Chemical Kinetics and the Rössler System. 1 Introduction. 2 The NH 3  HCl reaction. Dynamics at the Horsetooth Volume 2, 2010.


 Ginger Osborne
 1 years ago
 Views:
Transcription
1 Dynamics at the Horsetooth Volume 2, 21. Chemical Kinetics and the Rössler System Department of Mathematics Colorado State University Report submitted to Prof. P. Shipman for Math 4, Fall 21 Abstract. Chemical oscillations present questions about both the chemical mechanism behind the oscillations and the potential for mathematical models. The reaction of ammonia with hydrocloric acid, which is know to produce Liesegang ring patterns [1], exhibits oscillation patterns of mathematical interest [6]. This paper describes a mathematical model for this chemical system, as well as a comparison of experimental data to the wellknown Rössler System. Keywords: Diffusion Equation, Law of Mass Action, Rössler System, Periodicity, Chaos 1 Introduction During the summer of 21 at Colorado State University, I was awarded a CIMS grant and worked with Dr. Patrick Shipman of the Department of Mathematics and Dr. Stephen Thompson of the Department of Chemistry. We studied the reaction of ammonia with hyrdrocloric acid, a reaction that forms an interesting pattern which falls under the broad classification of Liesegang Rings [1]. My interest in this problem continued into the fall of 21, where I took Math 4, Dynamical Systems with Dr. Patrick Shipman, and I realized that interesting connections could be made from my summer work to concepts learned in class. In this paper, I will summarize some of the results obtained during my summer research which includes a mathematical model of the NH 3 HCl system. Then I will discuss the Rössler System, a wellknown system of equations which exhibits very nice periodic patterns. I will conclude with a comparison of actual experimental data from Tim Lenczycki s work [6] to both the mathematical model and the Rössler System. 2 The NH 3  HCl reaction For the modeling of this system, we considered a tube with ammonia (NH 3 ) on one end of the tube and hydrocloric acid (HCl) at the other end, as was the case in the data which we were trying to model [6]. We will denote [HCl] = a and [NH 3 ]=b. Since these two chemicals will react while in their gas phase, we certainly know that they will both obey the diffusion equation a t = D 2 a a t 2
2 b t = D 2 b b t 2 where D a and D b are diffusion coefficients. Now we consider the forward reaction of the system, where NH 3 (g) + HCl(g) NH 4 Cl(s) with reaction rate k 1. Denote [NH 4 Cl] = s. Then, by the Law of Mass Action, we have the following set of equations: a t = D 2 a a t 2 k 1ab b t = D 2 b b t 2 k 1ab s t = k 1ab. These equations are the basic building blocks of our model. We now proceed to include more of the details of this system. We know that both heterogeneous and homogeneous nucleation are important in this chemical reaction [6]. Mathematically, we needed to model the large jump in the reaction rate that occurs once ab exceeds a certain amount η. In Figure 1, we see an example of this where η = k 1 H(ab ) ab Figure 1: Jump in the reation rate as a result of nucleation. We modeled this behavior using the following equations a t = D 2 a a t 2 k 1H(ab η)ab b t = D 2 b b t 2 k 1H(ab η)ab s t = k 1H(ab η)ab. Dynamics at the Horsetooth 2 Vol. 2, 21
3 Using these equations, we were able to see oscillations in our solutions as seen in Figures 2, 3, and 4. Figure 2: The reaction front of the NH 3  HCl system as a function of position and time. Figure 3: A top view of the reaction front; the movement of the reaction front. Figure 4: Diffusion of NH 3 and HCl from either sides of the tube. The oscillations are a promising sign that we are at least heading in the correct direction, Dynamics at the Horsetooth 3 Vol. 2, 21
4 although we should certainly test our model against actual experimental data. Future modifications to this model are discussed in the conclusion of this paper. 3 The Rössler System We now turn to a wellknown system of equations, called the Rössler System. The Rössler equations are as follows dx = (y + z) dt dy dt = x + ay dz = b + xz cz dt where a, b, and c are parameters. If we let a = b =.2 and let c vary and we choose initial conditions x() = y() = z() = 1, we will see that we get very interesting patterns. When c =2.3, we can see that we get periodone behavior as shown in Figures and 6. Note that this periodone behavior, which begins after about t = 2, could be represented by a simple oscillatory function such as sin(x). The Rossler System 4 3 z(t) 2 1 y(t) Figure : Periodone limit cycle when c = 2.3. Dynamics at the Horsetooth 4 Vol. 2, 21
5 t Figure 6: Periodone behavior with respect to. When c =3.3, as shown in Figures 7 and 8, we see periodtwo behavior here. Note that we get two distinct amplitudes here as shown in Figure 8. The Rossler System 8 6 z(t) 4 2 y(t) Figure 7: Periodtwo limit cycle when c = 3.3. Dynamics at the Horsetooth Vol. 2, 21
