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Differential Equations and Dynamical Systems Differential Equations and Dynamical Systems P´eter L. Simon E¨otv¨os Lor´and University Institute of Mathematics Department of Applied Analysis and Computational Mathematics 2012 Contents 1 Introduction2 1.1 Qualitative theory of differential equations and dynamical systems....4 1.2 Topics of this lecture notes..........................5 2 Topological classification of dynamical systems9 2.1 Equivalences of dynamical systems..................... 10 2.1.1 Discrete time dynamical systems.................. 11 2.1.2 Continuous time dynamical systems................. 13 2.2 Ck classification of linear systems...................... 15 2.3 C0 classification of linear systems...................... 17 2.3.1 Continuous time case in n = 1 dimension.............. 17 2.3.2 Discrete time case in n = 1 dimension................ 17 2.3.3 Continuous time case in n dimension................ 19 2.3.4 Discrete time case in n dimension.................. 26 2.4 Exercises.................................... 30 3 Local classification, normal forms and the Hartman-Grobman theorem 33 3.1 Hartman{Grobman theorem......................... 35 3.1.1 STEP 1 of the proof of Theorem 3.5................. 37 3.1.2 STEP 2 of the proof of Theorem 3.5................. 38 3.1.3 STEP 3 of the proof of Theorem 3.5................. 39 3.1.4 STEP 4 of the proof of Theorem 3.5................. 40 3.2 Normal forms................................. 44 3.3 Exercises.................................... 50 4 Stable, unstable and center manifold theorems 51 4.1 Stable and unstable manifold theorem................... 51 4.1.1 General approach........................... 52 4.1.2 Global stable and unstable manifolds................ 57 4.2 Center manifold theorem........................... 58 4.2.1 General approach........................... 59 1 4.2.2 Approximation of the center manifold................ 60 4.3 Exercises.................................... 62 5 Global phase portrait, periodic orbits, index of a vector field 68 5.1 Investigating the global phase portrait by using the local ones...... 68 5.1.1 Global phase portraits of one dimensional systems......... 69 5.1.2 Global phase portraits of two dimensional systems......... 72 5.2 Periodic orbits................................. 86 5.2.1 Existence of periodic orbits..................... 86 5.2.2 Local phase portrait in a neighbourhood of a periodic orbit.... 90 5.3 Index theory for two dimensional systems.................. 96 5.4 Behaviour of trajectories at infinity..................... 101 5.5 Exercises.................................... 105 6 Introduction to bifurcation theory and structural stability 107 6.1 Normal forms of elementary bifurcations.................. 108 6.2 Necessary conditions of bifurcations..................... 120 6.3 Structural stability.............................. 123 6.3.1 Structural stability of one dimensional systems........... 125 6.3.2 Structural stability of higher dimensional systems......... 128 7 One co-dimensional bifurcations, the saddle-node and the Andronov- Hopf bifurcations 131 7.1 Saddle-node bifurcation............................ 131 7.2 Andronov{Hopf bifurcation......................... 134 7.2.1 Construction of the Lyapunov function............... 136 7.2.2 Andronov{Hopf bifurcation for linear parameter dependence... 140 7.2.3 Andronov{Hopf bifurcation with general parameter dependence and with linear part in Jordan canonical form........... 142 7.2.4 Andronov{Hopf bifurcation for arbitrary parameter dependence. 145 7.2.5 Case study for finding the Andronov{Hopf bifurcation....... 146 7.3 Computing bifurcation curves in two-parameter systems by using the para- metric representation method........................ 149 7.3.1 The parametric representation method............... 149 7.3.2 Bifurcation curves in two-parameter systems............ 154 8 Discrete dynamical systems, symbolic dynamics, chaotic behaviour 159 8.1 Discrete dynamical systems......................... 159 8.1.1 Logistic map............................. 161 8.1.2 Chaotic orbits............................. 163 8.1.3 Symbolic dynamics.......................... 166 2 9 Reaction-diffusion equations 169 9.1 Stationary solutions of reaction-diffusion equations............ 170 9.2 Travelling wave solutions of reaction-diffusion equations.......... 172 9.2.1 Existence of travelling waves..................... 173 9.2.2 Stability of travelling waves..................... 