The perturbed simple pendulums route to chaos

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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

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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 definidetec-tions 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´e and 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, tjallegalama@gmail.com, 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

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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

dt := ˙x = f (x), (1.1)

where x = x(t) ∈ Rn is a vector valued function of an independent variable (usually time) and f : U → Rn is a smooth (meaning sufficiently often differentiable) function defined on some subset U ⊂ Rn_{. We say that the vector field f generates a flow}

φt: U → Rn,

where φt(x) = φ(x, t) is a smooth function defined for all x ∈ U and t in some interval

I = (a, b) ⊂ R, and φ satisfies the differential equation in the sense that d

dt(φ(x, t))|t=τ = f (φ(x, τ ))

for all x ∈ U and τ ∈ 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∈ U,

in which case we seek a solution φ(x0, t) such that

φ(x0, 0) = x0.

In this case φ(x0, ·) : I → Rn defines a solution curve, trajectory or orbit of the

differential equation (1.1) based at x0.

Definition 1. The orbit of x ∈ M is the set {φ(x, t) := φt(x) | t ∈ I}, which we call

periodic if there exist t 6= s with s, t ∈ 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 t06= 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.

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Theorem 1. Let U ⊂ R be an open subset of real Euclidean space (or of a differentiable manifold M ), let f : U → Rn be a continuously differentiable (C1) map and let x0 ∈ U .

Then there is some constant c > 0 and a unique solution φ(x0, ·) : (−c, c) → U satisfying

the differential equation ˙x = f (x) with initial condition x(0) = x0.

This theorem becomes global when we work on compact manifolds M instead of open spaces like Rn_{, 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 → T M

where T M 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 concept, the vector field can be seen as a differential equation _dtdx = f (x).

Example 1. The vector field f (x, y) = (x2+ y, y) corresponds to the dynamical system

d

dtx = x2+ y , d dty = 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

d2_θ

dt2 +

g

lsin(θ) = 0

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 assum g_l = 1. We can rewrite this system into a pair of first order differential equations:

˙ θ = v

˙v = − sin(θ).

The so called phase space is given by plotting (θ(t),dθ_dt(t) = v(t)) such that dv_dt(t) = − sin(θ), coming from d_dt22θ = − sin(θ). We illustrate this in figure 1.1 where the vector

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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.

Definition 2. The point p ∈ Rn we call an equilibrium point for a differential equation

d

dtx = 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 differential equation _dtdx = 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 Ws(p) := {x ∈ M : lim

t→+∞φt(x) → p}.

And the unstable set of a fixed point p is defined by Wu(p) := {x ∈ M : lim

t→−∞φt(x) → p},

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 ∈ Γ we have that φt(x) → p for t → ∞ 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 ∈ Γ we have that φt(x) → p for t → ∞ and φt(x) → p for t → −∞

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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). Also have a look at figure 1.2.

Figure 1.2: At the “hanging motionless upside down” position we have a hyperbolic equilibrium, which is called unstable since a small force can make it fall into one of the two possible directions. This picture comes from [3].

Also very fundamental: the (here not included) Hartman - Grobman theorem states that the orbit structure of a dynamical system in a neighbourhood of a hyperbolic equilibrium point is topologically equivalent to the orbit structure of the linearized dynamical system. Example 2 (linearization). Given f (x, y) = (y, −x − x3_{− αy) such that}

dx dt = y

dy

dt = −x − x

3_{− αy} _{and α 6= 0.}

Then (0, 0) is the only equilibrium point. The linearization of f at the equilibrium is given by

Df(0, 0) =

0 1

−1 −α

which has eigenvalues −α ±

√ α2₋₄

2 . For all values of α 6= 0, the eigenvalues have non-zero

real part. Thus, this equilibrium point is a hyperbolic equilibrium point. The linearized system will behave similar to the non-linear system near (0, 0). When α = 0, the system has a nonhyperbolic equilibrium at (0, 0). 4

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1.5 First glimpse of chaos

This following is what we are heading towards.

Definition 7 (Devaneys Chaos). A dynamical system is chaotic on a given invariant set X for a flow φ when it satisfies the properties:

• φ is topologically mixing: for any pair of non-empty open set U, V ⊂ X, there is some k > 0 such that φl(U ) ∩ V 6= ∅ for all l > k.

• the periodic points of φ are dense in X.

• φ has sensitive dependence on initial conditions: for every q ∈ X and every (open) neighbourhood U ⊂ X of q, there exists a p ∈ U and there exists a δ > 0 such that for some time t0 we have that |φt0(p) − φt0(q)| > δ.

To be able to find out that our perturbed simple pendulum is chaotic we will first gather more relevant information about homoclinic bifurcation.