6 t Figure 8: Periodtwo behavior with respect to. In the same way, we can let c =.3 and get periodthree behavior, as shown in Figures 9 and 1. The Rossler System 2 1 z(t) 1 1 y(t) Figure 9: Periodthree limit cycle when c =.3. Dynamics at the Horsetooth 6 Vol. 2, 21
7 t Figure 1: Periodthree behavior with respect to. Finally, if we consider the case where c = 6.3, as in Figure 11, we can see that a slight change in the initial conditions has caused a great change in the solution. In the top part of Figure 11, we have initial conditions x() = y() = z() = 1 while in the bottom part, we have x() = 1.1,y() = z() = 1. Notice that the difference in the two solutions begins right around t =. z(t) z(t) y(t) y(t) The Rossler System The Rossler System t t Figure 11: The system exhibits chaotic behavior when c = 6.3. Dynamics at the Horsetooth 7 Vol. 2, 21
8 4 Highlights of the Experimental Data Experimental Data was collected for the NH 3  HCl reaction which measures the position of the reaction front as time progressed [6]. It was noted that the reaction front of this system oscillates and oftentimes in a very periodic manner. We now discuss some specific examples of this. During a specific time segment of the NH 3  HCl reaction, the reaction front oscillated as shown in Figure 12, which appears to exhibit periodone behavior. This is confirmed by the Fourier Analysis in Figure 13 since we obtain one clear peak.. Figure 12:. Figure 13: In a different time segment of the reaction, we find more periodic behavior. According to the Fourier Analysis, it appears that during this time segment, we have periodtwo behavior as shown in Figures 14 and 1. Dynamics at the Horsetooth 8 Vol. 2, 21
9 . Figure 14:. Figure 1: Finally, chaotic behavior of the oscillation front is apparent during another time segment of the reaction. This is shown in Figures 16 and 17. Dynamics at the Horsetooth 9 Vol. 2, 21
10 . Figure 16:. Figure 17: Conclusion This project certainly has many avenues of possiblity for future work. First, we would like to add a fourth equation to our NH 3  HCl model to account for the fact that the NH 4 Cl is, at first, believed to be in the gas phase until enough NH 4 Cl gas molecules form a solid. If we denote [NH 4 Cl (g)] = m (for monomer) and [NH 4 Cl (s)] = s (for solid), these equations should look something like a t = D 2 a a t 2 k 1ab b t = D 2 b b t 2 k 1ab Dynamics at the Horsetooth 1 Vol. 2, 21
11 m t = D 2 m m t 2 + k 1ab k 2 H(m η) s t = k 2H(m η) k 3 s. We would also like to consider the idea of autocatalysis within our model which, simply put, is the idea that the more NH 4 Cl (g) there is, the faster the reaction rate will be. We have also done simple experiment of this NH 3  HCl Reaction but in a vertical tube. This has resulted in a tornadolike reaction where oscillations are visible but a mathematical model for this would have to take into account things such as fluid flow and the effect that it has on these oscillations. Dynamics at the Horsetooth 11 Vol. 2, 21
12 References [1] Dee, G.T. Patterns Produced by Precipitation at a Moving Reaction Front. Physical Review Letters 7.3 (1986): [2] Hirsch, Morris W., Stephen Smale, and Robert L. Devaney. Differential Equations, Dynamical Systems, and an Introduction to Chaos. Elsevier, 24. [3] Holmes, Mark H. Introduction to the Foundations of Applied Mathematics. New York: Springer, 29. [4] Howard, P. Partial Differential Equations in MATLAB 7.. Spring 2. [] Lebedeva, M.I., D.G. Vlachos, and M. Tsapatsis. Bifurcation Analysis of Liesegang Ring Pattern Formation. Physical Review Letters 92.8 (24): 8831(1)8831(4). [6] Lenczycki, Timothy T. (Thesis) Oscillations in Gas Phase Periodic Precipitation Systems: The NH 3 HCl Story. Spring 23. [7] Lynch, Stephen. Dynamical Systems with Applications using MATLAB. Boston: Birkhauser, 24. [8] Strogatz, Steven H. Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry, and Engineering. AddisonWesley, [9] Verhulst, Ferdinand. Nonlinear Differential Equations and Dynamical Systems. Second edition. New York: SpringerVerlag Berlin Heidelberg, Dynamics at the Horsetooth 12 Vol. 2, 21
ME DYNAMICAL SYSTEMS SPRING SEMESTER 2009
ME 406  DYNAMICAL SYSTEMS SPRING SEMESTER 2009 INSTRUCTOR Alfred Clark, Jr., Hopeman 329, x54078; Email: clark@me.rochester.edu Office Hours: M T W Th F 1600 1800. COURSE TIME AND PLACE T Th 1400 1515
More informationDIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS
DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS Morris W. Hirsch University of California, Berkeley Stephen Smale University of California, Berkeley Robert L. Devaney Boston University
More informationDIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS
DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS Morris W. Hirsch University of California, Berkeley Stephen Smale University of California, Berkeley Robert L. Devaney Boston University
More informationWhy are Discrete Maps Sufficient?
