Part II The Path-Wise Deterministic Setting 185 The mathematician’s patterns, like the paint- er’s or the poet’s must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. G H H (1877 - 1947) Random Differential Equations in Scientific Computing Chapter 6 Recap: Theory of Ordinary Dif- ferential Equations (ODEs) This chapter serves as a holistic introduction to the theory of ordinary differ- ential equations (without singularities). After providing some preliminaries, integral curves in vector fields are discussed, i.e., ordinary differential equa- tions x_ = F (t; x). We start with continuous right hand sides F and their "- approximate solutions as well as the Peano-Cauchy existence theorem and its implications. Our discussion continues with Lipschitz-continuous functions and the existence and uniqueness theorem of Picard-Lindelöf. In particular, we analyse maximal integral curves, define the three types of maximal inte- gral curves that can occur in autonomous systems and show the transforma- tion of a d-th order equation into a first order system. We deal with the exis- tence of solutions in the extended sense where the right hand side function may be discontinuous on a set of Lebesgue-measure zero. Caratheodory’s existence theorem and its implications are studied together with maximum and minimum solutions. Then, we study the broad class of linear ordinary dif- ferential equations by discussing the unique existence of their solutions and their explicit construction. Applications of the theory focus on first integrals and oscillations in the deterministic pendulum and the Volterra-Lotka system. Finally, we provide a first glance into the existence, uniqueness and exten- sion of solutions of ordinary differential equations on infinite-dimensional Banach spaces. 6.1 Key Concepts As we have seen in Chap. 3, random (ordinary) differential equations can be interpreted path-wise as (deterministic) ordinary differential equations. Thus, it pays-off to recall the basic properties of these deterministic equations, and, as we will see, it is interesting to compare the results and proof-techniques we utilized in Chap. 3 with those that are relevant for (deterministic) ordinary differential equations. Before diving into the details, let us summarize the key results of this chap- d m ter: Let J ⊆ R, U ⊆ R (d 2 N) and Λ ⊆ R (m 2 N0) be open sub- sets and F a smooth function on the Cartesian product of these to Rd, i.e., F 2 Ck(J × U × Λ; Rn) for some 1 ≤ k ≤ 1. An ordinary differential equation Chapter 6 187 Tobias Neckel & Florian Rupp (ODE) is an equation of the form x_ = F (t; x; λ) ; (6.1) where the dot indicates the differentiation with respect to the time variable t 2 J, x 2 U is a vector of state variables and λ 2 Λ is a vector of param- eters. If d > 1, the equation (6.1) is called a system of d first order (ordinary) differential equations and the differentiation is considered component-wise. In particular, during the study of bifurcation/ change of stability scenarios (cf. Sec. 8.3) the dependence on parameters will become essential. For fixed λ 2 Λ a solution (or trajectory) of (6.1) is a function ' : J0 ⊂ J ! U, J0 open, given by t 7! '(t) such that the following holds for all times t 2 J0: d' = F (t; '(t); λ) : dt The fundamental issues in ODE theory concern existence, uniqueness, con- tinuity (with respect to the parameters and the initial conditions) and exten- sibility of solutions. Fortunately, all these issues are resolved, provided the right-hand side function F has enough regularity: Every initial value problem x_ = F (t; x; λ) with initial conditions x(t0) = x0 2 U (6.2) has a unique solution1 which is smooth with respect to the initial conditions and parameters2. Moreover, the solution of an initial value problem can be extended with respect to time until it either reaches the boundary @U of the domain of definition U 2 Rd of the differential equation or blows up to infin- ity3. Let us be more precise: If there is some finite T such that limt!T k'(t; x0; λ)k ! 1, we say that the solution blows up in finite time. Such solutions will leave every compact subset of U ⊆ Rd (see [162], p. 143). Moreover, every d-th order differential equation, d > 1, is equivalent to a system of d first order differential equations, see [12], p. 105 or [162], p. 148. Finally, as we have seen in Chap. 4, (deterministic) ordinary differential equations on Banach spaces play a crucial role in understanding the mean- square solutions of random differential equations. Because the Euclidean Rd is the “simplest” Banach space, we will basically gain the same results as 1 See, for instance, [56], p. 3, theorem 1.2, or [12], p. 93, corollary 3. In [162], pp. 140, the uniqueness and existence statements are provided for locally Lipschitz-continuous right-hand sides. 