing, China) 53 (2010) pp. 977 982 c Chinese Physical Society and IOP Publishing Ltd Vol. 53, No. 5, May 15, 2010 Multi-mode Spiral Wave in a Coupled Oscillatory Medium WANG Qun
More informationChaos in the Belousov-Zhabotinsky Reaction
Chaos in the Belousov-Zhabotinsky Reaction David Connolly Brad Nelson December 2, 2011 Abstract In this project, we investigate two different models of the Belousov-Zhabotinsky Reaction, the Oregonator
More informationModelling biological oscillations
Modelling biological oscillations Shan He School for Computational Science University of Birmingham Module 06-23836: Computational Modelling with MATLAB Outline Outline of Topics Van der Pol equation Van
More informationNature-inspired Analog Computing on Silicon
Nature-inspired Analog Computing on Silicon Tetsuya ASAI and Yoshihito AMEMIYA Division of Electronics and Information Engineering Hokkaido University Abstract We propose CMOS analog circuits that emulate
More informationarxiv:chao-dyn/ v1 12 Feb 1996
Spiral Waves in Chaotic Systems Andrei Goryachev and Raymond Kapral Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, ON M5S 1A1, Canada arxiv:chao-dyn/96014v1 12
More informationFestival of the Mind Chaotic Chemical Waves: Oscillations and waves in chemical systems. Dr. Jonathan Howse Dr.
Festival of the Mind 2014 Chaotic Chemical Waves: Oscillations and waves in chemical systems Dr. Jonathan Howse Dr. Annette Taylor Mark Fell www.markfell.com Some history The BZ reaction In 1951, Belousov,
More informationWave Pinning, Actin Waves, and LPA
Wave Pinning, Actin Waves, and LPA MCB 2012 William R. Holmes Intercellular Waves Weiner et al., 2007, PLoS Biology Dynamic Hem1 waves in neutrophils Questions? How can such waves / pulses form? What molecular
More informationLaurette TUCKERMAN Numerical Methods for Differential Equations in Physics
Laurette TUCKERMAN laurette@pmmh.espci.fr Numerical Methods for Differential Equations in Physics Time stepping: Steady state solving: 0 = F(U) t U = LU + N(U) 0 = LU + N(U) Newton s method 0 = F(U u)
More informationResonant Chemical Oscillations: Pattern Formation in Reaction-Diffusion Systems
Resonant Chemical Oscillations: Pattern Formation in Reaction-Diffusion Systems Anna L. Lin Department of Physics, Center for nonlinear and complex systems, Duke University, Durham, NC 778-5 Abstract.
More information6. Well-Stirred Reactors III
6. Well-Stirred Reactors III Reactors reaction rate or reaction velocity defined for a closed system of uniform pressure, temperature, and composition situation in a real reactor is usually quite different
More information2013 NSF-CMACS Workshop on Atrial Fibrillation
2013 NSF-CMACS Workshop on A Atrial Fibrillation Flavio H. Fenton School of Physics Georgia Institute of Technology, Atlanta, GA and Max Planck Institute for Dynamics and Self-organization, Goettingen,
More informationOregonator model of the Belousov-Zhabotinsky reaction Richard K. Herz,
Oregonator model of the Belousov-Zhabotinsky reaction Richard K. Herz, rherz@ucsd.edu Boris Belousov in the 1950's discovered that a mixture of malonic acid, potassium bromate, and cerium sulfate an acidic
More informationPHYSICAL REVIEW E VOLUME 62, NUMBER 6. Target waves in the complex Ginzburg-Landau equation
PHYSICAL REVIEW E VOLUME 62, NUMBER 6 DECEMBER 2000 Target waves in the complex Ginzburg-Landau equation Matthew Hendrey, Keeyeol Nam, Parvez Guzdar,* and Edward Ott University of Maryland, Institute for
More informationTracking the State of the Hindmarsh-Rose Neuron by Using the Coullet Chaotic System Based on a Single Input
ISSN 1746-7659, England, UK Journal of Information and Computing Science Vol. 11, No., 016, pp.083-09 Tracking the State of the Hindmarsh-Rose Neuron by Using the Coullet Chaotic System Based on a Single
More informationCHEM-UA 652: Thermodynamics and Kinetics
CHEM-UA 65: Thermodynamics and Kinetics Notes for Lecture I. THE COMPLEXITY OF MULTI-STEP CHEMICAL REACTIONS It should be clear by now that chemical kinetics is governed by the mathematics of systems of
More informationPattern Formation and Spatiotemporal Chaos in Systems Far from Equilibrium
Pattern Formation and Spatiotemporal Chaos in Systems Far from Equilibrium Michael Cross California Institute of Technology Beijing Normal University May 2006 Michael Cross (Caltech, BNU) Pattern Formation
More informationScroll Waves in Anisotropic Excitable Media with Application to the Heart. Sima Setayeshgar Department of Physics Indiana University
Scroll Waves in Anisotropic Excitable Media with Application to the Heart Sima Setayeshgar Department of Physics Indiana University KITP Cardiac Dynamics Mini-Program 1 Stripes, Spots and Scrolls KITP
More information1.2. Introduction to Modeling. P (t) = r P (t) (b) When r > 0 this is the exponential growth equation.