Why are Discrete Maps Sufficient? Why do dynamical systems specialists study maps of the form x n+ 1 = f ( xn), (time is discrete) when much of the world around us evolves continuously, and is thus well
More informationDifferential Equations Dynamical Systems And An Introduction To Chaos Solutions Manual Pdf
Differential Equations Dynamical Systems And An Introduction To Chaos Solutions Manual Pdf Math 134: Ordinary Differential Equations and Dynamical Systems and an Introduction to Chaos, third edition, Academic
More informationDynamical Systems with Applications
Stephen Lynch Dynamical Systems with Applications using MATLAB Birkhauser Boston Basel Berlin Preface xi 0 A Tutorial Introduction to MATLAB and the Symbolic Math Toolbox 1 0.1 Tutorial One: The Basics
More informationNonlinear Systems, Chaos and Control in Engineering
Coordinating unit: Teaching unit: Academic year: Degree: ECTS credits: 2018 205  ESEIAAT  Terrassa School of Industrial, Aerospace and Audiovisual Engineering 748  FIS  Department of Physics BACHELOR'S
More informationMATH 308 Differential Equations
MATH 308 Differential Equations Summer, 2014, SET 1 JoungDong Kim Set 1: Section 1.1, 1.2, 1.3, 2.1 Chapter 1. Introduction 1. Why do we study Differential Equation? Many of the principles, or laws, underlying
More informationDiscrete Time Coupled Logistic Equations with Symmetric Dispersal
Discrete Time Coupled Logistic Equations with Symmetric Dispersal Tasia Raymer Department of Mathematics araymer@math.ucdavis.edu Abstract: A simple two patch logistic model with symmetric dispersal between
More informationAn Application of Perturbation Methods in Evolutionary Ecology
Dynamics at the Horsetooth Volume 2A, 2010. Focused Issue: Asymptotics and Perturbations An Application of Perturbation Methods in Evolutionary Ecology Department of Mathematics Colorado State University
More informationCalculating Fractal Dimension of Attracting Sets of the Lorenz System
Dynamics at the Horsetooth Volume 6, 2014. Calculating Fractal Dimension of Attracting Sets of the Lorenz System Jamie Department of Mathematics Colorado State University Report submitted to Prof. P. Shipman
More informationMATH 415, WEEKS 7 & 8: Conservative and Hamiltonian Systems, Nonlinear Pendulum
MATH 415, WEEKS 7 & 8: Conservative and Hamiltonian Systems, Nonlinear Pendulum Reconsider the following example from last week: dx dt = x y dy dt = x2 y. We were able to determine many qualitative features
More informationCHEMUA 652: Thermodynamics and Kinetics
CHEMUA 65: Thermodynamics and Kinetics Notes for Lecture I. THE COMPLEXITY OF MULTISTEP CHEMICAL REACTIONS It should be clear by now that chemical kinetics is governed by the mathematics of systems of
More informationQuestion: Total. Points:
MATH 308 May 23, 2011 Final Exam Name: ID: Question: 1 2 3 4 5 6 7 8 9 Total Points: 0 20 20 20 20 20 20 20 20 160 Score: There are 9 problems on 9 pages in this exam (not counting the cover sheet). Make
More informationSome Dynamical Behaviors In Lorenz Model
International Journal Of Computational Engineering Research (ijceronline.com) Vol. Issue. 7 Some Dynamical Behaviors In Lorenz Model Dr. Nabajyoti Das Assistant Professor, Department of Mathematics, Jawaharlal
More informationCHEM 515: Chemical Kinetics and Dynamics
Alejandro J. Garza S01163018 Department of Chemistry, Rice University, Houston, TX email: ajg7@rice.edu, ext. 2657 Submitted December 12, 2011 Abstract Spontaneous antispiral wave formation was observed
More informationEen vlinder in de wiskunde: over chaos en structuur
Een vlinder in de wiskunde: over chaos en structuur Bernard J. Geurts Enschede, November 10, 2016 Tuin der Lusten (Garden of Earthly Delights) In all chaos there is a cosmos, in all disorder a secret
More informationDIFFERENTIAL EQUATIONS DYNAMICAL SYSTEMS AND AN INTRODUCTION TO CHAOS PDF
DIFFERENTIAL EQUATIONS DYNAMICAL SYSTEMS AND AN INTRODUCTION TO CHAOS PDF ==> Download: DIFFERENTIAL EQUATIONS DYNAMICAL SYSTEMS AND AN INTRODUCTION TO CHAOS PDF DIFFERENTIAL EQUATIONS DYNAMICAL SYSTEMS
More informationResearch Article Hidden Periodicity and Chaos in the Sequence of Prime Numbers
Advances in Mathematical Physics Volume 2, Article ID 5978, 8 pages doi:.55/2/5978 Research Article Hidden Periodicity and Chaos in the Sequence of Prime Numbers A. Bershadskii Physics Department, ICAR,
More informationONE DIMENSIONAL CHAOTIC DYNAMICAL SYSTEMS
Journal of Pure and Applied Mathematics: Advances and Applications Volume 0 Number 0 Pages 690 ONE DIMENSIONAL CHAOTIC DYNAMICAL SYSTEMS HENA RANI BISWAS Department of Mathematics University of Barisal
More informationWHAT IS A CHAOTIC ATTRACTOR?
