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The Perturbed Simple Pendulums Route to Chaos

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The Perturbed Simple Pendulums Route to Chaos The perturbed simple pendulums route to chaos Tjalle Rens Galama July 19, 2018 Bachelor thesis Mathematics Supervisor: prof. dr. Ale Jan Homburg Korteweg-de Vries Institute for Mathematics Faculty of Sciences University of Amsterdam Abstract The purpose of this thesis is to review the use of the Melnikov method in the detec- tion of chaotic dynamics. The first chapter focuses on definitions from the theory of dynamical systems and the mathematical objects necessary for the discussion. The second chapter continues this path and studies homoclinic bifurcation. The third chapter deals with the Melnikov method and chaotic dynamics. This type of dynamics is explained through the horseshoe map, invented by S. Smale in the 1960s. Some symbolic dynamics will be included. The Smale-Birkhoff homoclinic theorem will be presented, that connects the existence of transverse homoclinic points to chaotic dynamics. The conclusion concerns application of the Melnikov method. This method introduced by Poincar´eand developed by Melnikov is still used today to prove the existence of chaotic dynamics in various fields of analysis (and physics). This thesis will be ended with a comprehensible summary. Title: The perturbed simple pendulums route to chaos Authors: Tjalle Rens Galama, [email protected], 10542310 Supervisor: prof. dr. Ale Jan Homburg, Second grader: dr. Chris Stolk, End date: July 19, 2018 Korteweg-de Vries Institute for Mathematics University of Amsterdam Science Park 904, 1098 XH Amsterdam http://www.kdvi.uva.nl 2 Contents 1 Introduction 4 1.1 Basic setup . .4 1.2 The system that will be investigated . .5 1.3 Equilibria and stability . .6 1.4 Hyperbolicity . .7 1.5 First glimpse of chaos . .8 2 Homoclinic bifurcation 9 2.1 Come-from . .9 2.2 Unfolding around a hyperbolic equilibrium . .9 2.3 Premises and set-up for the homoclinic bifurcation theorem . 10 2.4 Homoclinic bifurcation theorem . 12 3 Melnikov's method for nonautonomous perturbations 14 3.1 Derivation of the Melnikov function . 14 3.2 Melnikov's method for nonautonomous perturbations . 16 3.2.1 Bounding and expanding the perturbed flow . 18 3.3 Expressions for the Melnikov function . 19 3.4 Finding transverse homoclinic points with the Melnikov function . 20 3.5 Homoclinic tangle . 21 3.6 Smale horseshoe map . 21 3.7 Cantor set of transverse homoclinic points . 22 3.8 Symbolic dynamics . 24 4 Application and conclusion 26 5 Summary 29 Bibliography 31 3 1 Introduction 1.1 Basic setup In this introductory chapter we review some basic topics in the theory of ordinary differ- ential equations. You can have a look at the introduction of [4] to find a more extensive approach, which has plenty of correlations with the following explanation. For the purposes of this thesis it is generally sufficient to regard a differential equation as a system dx :=x _ = f(x); (1.1) dt n where x = x(t) 2 R is a vector valued function of an independent variable (usually n time) and f : U ! R is a smooth (meaning sufficiently often differentiable) function n defined on some subset U ⊂ R . We say that the vector field f generates a flow n φt : U ! R ; where φt(x) = φ(x; t) is a smooth function defined for all x 2 U and t in some interval I = (a; b) ⊂ R, and φ satisfies the differential equation in the sense that d (φ(x; t))j = f(φ(x; τ)) dt t=τ for all x 2 U and τ 2 I. We note that φt satisfies the group properties φ0 = id and φt+s = φt ◦φs. Systems of the form (1.1), in which the vector field does not contain time explicitly, are called autonomous. Often we are given an initial condition x(0) = x0 2 U; in which case we seek a solution φ(x0; t) such that φ(x0; 0) = x0: n In this case φ(x0; ·): I ! R defines a solution curve, trajectory or orbit of the differential equation (1.1) based at x0. Definition 1. The orbit of x 2 M is the set fφ(x; t) := φt(x) j t 2 Ig, which we call periodic if there exist t 6= s with s; t 2 R for which φt(x) = φs(x). Since the vector field of the autonomous system is invariant with respect to translations in time, solutions based at times t0 6= 0 can always be translated to t0 = 0. We shall be concerned with individual solution curves and with families of such curves, such as the global behavior of the flow. We now state, without proof, the basic local existence and uniqueness theorem. 