G. NAGY ODE January 9, 2018 1 1.2. Introduction to Modeling Section Objective(s): Review of Exponential Growth. The Logistic Population Model. Competing Species Model. Overview of Mathematical Models.
More informationControl of spiral instabilities in reaction diffusion systems*
Pure Appl. Chem., Vol. 77, No. 8, pp. 1395 1408, 2005. DOI: 10.1351/pac200577081395 2005 IUPAC Control of spiral instabilities in reaction diffusion systems* Hui-Min Liao 1, Min-Xi Jiang 1, Xiao-Nan Wang
More informationComparative Analysis of Packet and Trigger Waves Originating from a Finite Wavelength Instability
11394 J. Phys. Chem. A 2002, 106, 11394-11399 Comparative Analysis of Packet and Trigger Waves Originating from a Finite Wavelength Instability Vladimir K. Vanag*, and Irving R. Epstein Department of Chemistry
More informationToward a Better Understanding of Complexity
Toward a Better Understanding of Complexity Definitions of Complexity, Cellular Automata as Models of Complexity, Random Boolean Networks Christian Jacob jacob@cpsc.ucalgary.ca Department of Computer Science
More informationLecture 18: Bistable Fronts PHYS 221A, Spring 2017
Lecture 18: Bistable Fronts PHYS 221A, Spring 2017 Lectures: P. H. Diamond Notes: Xiang Fan June 15, 2017 1 Introduction In the previous lectures, we learned about Turing Patterns. Turing Instability is
More informationTHE SYNCHRONIZATION OF TWO CHAOTIC MODELS OF CHEMICAL REACTIONS
ROMAI J., v.10, no.1(2014), 137 145 THE SYNCHRONIZATION OF TWO CHAOTIC MODELS OF CHEMICAL REACTIONS Servilia Oancea 1, Andrei-Victor Oancea 2, Ioan Grosu 3 1 U.S.A.M.V., Iaşi, Romania 2 Erasmus Mundus
More informationCompetition of Spatial and Temporal Instabilities under Time Delay near Codimension-Two Turing Hopf Bifurcations
Commun. Theor. Phys. 56 (2011) 339 344 Vol. 56, No. 2, August 15, 2011 Competition of Spatial and Temporal Instabilities under Time Delay near Codimension-Two Turing Hopf Bifurcations WANG Hui-Juan ( )
More informationPhysics: spring-mass system, planet motion, pendulum. Biology: ecology problem, neural conduction, epidemics
Applications of nonlinear ODE systems: Physics: spring-mass system, planet motion, pendulum Chemistry: mixing problems, chemical reactions Biology: ecology problem, neural conduction, epidemics Economy:
More informationIntroduction LECTURE 1
LECTURE 1 Introduction The source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in
More informationTWO DIMENSIONAL FLOWS. Lecture 5: Limit Cycles and Bifurcations
TWO DIMENSIONAL FLOWS Lecture 5: Limit Cycles and Bifurcations 5. Limit cycles A limit cycle is an isolated closed trajectory [ isolated means that neighbouring trajectories are not closed] Fig. 5.1.1
More informationOscillatory Turing Patterns in a Simple Reaction-Diffusion System
Journal of the Korean Physical Society, Vol. 50, No. 1, January 2007, pp. 234 238 Oscillatory Turing Patterns in a Simple Reaction-Diffusion System Ruey-Tarng Liu and Sy-Sang Liaw Department of Physics,
More informationSPIRAL WAVE GENERATION IN A DIFFUSIVE PREDATOR-PREY MODEL WITH TWO TIME DELAYS