174 Bibliography 185 3 Chapter 1 Introduction Preface These lecture notes are based on the series of lectures that were given by the author at the E¨otv¨os Lor´and University for master students in Mathematics and Applied Math- ematics about the qualitative theory of differential equations and dynamical systems. The prerequisite for this was an introductory differential equation course. Hence, these lecture notes are mainly at masters level, however, we hope that the material will be useful for mathematics students of other universities and for non-mathematics students applying the theory of dynamical systems, as well. The prerequisite for using this lecture notes is a basic course on differential equations including the methods of solving simple first order equations, the theory of linear sys- tems and the methods of determining two-dimensional phase portraits. While the basic existence and uniqueness theory is part of the introductory differential equation course at the E¨otv¨os Lor´and Universitynot, these ideas are not presented in these lecture notes. However, the reader can follow the lecture notes without the detailed knowledge of ex- istence and uniqueness theory. The most important definitions and theorems studied in the introductory differential equation course are briefly summarized in the Introduc- tion below. For the interested reader we suggest to consult a textbook on differential equations, for example Perko's book [19]. Acknowledgement The author is grateful to his colleagues at the Department of Applied Analysis and Computational Mathematics in the Institute of Mathematics at the E¨otv¨os Lor´and Uni- versity, who supported the creation of a dynamical system course at the University, and encouraged the efforts to make lecture notes electronically available to the students of the course. 4 The author appreciates the careful work of the referee, Tam´asKurics, who provided detailed comments, identified errors and typos, and helped to increase distinctness and consistency of the notes. The lecture notes were written in the framework of the project TAMOP-4.1.2.A/1- 11/1, titled Interdisciplinary and complex development of digital curriculum for science master programmes. 5 1.1 Qualitative theory of differential equations and dynamical systems The theory of differential equations is a field of mathematics that is more than 300 years old, motivated greatly by challenges arising from different applications, and leading to the birth of other fields of mathematics. We do not aim to show a panoramic view of this enormous field, we only intend to reveal its relation to the theory of dynamical systems. According to the author's opinion the mathematical results related to the investigation of differential equation can be grouped as follows. • Deriving methods for solving differential equations analytically and numerically. • Prove the existence and uniqueness of solutions of differential equations. • Characterize the properties of solutions without deriving explicit formulas for them. There is obviously a significant demand, coming from applications, for results in the first direction. It is worth to note that in the last 50 years the emphasis is on the numerical approximation of solutions. This demand created and motivated the birth of a new disci- pline, namely the numerical methods of differential equations. The question of existence and uniqueness of initial value problems for ordinary differential equations was answered completely in the first half of the twentieth century (motivating the development of fixed point theorems in normed spaces). Hence today's research in the direction of existence and uniqueness is aimed at boundary value problems for non-linear ordinary differen- tial equations (the exact number of positive solutions is an actively studied field) and initial-boundary value problems for partial differential equations, where the question is answered only under restrictive conditions on the type of the equation. The studies in the third direction go back to the end of the nineteenth century, when a considerable demand to investigate non-linear differential equations appeared, and it turned out that these kind of equations cannot be solved analytically in most of the cases. The start of the qualitative theory of differential equations was first motivated by Poincar´e,who aimed to prove some qualitative properties of solutions without deriving explicit formu- las analytically. The change in the attitude of studying differential equations can be easily interpreted by the simple examplex _ = x,_y = −y. This system can be obviously t solved analytically, the traditional approach is to determine the solutions as x(t) = e x0, −t y(t) = e y0. The new approach, called qualitative theory of differential equations, gives a different answer to this question. Instead of solving the differential equations it pro- vides the phase portrait as shown in Figure 1.1. This phase portrait does not show the time dependence of the solutions, but several important properties of the solutions can be obtained from it. Moreover, the phase portrait can be determined also for non-linear systems, for
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