WHAT IS A CHAOTIC ATTRACTOR? CLARK ROBINSON Abstract. Devaney gave a mathematical definition of the term chaos, which had earlier been introduced by Yorke. We discuss issues involved in choosing the properties
More informationQ.1 Write out equations for the reactions between...
1 CHEMICAL EQUILIBRIUM Dynamic Equilibrium not all reactions proceed to completion some end up with a mixture of reactants and products this is because some reactions are reversible; products revert to
More informationProblem Set Number 2, j/2.036j MIT (Fall 2014)
Problem Set Number 2, 18.385j/2.036j MIT (Fall 2014) Rodolfo R. Rosales (MIT, Math. Dept.,Cambridge, MA 02139) Due Mon., September 29, 2014. 1 Inverse function problem #01. Statement: Inverse function
More informationLab 5: Nonlinear Systems
Lab 5: Nonlinear Systems Goals In this lab you will use the pplane6 program to study two nonlinear systems by direct numerical simulation. The first model, from population biology, displays interesting
More informationSD  Dynamical Systems
Coordinating unit: 200  FME  School of Mathematics and Statistics Teaching unit: 749  MAT  Department of Mathematics Academic year: Degree: 2018 BACHELOR'S DEGREE IN MATHEMATICS (Syllabus 2009). (Teaching
More informationZEROHOPF BIFURCATION FOR A CLASS OF LORENZTYPE SYSTEMS
This is a preprint of: ZeroHopf bifurcation for a class of Lorenztype systems, Jaume Llibre, Ernesto PérezChavela, Discrete Contin. Dyn. Syst. Ser. B, vol. 19(6), 1731 1736, 214. DOI: [doi:1.3934/dcdsb.214.19.1731]
More informationHomogeneous Equations with Constant Coefficients
Homogeneous Equations with Constant Coefficients MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 General Second Order ODE Second order ODEs have the form
More informationMATH 312 Section 8.3: Nonhomogeneous Systems
MATH 32 Section 8.3: Nonhomogeneous Systems Prof. Jonathan Duncan Walla Walla College Spring Quarter, 2007 Outline Undetermined Coefficients 2 Variation of Parameter 3 Conclusions Undetermined Coefficients
More informationNONLINEAR DYNAMICS PHYS 471 & PHYS 571
NONLINEAR DYNAMICS PHYS 471 & PHYS 571 Prof. R. Gilmore 12918 X2779 robert.gilmore@drexel.edu Office hours: 14:00 Quarter: Winter, 20142015 Course Schedule: Tuesday, Thursday, 11:0012:20 Room: 12919
More informationOn the periodic logistic equation
On the periodic logistic equation Ziyad AlSharawi a,1 and James Angelos a, a Central Michigan University, Mount Pleasant, MI 48858 Abstract We show that the pperiodic logistic equation x n+1 = µ n mod
More informationA STUDY OF A VAN DER POL EQUATION
Journal of Mathematical Sciences: Advances and Applications Volume, 03, Pages 3943 A STUDY OF A VAN DER POL EQUATION A. M. MARIN, R. D. ORTIZ and J. A. RODRIGUEZ Faculty of Exact and Natural Sciences
More informationNonlinear Dynamics and Chaos Summer 2011
67717 Nonlinear Dynamics and Chaos Summer 2011 Instructor: Zoubir Benzaid Phone: 4247354 Office: Swart 238 Office Hours: MTWR: 8:309:00; MTWR: 12:001:00 and by appointment. Course Content: This course
More informationDynamical Systems with Applications using Mathematica
Stephen Lynch Dynamical Systems with Applications using Mathematica Birkhäuser Boston Basel Berlin Contents Preface xi 0 A Tutorial Introduction to Mathematica 1 0.1 A Quick Tour of Mathematica 2 0.2 Tutorial
More informationSymmetries 2  Rotations in Space
Symmetries 2  Rotations in Space This symmetry is about the isotropy of space, i.e. space is the same in all orientations. Thus, if we continuously rotated an entire system in space, we expect the system
More informationISSN X (print) BIFURCATION ANALYSIS OF FRACTIONALORDER CHAOTIC RÖSSLER SYSTEM