Bull. Korean Math. Soc. 52 (2015), No. 4, pp. 1113 1122 http://dx.doi.org/10.4134/bkms.2015.52.4.1113 SPIRAL WAVE GENERATION IN A DIFFUSIVE PREDATOR-PREY MODEL WITH TWO TIME DELAYS Wenzhen Gan and Peng
More informationSpectral Methods for Reaction Diffusion Systems
WDS'13 Proceedings of Contributed Papers, Part I, 97 101, 2013. ISBN 978-80-7378-250-4 MATFYZPRESS Spectral Methods for Reaction Diffusion Systems V. Rybář Institute of Mathematics of the Academy of Sciences
More informationShadow system for adsorbate-induced phase transition model
RIMS Kôkyûroku Bessatsu B5 (9), 4 Shadow system for adsorbate-induced phase transition model Dedicated to Professor Toshitaka Nagai on the occasion of his sixtieth birthday By Kousuke Kuto and Tohru Tsujikawa
More informationSelf-Organization in Nonequilibrium Systems
Self-Organization in Nonequilibrium Systems From Dissipative Structures to Order through Fluctuations G. Nicolis Universite Libre de Bruxelles Belgium I. Prigogine Universite Libre de Bruxelles Belgium
More informationHeterogeneous Sources of Target Patterns in Reaction-Diffusion Systems
J. Phys. Chem. 1996, 100, 19017-19022 19017 Heterogeneous Sources of Target Patterns in Reaction-Diffusion Systems Andrej E. Bugrim, Milos Dolnik, Anatol M. Zhabotinsky,* and Irving R. Epstein Department
More informationLocalized structures as spatial hosts for unstable modes
April 2007 EPL, 78 (2007 14002 doi: 10.1209/0295-5075/78/14002 www.epljournal.org A. Lampert 1 and E. Meron 1,2 1 Department of Physics, Ben-Gurion University - Beer-Sheva 84105, Israel 2 Department of
More informationFinal Project Descriptions Introduction to Mathematical Biology Professor: Paul J. Atzberger. Project I: Predator-Prey Equations
Final Project Descriptions Introduction to Mathematical Biology Professor: Paul J. Atzberger Project I: Predator-Prey Equations The Lotka-Volterra Predator-Prey Model is given by: du dv = αu βuv = ρβuv
More informationCHEMICAL OSCILLATIONS IN HOMOGENEOUS SYSTEMS 1. ESSENTIAL THERMODYNAMIC AND KINETIC CONDITIONS FOR THE OCCURRENCE OF OSCILLATIONS
CHEMICAL OSCILLATIONS IN HOMOGENEOUS SYSTEMS 1. ESSENTIAL THERMODYNAMIC AND KINETIC CONDITIONS FOR THE OCCURRENCE OF OSCILLATIONS Rodica Vîlcu and Daniela Bala abstract: This manuscript reviews the understanding
More informationarxiv:nlin/ v1 [nlin.ps] 21 Jan 2004
Resonance tongues and patterns in periodically forced reaction-diffusion systems Anna L. Lin, Aric Hagberg,, Ehud Meron,,, and Harry L. Swinney 5 Center for Nonlinear and Complex Systems and Department
More informationFront velocity in models with quadratic autocatalysis
JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 18 8 NOVEMBER 2002 Front velocity in models with quadratic autocatalysis Vladimir K. Vanag a) and Irving R. Epstein Department of Chemistry and Volen Center
More informationPractice Problems For Test 3
Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)
More informationNonlinear Analysis: Modelling and Control, Vilnius, IMI, 1998, No 3 KINK-EXCITATION OF N-SYSTEM UNDER SPATIO -TEMPORAL NOISE. R.