4 Theorem 1. Let U ⊂ R be an open subset of real Euclidean space (or of a differentiable n 1 manifold M), let f : U ! R be a continuously differentiable (C ) map and let x0 2 U. Then there is some constant c > 0 and a unique solution φ(x0; ·):(−c; c) ! U satisfying the differential equationx _ = f(x) with initial condition x(0) = x0. This theorem becomes global when we work on compact manifolds M instead of open n spaces like R , since there is no way in which solutions can escape from such manifolds. Therefore, we will also use a more general concept of a dynamical system as a flow on a differentiable compact manifold M arising from a vector field, regarded as a map f : M ! TM where TM is the tangent bundle of M. So here a vector field is an image that assigns to every point x in M a vector f(x) with basepoint x. Again, now within this more general d concept, the vector field can be seen as a differential equation dt x = f(x). Example 1. The vector field f(x; y) = (x2 + y; y) corresponds to the dynamical system d 2 d dt x = x + y , dt y = y. 1.2 The system that will be investigated Another example of such a system is the simple unperturbed planar pendulum. We will take such a system and perturb it later on but first we will start with understanding the unperturbed system. To get intuitive: a simple unperturbed planar pendulum has a fixed length and point mass and is acted on by a constant gravitational field. The values of these constants describe the parameters of the system. The phase space consists of possible values of the pendulum's position, represented by an angle and its angular velocity. Motion occurs only in two dimensions - the point mass does not trace an ellipse but an arc. The motion does not lose energy to friction or air resistance. The differential equation that represents the motion of this simple pendulum is d2θ g + sin(θ) = 0 dt2 l where g is acceleration due to gravity, l is the length of the pendulum, and θ angle between the pendulum and the (from the fixed centre) downward vertical direction. We g assum l = 1. We can rewrite this system into a pair of first order differential equations: θ_ = v v_ = − sin(θ): dθ dv The so called phase space is given by plotting (θ(t); dt (t) = v(t)) such that dt (t) = d2θ − sin(θ), coming from dt2 = − sin(θ). We illustrate this in figure 1.1 where the vector field as well as some solution curves are drawn. 5 Figure 1.1: Phase space of the simple unperturbed pendulum [8]. 1.3 Equilibria and stability We will now expand our knowledge with an important class of solutions of a differential equation, namely the fixed points, which are also called equilibria. n Definition 2. The point p 2 R we call an equilibrium point for a differential equation d dt x = f(t; x) if f(t; p) = 0 for all t. Differently put, we define it as a constant solution for the differential equation. So when f is not explicitly dependent on time, the ordinary d differential equation dt x = f(x) has an equilibrium solution x(t) = p if f(p) = 0. So, fixed points are defined by the vanishing of the vector field (when f(p) = 0). We continue with another fundamental definition. Definition 3. The stable set of a fixed point p is given by s W (p) := fx 2 M : lim φt(x) ! pg: t!+1 And the unstable set of a fixed point p is defined by u W (p) := fx 2 M : lim φt(x) ! pg; t→−∞ which are both invariant and not empty. These sets are connected when we consider a continuous differentiable dynamical system, such as the simple planar pendulum. And now we're ready to understand the following concepts, which will also be fre- quently used in the following chapters. Definition 4 (Heteroclinic orbit). Let p, q be equilibria and p 6= q. An orbit Γ is called heteroclinic if for each x 2 Γ we have that φt(x) ! p for t ! 1 and φt(x) ! q for t ! −∞ (where φt(p) = p and φt(q) = q for all t since they are fixed points). Definition 5 (Homoclinic orbit). Let p be an equilibrium. An orbit Γ is called homo- clinic if for each x 2 Γ we have that φt(x) ! p for t ! 1 and φt(x) ! p for t ! −∞ (where φt(p) = p for all t since it is a fixed point). 6 1.4 Hyperbolicity Definition 6. An equilibrium p0 of the vector field f0(x) is called hyperbolic if none of the eigenvalues of Df0(p0) have zero real part. We see that our system contains such a point. When the pendulum depicted on the front page is hanging motionless on the vertical axis, when the velocity equals 0 and the angle of the pendulum is −180◦ or 180◦ (which is the same position).
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