# MVE162 / MMG511 Ordinary differential equations and mathematical modelling

## MVE162 / MMG511 Ordinary differential equations and mathematical modelling

This page is under development and contains the program of the course: lectures and exercises, and modelling projects . Other information, such as learning outcomes, teachers, literature, lecture notes, records of streamed lectures, and examination are in a separate course PM.

### Program

The schedule of the course is in TimeEdit.

#### Lectures and exercises

 Week Day Topics, notions, theorems, methods Links to lecture notes, to recommended exercises, references W.1(12) Må03-2115:15 Course subject, structure, goals. Notion of I.V.P. for ODE. Integral form of an ODE. Phase plane, examples of orbits, equilibrium points, periodic orbits,Linear ODE with constant coefficients (autonomous)A simple version of Grönwall inequality, Lemma 2.4, p. 27, and uniqueness of solutions. Matrix exponential and general solution to a linear autonomous system.The space of solutions to a linear ODE and it's dimension. Appendix A.1,$2.1.3 Autonomous systems (we consider it with more details in lecture notes)Exercises 2.10, 2.11, p. 35, 2.12, p. 38. On03-2308:00 Properties of matrix exponent. Lemma 2.10 (1),(3),(4),(5), p. 34; Examples of linear systems and their phase portraits. Generalized eigenspaces and eigenvectors.Invariance of the generalized eigenspaces under the action of matrix A and exp(At) Lecture notes to first 6 lectures. Introduction and linear autonomous systemsan error in the Example 1.1 in the book is corrected,Matlab codes for illustrations with vector fields and phase portraits To03-2408:00 Structure of the general solution to linear ODE with constant coefficients;Th. 2.11; p.35 Examples of solutions to linear autonomous ODE: generalized eigenspaces and general solutions Examples of solutions in the case when there is no basis of eigenvectors. §2.1.3 Autonomous systems (we consider it with more details and examples in lecture notes) Download Exercises on linear autonomous ODE W. 2(13) Må03-2815:15 Exercises on solutions to linear autonomous ODE: generalized eigenspaces and general solutions. Real solutions to systems with real matrix having complex eigenvalues Th. 2.14, p. 38. Examples. §2.1.3, §2.4Appendix 1, 2 Shorter list with exercises for the Lecture 4 and home exercises On03-3008:00 Jordan canonical form of matrix. Theorem A.9 , p. 268 and lecture notes.Exponent of Jordan matrix.Examples and exercises on Jordan matrices Exercises on calculations of exponents of matrices and fundamental matrix solutions for linear autonomous ODEs. Lecture notes:Introduction and autonomous linear systems Exercises on linear autonomous ODEswith exercises on exponents of matrices and on Jordan' matrices with some solutions. To03-3108:00 Boundedness and limit properties of the norm of the matrix exponent: Corollary 2.13, p. 36 The first home assignment - modeling project is available in Canvas from the 6-th of April. Deadline is the 29th of April. W. 3(14) Må04-0415:15 Stability and asymptotic stability of equilibrium (stationary) points. Definitions 5.1, p.169, 5.14, p.182. Phase portraits for linear autonomous ODEs in plane and their classification. Exercises on phase portraits of autonomous linear systems in the plain and on exponents of matrices. Matlab codes giving analytical solutions and drawing phase portraits for autonomous linear ODEs Download problems on autonomous linear ODEs and phase portraits Exercises on linear autonomous ODEs with updated calculations of exponents of matrices On04-068:00 In this lecture an introduction to the first project - home assignment is given.Stability of the equilibrium point in the origin for linear systems with constant coefficients. Propositions 5.23, 5.24, 5.25, p.189, p.190.We do it in a simpler way on the lectures. Theorem on existence and uniqueness of solutions to general I.V.P. Non-homogeneous linear systems of ODEs. Variation of constant formula (Duhamel formula) for non-homogeneous linear equation, the case with constant coefficients. Corollary 2.17, p. 43.Stability of equilibrium points for a linear autonomous system perturbed by a “small” nonlinear right hand side. Th. 5.27, p.193. Proof by Grönwall inequality in lecture notes. (simpler than one in the book) Introduction to modelling project and lecture notes on stability by linearization. Matlab codes for illustrations with vector fields and phase portraits To04-078:00 Stability of stationary points by linearization. Simple criteria.Corollary 5.29, p.195 Formulation of the Grobman-Hartman theorem. Exercises on stability by linearization. Exercises on stability by linearization Notes on stability by linearization for the pendulum with friction.Exercises 5.20, 5.21, 5.22, 5.23. To04-07 15:15 Lecture on scientific writing (in Swedish) by Elin Götmark Lecture notes Examination and Easter vacation weeks 11-22 April W. 4(17) Må04-2515:15 Exercises on stability by linearization. Nonlinear systems of ODEs, Chapter 4. Peano existence theorem Th. 4.2, p. 102 (without proof) Existence and uniqueness theorems by Picard and Lindelöf. Th. 4.17, p. 118 (for continuous f(t,x), locally Lipschitz in x), Th.4.22, p.122 (for piecewise continuous f(t,x), locally Lipschitz in x) (proof will be given later, in the last week of the course). Prop. 4.15, p.115; on uniform Lipschitz property on the compact.Maximal solutions. Continuation of solutions.Existence of maximal solutions. Th. 4.8, p.108. Extension of bounded solutions. Lemma 4.9, p. 110; Cor. 4.10, p. 111. Lecture notes on existence and maximal solutions §1.2.1, §1.2.3§4.1, Existence of solutions§4.2, Maximal solutions§4.3, 4.4, Existence and uniqueness of solutions.Exercises 1.3,p. 15; 1.4,1.5, p. 18-19Exercise 4.2, 4.3, p. 109; Exercise 4.4, p. 110§4.3 Exercise 4.8*,p. 114-115 On.04-278:00 Limits of maximal solutions. Th. 4.11, p. 112. (escaping a compact property)On"global" extensibility of solutions for an ODE with a linear bound for the right hand side. Prop. 4.12, p.114, Examples 4.6, 4.7, p.108 on extensibility of solutionsTransition map. Def. p.126.The openness of the domain and the continuity of transition map.Theorem 4.29, p. 129; Theorem 4.34, p.139 (autonomous case)Transition property of transition map. Th. 4.26, p.126; Th. 4.35, p.140 (autonomous case) Autonomous differential equations §4.6 Transition maps are called also flows in autonomous case. §4.6.1, Flows and continuous dependence§4.6.2, Limit sets$4.6.3, Equilibrium points and periodic pointsExercise 4.15, p. 140, Exercise 4.16, p. 140, Exercise 4.17, p. 140Lecture notes on non-linear systems. Existence, extension Tors.04-288:00 Positive, negative semi-orbits.Positively invariant sets. p. 141, Omega limit points, omega - limit sets, 4.6.2, p. 141, Examples on two methods to find positively invariant sets. Periodic solutions of autonomous systems. §4.7.1, 4.7.2.Poincare- Bendixson theorem 4.46, p. 151 (Only an idea of the proof is discussed). Applications of Poincare- Bendixson theorem, p. 157Example 4.57, p. 165 Lecture notes on limit sets and Poincare Bendixson theorem.User guide on  invariant and limit sets.Download problems on invariant sets with answersExamples on transition maps and limit sets: Exercise 4.16, p. 140, Example 4.37, p. 142, see lecture notes for solutions. §4.7.1  Poincare- Bendixson theorem, Exercise 4.21, p.158 The second modeling project - home assignment is available at Canvas on Monday 05-02 W. 5(18) Må05-0215:15 Existence of an equilibrium point in a compact positively invariant set homeomorphic to a ball. Theorem 4.45, p. 150., Exercises on Poincare-Bendixsons theory. Download exercises on periodic solutions and limit cycles  HERE On05-048:00 Limit cycles. 4.7.3, p. 167. c Prop. 4.5.6, p. 165  on existence of limit cycles.Bendixson's criterion for non-existence of periodic solutions:  div(f) >0 or div(f)<0  on a simply connected domain in plane - without holes (after lecture notes) First integrals. Examples with Newton equation and pendulum. §4.7.3 Limit cycles. Lecture notes on Bendixson's criterion for non-existence of periodic orbits §4.7.2, First integrals and periodic orbits p. 161 Tors05-058:00 Stability and asymptotic behavior of equilibrium points. Stability by Lyapunov functions. Th.5.2, p.170 Asymptotic stability by Lyapunov functions. Cor. 5.17, p.185Region of attraction. §5.1 Lyapunov stability theoryExercise 5.16, p. 188, Exercise 5.17, ,189 W. 6(19) Må05-098:00 Proof of Cor. 5.17, p.185 on asymptotic stability by strong Lyapunov functions. Instability by Lyapunov functions. Th. 5.7, p. 174  Examples on using Lyapunov functions. Young inequality, Cauchy inequality. Elementary introduction to LaSalle's invariance principle \$5.2 Invariance principles. Lecture notes on omega-limit sets and LaSalle's invariance principle with applications Exercise 5.7, 5.8, 5.10. Ons05-118:00 Main theorem on the properties of limit sets. Omega- limit sets are connected and consist of orbits. Th. 4.38, p.143 LaSalle's invariance principle  Th.5.12, p.180; we take the proof from the solution to Exercise 5.9, p. 312. Asymptotic stability by "weak" Lyapunov's functon. Th. 5.15, p. 183.Example 5.13, p. 181 Theorem 5.22 , p. 188, on global asymptotic stability.Exponential stability by Lyapunov functions.Th.5.35, p.200 Lecture notes on omega-limit sets and LaSalle's invariance principle with applications Tors. 05-12 8:00 Examples and exercises on stability and instability by Lyapunov functions.Exercises on application of LaSalle's invariance principle Matlab code for drawing a phase portrait in plane Deadline for the second project-home assignment is the 20-th of May W. 7(20) Må05-1615:15 Homogeneous linear non-autonomous ODEs. Transition matrix function  Lemma 2.1, p.24; Corollary 2.3, p. 26Uniqueness of solutions to general systems of linear ODEs. Th. 2.5, p.28Space of solutions to non-autonomous systems of linear ODEs and its dimension : Prop. 2.7 first statement , p.30. Example 2.2, p.26.Group properties of the transition matrix function (Chapman - Kolmogorov relations): Corollary 2.6, p.29Fundamental matrix solution for linear homogeneous ODE, Prop. 2.8, p. 33 Linear systems with periodic coefficients. Floquet's theory.Shifting invariance property of transition matrix for periodic systems: (formula 2.31 , p. 45):  Φ(t+p,T+p)=Φ(t,T) Structure of the transition matrix for a time interval including a finite number of periods. (formula 2.32, p. 45 ): Φ(t+p,T)=Φ(t,0)[Φ(p,0) ]Φ(0,T) Monodromy matrix:  Φ(p,0). §2.1.1 Homogeneous linear systems Appendix 2,3Exercises 2.1, 2.2, pp. 22-23Exercise 2.9, p. 33.Exercises 2.13, 2.14, p. 42-43 §2.1.2 Solution space §2.3, Floquet theory, §2.4  Exercises on non-autonomous linear systems. Ons05-188:00 Example on calculation of transition matrix.       Motivating example and reflections on main ideas of Floquet theory. Floquet's theorem on factorisation of the transition matrix for a linear system with periodic coefficients. Th. 2.30, p. 53 Lecture notes linear systems of ODE with variable coefficients and Floquet theory §2.3, Floquet theory, examples Tors 05-19 8:00 Logarithm of a matrix. Prop. 2.29, p.53 Existence of the logarithm of a matrixSpectral mapping theorem. Th. 2.19, mainly for f(x)=exp(x), and f(x)=log(x)Floquet multipliers p.48.Floquet's theorem on zero limit and on boundedness of solutions to linear systems with periodic coefficients. Th. 2.31, p. 54. Examples and exercises on periodic linear systems.Existence of periodic solutions. Prop. 2.20, p.45 §2.3, Floquet theory, examples Download exercises on linear periodic systems.Exercise 2.9, p. 33Exercise 2.16, 2.17, p. 47. Alternative proof to the existence of matrix logarithm W. 8(21) Må05-2315:15 Banach spaces. C(I) Banach space. Fixed point problems.Contraction mapping principle by Banach. Theorem A.25, p. 277Lemma 4.21, p.121Picard-Lindelöf existence and uniquness theorem with proof; Picard iterations Th. 4.22, p. 122. Exercises: Picard iterations.Exercises on contraction principle. Exercises with solutions and hintsfor Banach's contraction principle On05-258:00 Repetition of key ideas and methods in the course.Preparation to examination A detailed list of Definitions, Methods, Theorems, and Typical Problems (Links to an external site.) with proofs required at the exam marked. Lecture notes on main techniques studied in the course with examples from exams Examples of theoretical questions to the exam. Examination.