Nonlinear Analysis: Modelling and Control Vilnius IMI 1998 No 3 KINK-EXCITATION OF N-SYSTEM UNDER SPATIO -TEMPORAL NOISE R. Bakanas Semiconductor Physics Institute Go štauto 11 6 Vilnius Lithuania Vilnius
More informationObservations on the ponderomotive force
Observations on the ponderomotive force D.A. Burton a, R.A. Cairns b, B. Ersfeld c, A. Noble c, S. Yoffe c, and D.A. Jaroszynski c a University of Lancaster, Physics Department, Lancaster LA1 4YB, UK b
More informationMath 216 Final Exam 14 December, 2012
Math 216 Final Exam 14 December, 2012 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More informationPractice Problems For Test 3
Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)
More informationLokta-Volterra predator-prey equation dx = ax bxy dt dy = cx + dxy dt
Periodic solutions A periodic solution is a solution (x(t), y(t)) of dx = f(x, y) dt dy = g(x, y) dt such that x(t + T ) = x(t) and y(t + T ) = y(t) for any t, where T is a fixed number which is a period
More informationChemical Kinetics and the Rössler System. 1 Introduction. 2 The NH 3 - HCl reaction. Dynamics at the Horsetooth Volume 2, 2010.
Dynamics at the Horsetooth Volume 2, 21. Chemical Kinetics and the Rössler System Department of Mathematics Colorado State University shinn@math.colostate.edu Report submitted to Prof. P. Shipman for Math
More information2D-Volterra-Lotka Modeling For 2 Species
Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya 2D-Volterra-Lotka Modeling For 2 Species Alhashmi Darah 1 University of Almergeb Department of Mathematics Faculty of Science Zliten Libya. Abstract The purpose
More informationMarch 9, :18 Int J. Bifurcation and Chaos/INSTRUCTION FILE Morfu2v2 EFFECT OF NOISE AND STRUCTURAL INHOMOGENEITIES IN REACTION DIFFUSION MEDIA
March 9, 2007 10:18 Int J. Bifurcation and Chaos/INSTRUCTION FILE Int J. Bifurcation and Chaos Submission Style EFFECT OF NOISE AND STRUCTURAL INHOMOGENEITIES IN REACTION DIFFUSION MEDIA S. Morfu Laboratoire
More informationModels Involving Interactions between Predator and Prey Populations
Models Involving Interactions between Predator and Prey Populations Matthew Mitchell Georgia College and State University December 30, 2015 Abstract Predator-prey models are used to show the intricate
More informationCompactlike Kink Solutions in Reaction Diffusion Systems. Abstract
Compactlike Kink Solutions in Reaction Diffusion Systems J.C. Comte Physics Department, University of Crete and Foundation for Research and Technology-Hellas P. O. Box 2208, 71003 Heraklion, Crete, Greece
More informationSpatiotemporal pattern formation in a prey-predator model under environmental driving forces
Home Search Collections Journals About Contact us My IOPscience Spatiotemporal pattern formation in a prey-predator model under environmental driving forces This content has been downloaded from IOPscience.
More informationIntroduction to multiscale modeling and simulation. Explicit methods for ODEs : forward Euler. y n+1 = y n + tf(y n ) dy dt = f(y), y(0) = y 0
Introduction to multiscale modeling and simulation Lecture 5 Numerical methods for ODEs, SDEs and PDEs The need for multiscale methods Two generic frameworks for multiscale computation Explicit methods
More informationBounds on Surface Stress Driven Flows
Bounds on Surface Stress Driven Flows George Hagstrom Charlie Doering January 1, 9 1 Introduction In the past fifteen years the background method of Constantin, Doering, and Hopf has been used to derive
More informationComplex periodic spiral waves
Complex periodic spiral waves (in a BZ system & in vitro cardiac system) Kyoung J. Lee NCRI Center for Neuro-dynamics and Department of Physics, Korea University http://turing.korea.ac.kr 1 Coupled network
More informationNumerical solution of stiff systems of differential equations arising from chemical reactions
Iranian Journal of Numerical Analysis and Optimization Vol 4, No. 1, (214), pp 25-39 Numerical solution of stiff systems of differential equations arising from chemical reactions G. Hojjati, A. Abdi, F.