Matematiqki Bilten ISSN 0351336X (print) 42(LXVIII) No 1 ISSN 18579914 (online) 2018(2736) UDC: 517938:5198765 Skopje, Makedonija BIFURCATION ANALYSIS OF FRACTIONALORDER CHAOTIC RÖSSLER SYSTEM GJORGJI
More informationReconstruction Deconstruction:
Reconstruction Deconstruction: A Brief History of Building Models of Nonlinear Dynamical Systems Jim Crutchfield Center for Computational Science & Engineering Physics Department University of California,
More informationPerformance Evaluation of Generalized Polynomial Chaos
Performance Evaluation of Generalized Polynomial Chaos Dongbin Xiu, Didier Lucor, C.H. Su, and George Em Karniadakis 1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA, gk@dam.brown.edu
More informationLogistic Map f(x) = x(1 x) is Topologically Conjugate to the Map f(x) =(2 ) x(1 x)
Tamkang Journal of Science and Engineering, Vol. 10, No 1, pp. 8994 (2007) 89 Logistic Map f(x) =x(1 x) is Topologically Conjugate to the Map f(x) =(2) x(1 x) ChyiLung Lin* and MonLing Shei Department
More informationNonlinear Dynamics. Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna.
Nonlinear Dynamics Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna http://www.moreno.marzolla.name/ 2 Introduction: Dynamics of Simple Maps 3 Dynamical systems A dynamical
More informationME8230 Nonlinear Dynamics
ME8230 Nonlinear Dynamics Lecture 1, part 1 Introduction, some basic math background, and some random examples Prof. Manoj Srinivasan Mechanical and Aerospace Engineering srinivasan.88@osu.edu Spring mass
More informationAPPM 2460 CHAOTIC DYNAMICS
APPM 2460 CHAOTIC DYNAMICS 1. Introduction Today we ll continue our exploration of dynamical systems, focusing in particular upon systems who exhibit a type of strange behavior known as chaos. We will
More informationComplex system approach to geospace and climate studies. Tatjana Živković
Complex system approach to geospace and climate studies Tatjana Živković 30.11.2011 Outline of a talk Importance of complex system approach Phase space reconstruction Recurrence plot analysis Test for
More informationEntrainment Alex Bowie April 7, 2004
Entrainment Alex Bowie April 7, 2004 Abstract The driven Van der Pol oscillator displays entrainment, quasiperiodicity, and chaos. The characteristics of these different modes are discussed as well as
More informationAn Undergraduate s Guide to the HartmanGrobman and PoincaréBendixon Theorems
An Undergraduate s Guide to the HartmanGrobman and PoincaréBendixon Theorems Scott Zimmerman MATH181HM: Dynamical Systems Spring 2008 1 Introduction The HartmanGrobman and PoincaréBendixon Theorems
More informationDynamics at the Horsetooth Volume 2A, Focused Issue: Asymptotics and Perturbations, 2010.
Dynamics at the Horsetooth Volume 2A, Focused Issue: Asymptotics Perturbations, 2010. Perturbation Theory the WKB Method Department of Mathematics Colorado State University shinn@math.colostate.edu Report
More informationMath 240: Springmass Systems
Math 240: Springmass Systems Ryan Blair University of Pennsylvania Tuesday March 1, 2011 Ryan Blair (U Penn) Math 240: Springmass Systems Tuesday March 1, 2011 1 / 15 Outline 1 Review 2 Today s Goals
More informationExperimental Characterization of Nonlinear Dynamics from Chua s Circuit
Experimental Characterization of Nonlinear Dynamics from Chua s Circuit John Parker*, 1 Majid Sodagar, 1 Patrick Chang, 1 and Edward Coyle 1 School of Physics, Georgia Institute of Technology, Atlanta,
More informationReport submitted to Prof. P. Shipman for Math 540, Fall 2009
Dynamics at the Horsetooth Volume 1, 009. ThreeWave Interactions of Spin Waves Aaron Hagerstrom Department of Physics Colorado State University aaronhag@rams.colostate.edu Report submitted to Prof. P.
