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MA371 the Qualitative Theory of Ordinary Differential Equations

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MA371 the Qualitative Theory of Ordinary Differential Equations MA371 The Qualitative Theory of Ordinary Differential Equations As Lectured by Professor C. Sparrow Typeset by D. Kitson Assisted by K. Crooks October 2009 to Jan 2010 Contents 1 Introduction 2 2 One Dimensional ODES (X = R1) 5 2.1 Flows, Existence and Uniqueness . 5 2.2 Orbits . 7 2.3 Stability . 9 3 Two Dimensional Flows (X = R2) 11 3.1 Hamiltonian Flows and Hamiltonians . 11 3.2 Liaponov Functions . 13 3.3 Fixed Points and Nearby Behaviour . 15 3.3.1 The Linear Part . 15 3.3.2 The Non-Linear Part . 16 3.4 Stable and Unstable Manifolds . 16 3.5 Stable Manifold Theorem (SMT) . 18 3.6 Hartman-Grobman Theorem . 20 3.7 Non-Hyperbolic Fixed points . 20 4 Periodic Orbits 23 4.1 The Poincar´e-BendixsonTheorem . 23 4.2 Dulac's Criterion or the Negative Divergence Test . 24 4.3 Poincar´eIndex in R2 ................................... 26 4.4 Stability of Periodic Orbits . 27 4.5 Van de Pol Oscillators . 28 5 Bifurcations 34 5.1 Bifurcations in R1 .................................... 34 5.1.1 Summary . 37 5.2 Bifurcations in R2 and the Central Manifold Theorem . 37 5.3 Hopf Bifurcations . 40 5.4 Co-Dimension 1 Bifurcations of Periodic Orbits . 41 5.5 Global Bifurcations . 42 5.5.1 Saddle Node on a cycle . 43 5.5.2 Homoclinic Orbits . 44 Warning: I cannot guarantee that these notes are accurate or complete. Note: Definitions will be in Bold Text. Chapter 1 Introduction We will study Ordinary Differential Equations, or ODEs, of the formx _ = f(x). The State Variables x vary in the the State Space (or phase space) X. In this course X will be finite dimensional (Partial Differential Equations live in infinite dimensional spaces). X is a metric space, usually Rn. x(t) is the State of the system x 2 X at time t. t 2 R is the Independent Variable. f : X ! X is a vector field. f(x) gives the speed and direction of the motion at x 2 X. The graph of (x(t); t) in X is the Phase Portrait. The graph of x(t) is called the Solution, the trajectory, the orbit of the system or the flow.1 Figure 1.1: A phase portrait. x_ = f(x) gives an autonomous DE, i.e. one not depending on t. This means that given any point (x1; : : : ; xn) 2 X, f(x1; : : : ; xn) will take the same value 8 t 2 R. This course is concerned with the study of first order autonomous ODEs. Qualitative Theory helps one to understand the local and global behaviour of an ODE without actually having to find explicit solutions to them; thus the subject is of great importance as many ODEs have no explicit solutions. t Example 1.0.1: Take X = R,_x = x has solution x(t) = x0e , with x(0) = x0. Looking at the phase portrait, one can see that 0 is an unstable fixed point, which is also a global repeller and all orbits not starting at 0 go to ±∞. This is the qualitative description of the the behaviour of the system. Thus, if we know f or its graph, we don't need a solution to understand the behaviour of the system. This works for lots of \nice" fs in R, but this becomes very difficult in Rn≥3. Example 1.0.2: The below cubic-like function f displays the behaviour one would expect as the ODEx _ = f(x), and is similar to the type of thing studied in first year ODEs. 2 1 Example 1.0.3: Considerx _ = f(x) = Sign(x)x . f(x) = 0 at x = 0. For x > 0, x(t) = −1 . x0 −t The solution goes to infinity as t ! 1 . This is called Finite Time Blowup (FTB), i.e. x ! 1 as x0 t ! T < 1. This happens but we do not worry about it. It doesn't change the phase portrait. However, we do worry about the existence and uniqueness of solutions. 1From now on the vector x will not be underlined unless we have something like x = (x; y). 2 Figure 1.2: Two \nice" functions giving the same result, but only one has an explicit solution. Figure 1.3: Example 1.0.2: a cubic-like function giving two \unstable" fixed points and one \stable" fixed point, as per MA133 Differential Equations. 