More informationarxiv: v1 [nlin.cd] 6 Mar 2018
Constructive approach to limiting periodic orbits with exponential and power law dynamics A. Provata Institute of Nanoscience and Nanotechnology, National Center for Scientific Research Demokritos, GR-15310
More informationMulti-physics Modeling Using Cellular Automata
Multi-physics Modeling sing Cellular Automata Brian Vic Mechanical Engineering Department, Virginia Tech, Blacsburg, VA 246-238 This paper proposes a new modeling and solution method that is relatively
More informationStochastic models, patterns formation and diffusion
Stochastic models, patterns formation and diffusion Duccio Fanelli Francesca Di Patti, Tommaso Biancalani Dipartimento di Energetica, Università degli Studi di Firenze CSDC Centro Interdipartimentale per
More information6.3.4 Action potential
I ion C m C m dφ dt Figure 6.8: Electrical circuit model of the cell membrane. Normally, cells are net negative inside the cell which results in a non-zero resting membrane potential. The membrane potential
More informationQuasipatterns in surface wave experiments
Quasipatterns in surface wave experiments Alastair Rucklidge Department of Applied Mathematics University of Leeds, Leeds LS2 9JT, UK With support from EPSRC A.M. Rucklidge and W.J. Rucklidge, Convergence
More informationFinal Exam December 20, 2011
Final Exam December 20, 2011 Math 420 - Ordinary Differential Equations No credit will be given for answers without mathematical or logical justification. Simplify answers as much as possible. Leave solutions
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationChimera states in networks of biological neurons and coupled damped pendulums
in neural models in networks of pendulum-like elements in networks of biological neurons and coupled damped pendulums J. Hizanidis 1, V. Kanas 2, A. Bezerianos 3, and T. Bountis 4 1 National Center for
More informationNeural Modeling and Computational Neuroscience. Claudio Gallicchio
Neural Modeling and Computational Neuroscience Claudio Gallicchio 1 Neuroscience modeling 2 Introduction to basic aspects of brain computation Introduction to neurophysiology Neural modeling: Elements
More information1. Consider the initial value problem: find y(t) such that. y = y 2 t, y(0) = 1.
Engineering Mathematics CHEN30101 solutions to sheet 3 1. Consider the initial value problem: find y(t) such that y = y 2 t, y(0) = 1. Take a step size h = 0.1 and verify that the forward Euler approximation
More informationAn Analysis of the Belousov-Zhabotinskii Reaction
An Analysis of the Belousov-Zhabotinskii Reaction Casey R. Gray Calhoun High School Port Lavaca, TX 77979 and The High School Summer Science Research Program Department of Mathematics Baylor University
More informationOscillatory pulses in FitzHugh Nagumo type systems with cross-diffusion
Mathematical Medicine and Biology (2011) 28, 217 226 doi:10.1093/imammb/dqq012 Advance Access publication on August 4, 2010 Oscillatory pulses in FitzHugh Nagumo type systems with cross-diffusion E. P.