More informationUnit VI Stoichiometry. Applying Mole Town to Reactions
Unit VI Stoichiometry Applying Mole Town to Reactions Learning Goals I can apply mole town to reactions to determine the amount of product based on the amount of a reactant. I can apply mole town to reaction
More informationMole Ratios. How can the coefficients in a chemical equation be interpreted? (g) 2NH 3. (g) + 3H 2
Why? Mole Ratios How can the coefficients in a chemical equation be interpreted? A balanced chemical equation can tell us the number of reactant and product particles (ions, atoms, molecules or formula
More informationNonlinear Dynamics And Chaos By J. M. T. Thompson;H. B. Stewart
Nonlinear Dynamics And Chaos By J. M. T. Thompson;H. B. Stewart If you are searched for a book Nonlinear Dynamics and Chaos by J. M. T. Thompson;H. B. Stewart in pdf format, then you've come to the right
More informationDYNAMICS OF A DISCRETE BRUSSELATOR MODEL: ESCAPE TO INFINITY AND JULIA SET
DYNAMICS OF A DISCETE BUSSELATO MODEL: ESCAPE TO INFINITY AND JULIA SET HUNSEOK KANG AND YAKOV PESIN Abstract. We consider a discrete version of the Brusselator Model of the famous BelousovZhabotinsky
More informationMATH The Chain Rule Fall 2016 A vector function of a vector variable is a function F: R n R m. In practice, if x 1, x n is the input,
MATH 20550 The Chain Rule Fall 2016 A vector function of a vector variable is a function F: R n R m. In practice, if x 1, x n is the input, F(x 1,, x n ) F 1 (x 1,, x n ),, F m (x 1,, x n ) where each
More informationMath 266: Ordinary Differential Equations
Math 266: Ordinary Differential Equations Long Jin Purdue University, Spring 2018 Basic information Lectures: MWF 8:309:20(111)/9:3010:20(121), UNIV 103 Instructor: Long Jin (long249@purdue.edu) Office
More informationFundamentals of Transport Processes Prof. Kumaran Indian Institute of Science, Bangalore Chemical Engineering
Fundamentals of Transport Processes Prof. Kumaran Indian Institute of Science, Bangalore Chemical Engineering Module No # 05 Lecture No # 25 Mass and Energy Conservation Cartesian Coordinates Welcome
More informationOscillations in Damped Driven Pendulum: A Chaotic System
International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 3, Issue 10, October 2015, PP 1427 ISSN 2347307X (Print) & ISSN 23473142 (Online) www.arcjournals.org Oscillations
More informationTurbulence and Devaney s Chaos in Interval
Global Journal of Pure and Applied Mathematics. ISSN 09731768 Volume 13, Number 2 (2017), pp. 713717 Research India Publications http://www.ripublication.com Turbulence and Devaney s Chaos in Interval
More informationCHAOTIC BEHAVIOR IN A FORECAST MODEL
CHAOTIC BEHAVIOR IN A FORECAST MODEL MICHAEL BOYLE AND MARK TOMFORDE Abstract. We examine a certain interval map, called the weather map, that has been used by previous authors as a toy model for weather
More informationFigure 1: Schematic of ship in still water showing the action of bouyancy and weight to right the ship.
MULTIDIMENSIONAL SYSTEM: In this computer simulation we will explore a nonlinear multidimensional system. As before these systems are governed by equations of the form x 1 = f 1 x 2 = f 2.. x n = f n
More informationOn new chaotic mappings in symbol space
Acta Mech Sin DOI 10.1007/s1040901104081 RESEARCH PAPER On new chaotic mappings in symbol space Inese Bula Jānis Buls Irita Rumbeniece Received: 24 August 2010 / Revised: 15 September 2010 / Accepted:
More informationApplied Dynamical Systems
Applied Dynamical Systems Recommended Reading: (1) Morris W. Hirsch, Stephen Smale, and Robert L. Devaney. Differential equations, dynamical systems, and an introduction to chaos. Elsevier/Academic Press,
More informationTimeSeries Based Prediction of Dynamical Systems and Complex. Networks
TimeSeries Based Prediction of Dynamical Systems and Complex Collaborators Networks YingCheng Lai ECEE Arizona State University Dr. Wenxu Wang, ECEE, ASU Mr. Ryan Yang, ECEE, ASU Mr. Riqi Su, ECEE, ASU
More informationTopic 5.1: Line Element and Scalar Line Integrals
Math 275 Notes Topic 5.1: Line Element and Scalar Line Integrals Textbook Section: 16.2 More Details on Line Elements (vector dr, and scalar ds): http://www.math.oregonstate.edu/bridgebook/book/math/drvec
More informationINVESTIGATION OF CHAOTICITY OF THE GENERALIZED SHIFT MAP UNDER A NEW DEFINITION OF CHAOS AND COMPARE WITH SHIFT MAP
ISSN 2411247X INVESTIGATION OF CHAOTICITY OF THE GENERALIZED SHIFT MAP UNDER A NEW DEFINITION OF CHAOS AND COMPARE WITH SHIFT MAP Hena Rani Biswas * Department of Mathematics, University of Barisal, Barisal
More informationUnit Ten Summary Introduction to Dynamical Systems and Chaos
Unit Ten Summary Introduction to Dynamical Systems Dynamical Systems A dynamical system is a system that evolves in time according to a welldefined, unchanging rule. The study of dynamical systems is
More informationCOLLATZ CONJECTURE: IS IT FALSE?