1 Example 1.0.4: Considerx _ = jxj 2 Sign(x). We find that for any T ≥ 0, we have a solution x = 0 t 2 [0;T ] x(t) = 1 2 x = 4 (t − T ) t 2 [T; 1) so the solution is not unique - what about 0 and T ? Example 1.0.5: With X = R2. Consider 1-dimensional Simple Harmonic Motion (SHM), given byx ¨ = −x. Write y =x _ to gety _ = −x,_x = y, which is a first order ODE in R2. We can write this in the form x_ 0 1 x = y_ −1 0 y We can find solutions of the form x(t) = r0 cos(t + φ); y(t) = r0 sin(t + φ) The phase portrait shows that all solutions are periodic orbits around a fixed point (0; 0), which is structurally unstable. Aside: One method of solving these is a change of coordinates. In X = R2 wherex _ = Ax with complex eigenvalues for A, write x = r cos(θ), y = r sin(θ). Then try to get to the formr _ = 0, θ_ = 1. Example 1.0.6: In X = R2, x_ 2 0 x = y_ 0 −1 y gives x ! ±∞ as t ! 1 and y ! 0 as t ! 1. So (0; 0) is a fixed point. but suppose we had some non-linear, i.e. higher order terms as well? Then the equations would look like x_ 2 0 x = + h: o: t: y_ 0 −1 y This usually can't be solved, but the qualitative question is \what does the phase portrait look like near (0; 0) now?". We will show that it is the same under certain conditions. 3 Figure 1.4: Example 1.0.5: the phase portrait of Simple Harmonic Motion. Example 1.0.7: With X = R3. x_ = 10(y − x) y_ = 28x − y − xz 8 z_ = z − xy 3 The Lorenz equations have only two quadratic but terms, but the phase portrait is chaotic with a strange attractor: Figure 1.5: The well-known Lorenz attractor. We are interested in behaviour (such as periodicity, going to a limit or infinity, etc) which is invariant under \nice" changes of coordinates. By a \nice" change of coordinates h with y = h(x) we mean a diffeomorphism, i.e. a differentiable bijection with differentiable inverse. We are also interested in properties invariant under parameterisation of time, e.g. new time s = α(t), where α : R ! R. Figure 1.6: These systems have been subjected to a diffeomorphism, but their phase portraits display the same sort of behaviors. What does it mean to say that there does not exist explicit solutions? Considery _ = y2 − t - this has no explicit solution, but we could define a new function L(y0; t) as being the solution, but obviously this is not helpful. Similarly the pendulum solution θ¨ = − sin(θ) can be solved using elliptic functions, but these are not \normal". 4 Chapter 2 1 One Dimensional ODES (X = R ) 2.1 Flows, Existence and Uniqueness It is more natural to write \solutions" as functions of starting position and time. Definition: φ : X × R ! R is called the Flow; φ(x0; t) = x(t) the place one gets to starting at x0 at t = 0 and solving until time t. We need φ to have some obvious properties: (I) φ(x0; 0) = x0 (II) φ(φ(x0; s); t) = φ(x0; s + t) A function φ satisfying these properties is a candidate for the solution of the ODE.1 To solve x_ = f(x), then it should satisfy d (φ(x; t)) = f(φ(x; t)) dt Equivalently, integrate both sides to get Z t φ(x0; t) = x0 + f (φ(x0; s)) ds 0 So, given f, does φ exist and is it unique? Definition: A function f : X ! Y metric spaces is Lipschitz with Lipschitz constant L if kf(x) − f(y)k ≤ Lkx − yk for some L 2 R, and for all x; y 2 X. Example 2.1.1: If f : R ! R, f is Lipschitz with constant L, the function remains in the region bounded by lines of slope ±L through the point (x; f(x)). Figure 2.1: Example 2.1.1: a Lipschitz function. f cannot escape the lines of slope ±L. 1If it does so for s; t ≥ 0 then it is called a semi flow in some books. 5 Note: Lipschitz ) Continuous, but Lipschitz ; Differentiable. Definition: f is Locally Lipschitz at the point x∗ with constant L if kf(x∗) − f(y)k ≤ Lkx∗ − yk for all y in a neighborhood of x∗. 1 Example 2.1.2: f(x) = jxj 2 is not locally Lipschitz at 0 because L would have to be ±∞. Outline of what we will do next: (1) Theorem: f Lipschitz ) 9! φ solvingx _ = f(x). (2) Worry: How long does φ exist for? The theorem is only local. (3) Proof: An algorithm to find a sequence of functions converging to φ. How to solvex _ = f(x)?2 We shall try to approximate the solution with a sequence of functions fuig: u0(x0; t) = x0 Z t u1(x0; t) = x0 + f (u0(x0; s)) ds 0 .
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