More informationBlack spots in a surfactant-rich Belousov Zhabotinsky reaction dispersed in a water-in-oil microemulsion system
THE JOURNAL OF CHEMICAL PHYSICS 122, 174706 2005 Black spots in a surfactant-rich Belousov Zhabotinsky reaction dispersed in a water-in-oil microemulsion system Akiko Kaminaga, Vladimir K. Vanag, and Irving
More informationSymmetry Properties of Confined Convective States
Symmetry Properties of Confined Convective States John Guckenheimer Cornell University 1 Introduction This paper is a commentary on the experimental observation observations of Bensimon et al. [1] of convection
More informationSuppression of Spiral Waves and Spatiotemporal Chaos Under Local Self-adaptive Coupling Interactions
Commun. Theor. Phys. (Beijing, China) 45 (6) pp. 121 126 c International Academic Publishers Vol. 45, No. 1, January 15, 6 Suppression of Spiral Waves and Spatiotemporal Chaos Under Local Self-adaptive
More informationFinite Differences for Differential Equations 28 PART II. Finite Difference Methods for Differential Equations
Finite Differences for Differential Equations 28 PART II Finite Difference Methods for Differential Equations Finite Differences for Differential Equations 29 BOUNDARY VALUE PROBLEMS (I) Solving a TWO
More informationBasic Theory of Dynamical Systems
1 Basic Theory of Dynamical Systems Page 1 1.1 Introduction and Basic Examples Dynamical systems is concerned with both quantitative and qualitative properties of evolution equations, which are often ordinary
More informationReactions. John Vincent Department of Chemistry University of Alabama
Oscillating Chemical Reactions John Vincent Department of Chemistry University of Alabama Kinetics In kinetics we study the rate at which a chemical process occurs. Besides information about the speed
More informationDiffusion of a density in a static fluid
Diffusion of a density in a static fluid u(x, y, z, t), density (M/L 3 ) of a substance (dye). Diffusion: motion of particles from places where the density is higher to places where it is lower, due to
More informationBifurcation and Stability Analysis of a Prey-predator System with a Reserved Area
ISSN 746-733, England, UK World Journal of Modelling and Simulation Vol. 8 ( No. 4, pp. 85-9 Bifurcation and Stability Analysis of a Prey-predator System with a Reserved Area Debasis Mukherjee Department
More informationReactive Lattice Gas Model for FitzHugh-Nagumo Dynamics
Reactive Lattice Gas Model for FitzHugh-Nagumo Dynamics Anatoly Malevanets and Raymond Kapral Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada
More informationLIMIT CYCLE OSCILLATORS
MCB 137 EXCITABLE & OSCILLATORY SYSTEMS WINTER 2008 LIMIT CYCLE OSCILLATORS The Fitzhugh-Nagumo Equations The best example of an excitable phenomenon is the firing of a nerve: according to the Hodgkin
More informationBio-inspired materials: an electrochemically controlled polymeric system which mimics biological learning behavior
Bio-inspired materials: an electrochemically controlled polymeric system which mimics biological learning behavior Victor Erokhin Institute of Crystallography Russian Academy of Sciences Department of
More informationNumerical solutions for a coupled non-linear oscillator
Journal of Mathematical Chemistry Vol. 28, No. 4, 2000 Numerical solutions for a coupled non-linear oscillator A.B. Gumel a,, W.F. Langford a,,e.h.twizell b and J. Wu c a The Fields Institute for Research
More informationMinimal periods of semilinear evolution equations with Lipschitz nonlinearity
Minimal periods of semilinear evolution equations with Lipschitz nonlinearity James C. Robinson a Alejandro Vidal-López b a Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. b Departamento
More informationA Model of Evolutionary Dynamics with Quasiperiodic Forcing
paper presented at Society for Experimental Mechanics (SEM) IMAC XXXIII Conference on Structural Dynamics February 2-5, 205, Orlando FL A Model of Evolutionary Dynamics with Quasiperiodic Forcing Elizabeth
More informationSpatio-Temporal Chaos in Pattern-Forming Systems: Defects and Bursts
Spatio-Temporal Chaos in Pattern-Forming Systems: Defects and Bursts with Santiago Madruga, MPIPKS Dresden Werner Pesch, U. Bayreuth Yuan-Nan Young, New Jersey Inst. Techn. DPG Frühjahrstagung 31.3.2006
More informationLecture No 1 Introduction to Diffusion equations The heat equat
Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and
More informationExcitation of Waves in a Belousov-Zhabotinsky System in Emulsion Media
Excitation of Waves in a Belousov-Zhabotinsky System in Emulsion Media Bull. Korean Chem. Soc. 2008, Vol. 29, No. 11 2241 Excitation of Waves in a Belousov-Zhabotinsky System in Emulsion Media C. Basavaraja,
More informationDynamical Systems and Chaos Part II: Biology Applications. Lecture 10: Coupled Systems. Ilya Potapov Mathematics Department, TUT Room TD325
Dynamical Systems and Chaos Part II: Biology Applications Lecture 10: Coupled Systems. Ilya Potapov Mathematics Department, TUT Room TD325 Foreword In order to model populations of physical/biological
More informationLectu re Notes in Biomathematics
Lectu re Notes in Biomathematics Managing Editor: S. Levin 10 John J. Tyson The Belousov-Zhabotinskii Reaction Springer-Verlag Berlin Heidelberg New York 1976 Editorial Board W. Bossert H. J. Bremermann.