2 See, for instance, [56], p. 3, theorem 1.3, or [12], p. 93, corollary 4. 3 See, for instance, [56], p. 3, theorem 1.4, and [162], pp. 143. Again, [162] provides his statements for locally Lipschitz-continuous right-hand sides. 188 Chapter 6 Random Differential Equations in Scientific Computing cited above, though the methods for establishing them will take the infinite- dimensional nature of arbitrary Banach spaces into account. When reading this chapter note the answers to the following key concepts 1. Vector fields, local Lipschitz-continuity, and integral curves. 2. Estimating bounds for the solution of ordinary differential equations by Gronwall’s lemma. 3. The theorems of (a) Cauchy-Peano (local existence w.r.t. continuous right-hand sides), (b) Picard-Lindelöf (local existence and uniqueness w.r.t. Lipschitz- continuous right-hand sides), (c) Caratheodry (local existence of solutions in the extended sense), and their applications. 4. "-approximate solutions. 5. Life-time and extension of maximal integral curves. 6. Classification of maximal integral curves in autonomous/ time- independent systems. 7. Maximum and minimum solutions of ordinary differential equations in the extended sense. 8. Fundamental system and matrix for homogeneous linear ordinary differ- ential equations. 9. Method of the variation of constant. 10. First integrals and oscillations (deterministic pendulum equation and Volterra-Lotka equation). 11. Existence, uniqueness and extension of solutions of ordinary differential equations on infinite-dimensional Banach spaces. as well as the following questions 1. How are vector fields and integral curves connected? 2. What is the method of successive approximation? 3. How and under which conditions can solutions be extended beyond a given open interval of existence? Section 6.1 189 Tobias Neckel & Florian Rupp 4. What does the lemma of Arzela-Ascoli state? 5. How can we use MATLAB to symbolically solve ordinary differential equa- tions and display the solutions graphically? 6. How can a d-th order equation be transformed into a first order system? 7. What are solutions in the extended sense? 8. How are the solution of a inhomogeneous linear ordinary differential equations constructed? The main sources for this chapter are [162], Chap. 4, for the numerics- friendly introduction of ordinary differential equations on Rd as integral curves of vector fields, as well as [82], Chap. 1, for the brief discussion of ordinary differential equations on arbitrary Banach spaces. This chapter is structured as follows: First, Sec. 6.2 equips us with the es- sential information about vector fields and their representation as well as the notion of local Lipschitz continuity and Gronwall’s lemma. Next inte- gral curves in vector fields are discussed, i.e., ordinary differential equations x_ = F (t; x). We start in Sec. 6.3 by analyzing ordinary differential equations with continuous right hand side F and studying "-approximate solutions as well as the Peano-Cauchy existence theorem and its implications. Then, in Sec. 6.4, we continue our discussion with Lipschitz-continuous functions F and the existence and uniqueness theorem of Picard-Lindelöf. It is here that we give the MATLAB commands for symbolically solving ordinary differen- tial equations and analyse maximal integral curves. In particular, we give the three types of maximal integral curves that can occur in autonomous systems and show the transformation of a d-th order equation into a first order system. Sec. 6.5 deals with existence of solutions in the extended sense where the right hand side function may be continuous except for a set of Lebesgue- measure zero. Caratheodory’s existence theorem and its implications are studied together with maximum and minimum solutions. Section 6.6 analy- ses the broad class of linear ordinary differential equations by discussing the unique existence of their solutions and explicitly constructing them. Appli- cations are given in Sec. 6.7, where we discuss in particular the deterministic pendulum and the Volterra-Lotka system. Sec. 6.8 rounds up our discussion by providing a first glance into the existence, uniqueness and extension of solutions of ordinary differential equations on infinite-dimensional Banach spaces. Finally, Sec. 6.9 wraps up the contents of this chapter. 190 Chapter 6 Random Differential Equations in Scientific Computing v(x) x Ω Figure 6.1. Sketch of the geometric/ physical interpretation of a vector field on a domain Ω. 6.2 Preliminaries Following [162], pp. 131, vector fields occur naturally in many physics and engineering contexts. Here, we recall their basic definitions and properties together with some illustrative examples. We lay the foundations for the dis- cussion of integral curves in vector fields, i.e.