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### Modelling projects

Two modeling projects - home assignments are obligatory and consist of a theoretical part that requires some mathematical reasoning and an implementation part including writing a simple Matlab code solving an ODE, graphical output, and analysis of numerical solutions and conclusions.
Each of you must write an own report but you are encouraged to work in small groups of 2-3 people discussing theoretical and programming problems.
The report must we written as a small scientific article that a person who did not study the course can understand. It must include: 1) theoretical argumentation, with necessary references 2) numerical results with graphical output and 3) interpretation of results, and names of all group members.

Second year batchelor students supply reports in Swedish and are advised to watch video (or streaming) of the the  lecture on scientific writing that will be given by Elin Götmark on the 7-th of April at 15:15. Students will get a feedback on scientific writing aspects of the preliminary version of the report to the first home assignment from Elin Götmark. Students will need to correct reports to the first home assignment according to this feedback.
Students will supply reports and Matlab codes with clear comments via an assignment in Canvas to be checked by examiner Alexei Heintz, on mathematical qualities and (only the first project) by Elin Götmark,  who will check the quality of scientific writing.
Grades for your reports on two home assignments will contribute 16% each to the final marks for the course.

One must pass independently both the exam and modeling projects to pass the course.

#### Reference literature:

1. Learning MATLAB, Tobin A. Driscoll. Provides a brief introduction to Matlab to the one who already knows computer programming. Available as e-book from Chalmers library.
2. Physical Modeling in MATLAB 3/E, Allen B. Downey
The book is free to download from the web. The book gives an introduction for those who have not programmed before. It covers basic MATLAB programming with a focus on modeling and simulation of physical systems.

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Date Details Due