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2 Homoclinic bifurcation

2.1 Come-from

A bifurcation occurs when a small change made to the parameter values (the bifurca-tion parameters) of a system causes a sudden ‘qualitative’ or topological change in its behaviour. We will show that a homoclinic bifurcation occurs when a periodic orbit collides with a saddle point. We will now study this global bifurcation. Recall that we call a function smooth when it is infinitely (or sufficiently) differentiable. Bifurcation theory begins with a particular smooth vector field, say f0(x). To study the dependence

of the dynamics on parameters, this vector field is then unfolded.

2.2 Unfolding around a hyperbolic equilibrium

Definition 8. A family of vector fields f (x, µ) (for µ ∈ (−δ, δ)) is an unfolding of the vector field f0(x) if f (x, 0) = f0(x).

Definition 9. A hyperbolic equilibrium of a continuously differentiable vector field f0(x)

is a sink if all of the eigenvalues of Df0(p0) have negative real part, is a source if all of

the eigenvalues of Df0(p0) have positive real part and is a saddle if it isn’t a source or

a sink.

Example 3. Have a look at the following system. ˙

x = µ + x2− xy y = y˙ 2− x2− 1

When µ = 0 the system has two equilibria (0, ±1). Both are saddle points since the Jacobian matrix (total derivative) is

Df (x, y) =2x − y −x −2x 2y (x,y)=(0,±1) −−−−−−−−→∓1 0 0 ±2 .

The y-axis of the system with is invariant when µ = 0 and on this axis the y-dynamic follows ˙y = y2− 1, which has a stable (attracting) equilibrium at −1 and an unstable (repelling) equilibrium at +1. Thus the segment {(0, y) : −1 < y < 1} is a heteroclinic orbit when µ = 0. The soon to be introduced implicit function theorem guarantees that the saddle points persist when µ is small. They can be found by series expansion in µ. For this system we can also find the equilibria explicitly,

x = ±√ µ

1 − 2µ, y = ± 1 − µ √

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which shows us that the equilibria persist until µ = 1₂.

To study the change in the manifolds with µ, note that when x ≈ 0, we have ˙x ≈ µ, while if |y| < 1, then ˙y < 0. Thus if µ > 0, the upper saddle is to the right of the y-axis, and its unstable manifold begins moving downward but necessarily moves to the right. The lower saddle, by contrast, is to the left of the y-axis, and its downward moving stable manifold comes from larger negative x. Thus the two manifolds no longer join, and there is no longer a saddle connection.

The heteroclinic connection is destroyed for µ < 0 too, since the equilibria move to the opposite sides of the y-axis, but in this case ˙x < 0. 4

The implication of this example is that if an ordinary differential equation in the plane has a heteroclinic connection, then changing a single paramater can destroy it.

We will now develop the general theory for bifurcations of a homoclinic orbit in the plane. Our results will show that when the system is non-Hamiltonian (∇ · f 6≡ 0), the destruction of a homoclinic orbit is a codimension-one bifurcation (which means that the number of parameters which must be varied for the bifurcation to occur is 1) and is associated with the creation of a periodic orbit.

2.3 Premises and set-up for the homoclinic bifurcation

theorem

Suppose that f (x, µ) (where µ ∈ I, which is an interval around 0) is an unfolding of a planar vector field f0(x) that has a saddle equilibrium p(0) = p0 = (0, 0), with a

homoclinic connection γ0.

The following theorem will be a very useful theorem in the upcoming theory.

Theorem 2 (Implicit Function Theorem). For U ⊂ Rn× Rk _{an open subset, let f :}

Rn× Rk⊃ U → Rnbe (r-times with r ≥ 1) continuously differentiable. Let (x0, µ0) ∈ U

for which f (x0, µ0) = c ∈ Rn. If the Jacobian matrix of f (in (x0, µ0)) restricted to the

first n coordinates Rn×n 3 Dxf (x0, µ0) = ∂fi ∂xj (x0, µ0) n i,j=1

is invertible then there exist open sets (x0 ∈)V ⊂ Rnand (µ0 ∈)W ⊂ Rkfor which there

exists an unique (r-times) continuously differentiable function g : W → V such that g(µ0) = x0 and f (g(w), w) = c ∀w ∈ W .

Definition 10. An equilibrium p0 of the vector field f is called nondegenerate if none

of the eigenvalues of the linearization(matrix) of f evaluated in the point p0 (which we

denote by Df (p0)) equals 0.

Note that a hyperbolic equilibrium is nondegenerate.

Lemma 1 (Preservation of nondegenerate equilibrium). Suppose that a vector field f (x, µ) is continuously differentiable in x and µ and that x0 is a nondegenerate

equilib-rium point for parameter value µ0. Then there exist a unique continuously differentiable

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Proof. The matrix A = Dxf (x0, µ0) governs the stability of the equilibrium x0. All the

eigenvalues of A are nonzero (since x0 is nondegenerate) so A is invertible. The implicit

function theorem implies that there is a neighborhood of µ0 denoted by I for which there

is a curve of equilibria x∗(µ).