COLLATZ CONJECTURE: IS IT FALSE? JUAN A. PEREZ arxiv:1708.04615v2 [math.gm] 29 Aug 2017 ABSTRACT. For a long time, Collatz Conjecture has been assumed to be true, although a formal proof has eluded all
More informationAPPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS
LECTURE 10 APPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS Ordinary Differential Equations Initial Value Problems For Initial Value problems (IVP s), conditions are specified
More informationFreeenergy change ( G) and entropy change ( S)
Freeenergy change ( G) and entropy change ( S) A SPONTANEOUS PROCESS (e.g. diffusion) will proceed on its own without any external influence. A problem with H A reaction that is exothermic will result
More informationNonlinear Dynamics And Chaos PDF
Nonlinear Dynamics And Chaos PDF This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods,
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 informationNotes on the prion model
Notes on the prion model Introduction Written by Jeremy Gunawardena (jeremy@hmsharvardedu) for MCB 195, A systems approach to biology, on 10 February 2005, with thanks to Shai ShenOrr and Becky Ward for
More informationChain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics
3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)
More information2DVolterraLotka Modeling For 2 Species
Majalat AlUlum AlInsaniya wat  Tatbiqiya 2DVolterraLotka Modeling For 2 Species Alhashmi Darah 1 University of Almergeb Department of Mathematics Faculty of Science Zliten Libya. Abstract The purpose
More informationThe Kinetic Molecular Theory of Gases
The Kinetic Molecular Theory of Gases Background: It is straightforward to observe that there is an inverse relationship between pressure and volume for a gas at constant temperature. Curious scientists
More informationLectures on Periodic Orbits
Lectures on Periodic Orbits 11 February 2009 Most of the contents of these notes can found in any typical text on dynamical systems, most notably Strogatz [1994], Perko [2001] and Verhulst [1996]. Complete
More informationKeble College  Hilary 2014 CP3&4: Mathematical methods I&II Tutorial 5  Waves and normal modes II
Tomi Johnson 1 Keble College  Hilary 2014 CP3&4: Mathematical methods I&II Tutorial 5  Waves and normal modes II Prepare full solutions to the problems with a self assessment of your progress on a cover
More informationSolving Zhou Chaotic System Using FourthOrder RungeKutta Method
World Applied Sciences Journal 21 (6): 939944, 2013 ISSN 114952 IDOSI Publications, 2013 DOI: 10.529/idosi.wasj.2013.21.6.2915 Solving Zhou Chaotic System Using FourthOrder RungeKutta Method 1 1 3
More informationNonlinear systems, chaos and control in Engineering
Nonlinear systems, chaos and control in Engineering Module 1 block 3 Onedimensional nonlinear systems Cristina Masoller Cristina.masoller@upc.edu http://www.fisica.edu.uy/~cris/ Schedule Flows on the
More informationAbout Some Features of a Magma Flow Structure at Explosive Volcano Eruptions
About Some Features of a Magma Flow Structure at Explosive Volcano Eruptions V. Kedrinskiy 1 Introduction The cyclic character of magma ejections is one of the basic aspects in the research field of the
More informationDynamics of a massspringpendulum system with vastly different frequencies
Dynamics of a massspringpendulum system with vastly different frequencies Hiba Sheheitli, hs497@cornell.edu Richard H. Rand, rhr2@cornell.edu Cornell University, Ithaca, NY, USA Abstract. We investigate
More informationMath 308 Week 8 Solutions
Math 38 Week 8 Solutions There is a solution manual to Chapter 4 online: www.pearsoncustom.com/tamu math/. This online solutions manual contains solutions to some of the suggested problems. Here are solutions
More informationReport submitted to Prof. P. Shipman for Math 641, Spring 2012
Dynamics at the Horsetooth Volume 4, 2012. The WeierstrassEnneper Representations Department of Mathematics Colorado State University mylak@rams.colostate.edu drpackar@rams.colostate.edu Report submitted
More informationLecture 3 : Bifurcation Analysis
Lecture 3 : Bifurcation Analysis D. Sumpter & S.C. Nicolis October  December 2008 D. Sumpter & S.C. Nicolis General settings 4 basic bifurcations (as long as there is only one unstable mode!) steady state
More informationSimple conservative, autonomous, secondorder chaotic complex variable systems.