More information6.2 Brief review of fundamental concepts about chaotic systems
6.2 Brief review of fundamental concepts about chaotic systems Lorenz (1963) introduced a 3-variable model that is a prototypical example of chaos theory. These equations were derived as a simplification
More informationDispersion relations, linearization and linearized dynamics in PDE models
Dispersion relations, linearization and linearized dynamics in PDE models 1 Dispersion relations Suppose that u(x, t) is a function with domain { < x 0}, and it satisfies a linear, constant coefficient
More informationLattice Bhatnagar Gross Krook model for the Lorenz attractor
Physica D 154 (2001) 43 50 Lattice Bhatnagar Gross Krook model for the Lorenz attractor Guangwu Yan a,b,,liyuan a a LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences,
More informationScroll Waves in Anisotropic Excitable Media with Application to the Heart. Sima Setayeshgar Department of Physics Indiana University
Scroll Waves in Anisotropic Excitable Media with Application to the Heart Sima Setayeshgar Department of Physics Indiana University KITP Cardiac Dynamics Mini-Program 1 Stripes, Spots and Scrolls KITP
More informationTURING AND HOPF PATTERNS FORMATION IN A PREDATOR-PREY MODEL WITH LESLIE-GOWER-TYPE FUNCTIONAL RESPONSE
Dynamics of Continuous, Discrete and Impulsive Systems Series B: Algorithms and Applications 16 2009) 479-488 Copyright c 2009 Watam Press http://www.watam.org TURING AND HOPF PATTERNS FORMATION IN A PREDATOR-PREY
More informationBifurcations of Traveling Wave Solutions for a Generalized Camassa-Holm Equation
Computational and Applied Mathematics Journal 2017; 3(6): 52-59 http://www.aascit.org/journal/camj ISSN: 2381-1218 (Print); ISSN: 2381-1226 (Online) Bifurcations of Traveling Wave Solutions for a Generalized
More informationz x = f x (x, y, a, b), z y = f y (x, y, a, b). F(x, y, z, z x, z y ) = 0. This is a PDE for the unknown function of two independent variables.
Chapter 2 First order PDE 2.1 How and Why First order PDE appear? 2.1.1 Physical origins Conservation laws form one of the two fundamental parts of any mathematical model of Continuum Mechanics. These
More informationPhysics Dec Time Independent Solutions of the Diffusion Equation
Physics 301 10-Dec-2004 33-1 Time Independent Solutions of the Diffusion Equation In some cases we ll be interested in the time independent solution of the diffusion equation Why would be interested in
More informationTime-periodic forcing of Turing patterns in the Brusselator model
Time-periodic forcing of Turing patterns in the Brusselator model B. Peña and C. Pérez García Instituto de Física. Universidad de Navarra, Irunlarrea, 1. 31008-Pamplona, Spain Abstract Experiments on temporal
More informationReport submitted to Prof. P. Shipman for Math 540, Fall 2009
Dynamics at the Horsetooth Volume 1, 009. Three-Wave Interactions of Spin Waves Aaron Hagerstrom Department of Physics Colorado State University aaronhag@rams.colostate.edu Report submitted to Prof. P.
More information