Simple conservative, autonomous, secondorder chaotic complex variable systems. Delmar Marshall 1 (Physics Department, Amrita Vishwa Vidyapeetham, Clappana P.O., Kollam, Kerala 690525, India) and J. C.
More informationEnhanced sensitivity of persistent events to weak forcing in dynamical and stochastic systems: Implications for climate change. Khatiwala, et.al.
Enhanced sensitivity of persistent events to weak forcing in dynamical and stochastic systems: Implications for climate change Questions What are the characteristics of the unforced Lorenz system? What
More informationBifurcation and chaos in simple jerk dynamical systems
PRAMANA c Indian Academy of Sciences Vol. 64, No. 1 journal of January 2005 physics pp. 75 93 Bifurcation and chaos in simple jerk dynamical systems VINOD PATIDAR and K K SUD Department of Physics, College
More informationExample 4.1 Let X be a random variable and f(t) a given function of time. Then. Y (t) = f(t)x. Y (t) = X sin(ωt + δ)
Chapter 4 Stochastic Processes 4. Definition In the previous chapter we studied random variables as functions on a sample space X(ω), ω Ω, without regard to how these might depend on parameters. We now
More informationIntroduction to Dynamical Systems Basic Concepts of Dynamics
Introduction to Dynamical Systems Basic Concepts of Dynamics A dynamical system: Has a notion of state, which contains all the information upon which the dynamical system acts. A simple set of deterministic
More informationMaps and differential equations
Maps and differential equations Marc R. Roussel November 8, 2005 Maps are algebraic rules for computing the next state of dynamical systems in discrete time. Differential equations and maps have a number
More informationSingular Perturbations in the McMullen Domain
Singular Perturbations in the McMullen Domain Robert L. Devaney Sebastian M. Marotta Department of Mathematics Boston University January 5, 2008 Abstract In this paper we study the dynamics of the family
More informationStates of matter. Particles in a gas are widely spread out and can both vibrate and move around freely. They have the most energy of the three states.
States of matter Particles in a solid are closely packed and can vibrate but cannot move around, they have low energies. Particles in a liquid are still closely packed, but can both vibrate and move around
More informationChapter 14, Chemical Kinetics
Last wee we covered the following material: Review Vapor Pressure with two volatile components Chapter 14, Chemical Kinetics (continued) Quizzes next wee will be on Chap 14 through section 14.5. 13.6 Colloids
More informationChapter 1: Introduction
Chapter 1: Introduction Definition: A differential equation is an equation involving the derivative of a function. If the function depends on a single variable, then only ordinary derivatives appear and
More informationUPPER AND LOWER SOLUTIONS FOR A HOMOGENEOUS DIRICHLET PROBLEM WITH NONLINEAR DIFFUSION AND THE PRINCIPLE OF LINEARIZED STABILITY
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 30, Number 4, Winter 2000 UPPER AND LOWER SOLUTIONS FOR A HOMOGENEOUS DIRICHLET PROBLEM WITH NONLINEAR DIFFUSION AND THE PRINCIPLE OF LINEARIZED STABILITY ROBERT
More informationChapter 14 Chemical Kinetics
How fast do chemical processes occur? There is an enormous range of time scales. Chapter 14 Chemical Kinetics Kinetics also sheds light on the reaction mechanism (exactly how the reaction occurs). Why
More informationBinarycoded and realcoded genetic algorithm in pipeline flow optimization
Mathematical Communications 41999), 3542 35 Binarycoded and realcoded genetic algorithm in pipeline flow optimization Senka Vuković and Luka Sopta Abstract. The mathematical model for the liquidgas
More informationPhysics 250 Green s functions for ordinary differential equations
Physics 25 Green s functions for ordinary differential equations Peter Young November 25, 27 Homogeneous Equations We have already discussed second order linear homogeneous differential equations, which
More informationLecture 2. Classification of Differential Equations and Method of Integrating Factors
Math 245  Mathematics of Physics and Engineering I Lecture 2. Classification of Differential Equations and Method of Integrating Factors January 11, 2012 Konstantin Zuev (USC) Math 245, Lecture 2 January
More informationMath Partial Differential Equations
Math 531  Partial Differential Equations to Partial Differential Equations Joseph M. Mahaffy, jmahaffy@sdsu.edu Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences
More informationM.Sc. in Meteorology. Numerical Weather Prediction
M.Sc. in Meteorology UCD Numerical Weather Prediction Prof Peter Lynch Meteorology & Climate Centre School of Mathematical Sciences University College Dublin Second Semester, 2005 2006. In this section
More information