By applying this lemma it follows that there is an interval I around 0 such that for µ ∈ I the saddle equilibrium p(µ) persists and remains a saddle.

Theorem 3 (Stable manifold ). Let f : U ⊂ Rn → Rn _{be smooth with hyperbolic}

fixed point p. Then the previously defined stable set of p (defined as Ws(p)) is a smooth manifold and its tangent space equals the stable space of the linearization of f at p. Wu(p) is a smooth manifold and its tangent space equals unstable space of the linearization of f at p. Accordingly Ws(p) is called the stable manifold and Wu(p) the unstable manifold.

From this it follows that p(µ) (the fixed point is moving continuously as the parameter µ changes) has stable and unstable manifolds Ws(p(µ)) and Wu(p(µ)), respectively. To study the way in which these manifolds move with µ we choose any point q ∈ γ0. Let S

be a section (a line segment) through q that is perpendicular to the vector field f0(q).

We denote the points of intersection of Ws(p(µ)) and Wu(p(µ)) with S that continue from q by u(µ) and s(µ) (respectively), such that s(0) = u(0) = q. These functions u(µ) and s(µ) exist for some interval I in µ because the stable and unstable manifolds move continuously with µ, which can also be derived with the implicit function theorem. To study the homoclinic bifurcation theorem we will study some more needed definitions. Definition 11 (First return map). The first return map or also called Poincar´e map for a section S is obtained by choosing an x ∈ S and following the flow to find the first return to S. Let τ (x) be the first positive time for which φt(x) ∈ S. The first return

map is defined by P (x) = φ_{τ (x)}(x). Note that τ (x) might not exist for all x ∈ S, in which case the first return map is not well defined.

Definition 12 (Transversality). The stable and unstable manifold intersect transversely when they cross instead of intersecting tangently. See figure 2.1.

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2.4 Homoclinic bifurcation theorem

Theorem 4. Let f ∈ C2(R2× R, R2_{) and suppose f}

0(x) = f (x, 0) has a saddle

equi-librium p0 such that there is a homoclinic orbit γ0 (such that ϕt(q) −−−−→

t→±∞ p0 for all

q ∈ γ0) and

τ ≡ Tr(Df0(p0)) = h ∇ , f0(p0) i 6= 0.

Let p(µ) be the saddle equilibrium of f that continues from p0and denote its manifolds by

Wu(p) and Ws(p). Define a section S to f0 at a point q ∈ γ0 and let s(µ) = S ∩ Ws(p)

and u(µ) = S ∩ Wu(p) be the continuous functions of µ such that s(0) = u(0) = q. Suppose that

4 = d

dµ(s(µ) − u(µ))µ=06= 0.

Then if τ < 0 there is a family γ(µ) of stable periodic orbits that bifurcate from γ0. The

periods of these orbits are unbounded as µ → 0. Moreover, there is an (which may be negative) such that there is exactly one periodic orbit in a neighborhood of γ0 when

µ ∈ (0, ).

Proof. (sketch) We consider the stable case τ < 0 and assume (for simplicity) ∆ > 0. Since S is transverse to f (x, 0) it will be transverse to f (x, µ) for µ small enough and x close to q. Let Pµ: S → S be the Poincar´e map for f (x, µ) to the section S. This map

is defined only for points on S that return to S. So, in figure 2.2 the map Pµ is defined

only for the points that are closer to the equilibrium then s(µ) since points that are further away typically escape and do not return. As x → s(µ) we have Pµ(x) → u(µ).

Note that P0 is defined for all points in the interior of the homoclinic loop γ0. The

transversality assumption ∆ 6= 0 implies s and u change at different rates with µ. We define d(µ) = s(µ) − u(µ) so that ∆ = _dµdd(µ)µ=0. Since ∆ > 0, if µ < 0 we

have that u = Pµ(s) > s. This means that the value of Pµ(s(µ)) = s(µ) − d(µ) is

above the diagonal. When µ > 0 we have that u < s, so the value of Pµ(s(µ)) is

below the diagonal. Therefore, since Pµ is defined for some interval to the left of s

(again, as visualized in Figure 2.2), the graph of Pµ must intersect the diagonal at

some point x∗ < u(µ) when µ > 0. This fixed point (where Pµ(x∗) = x∗) corresponds

to a periodic orbit γ(µ) of the flow. To study the stability of the periodic orbit, we must evaluate the slope of Pµ at x∗. The fact that τ < 0 implies that the stable

eigenvalue of Df0(p0) is stronger than the unstable eigenvalue, and this means that

orbits inside the homoclinic loop at µ = 0 are strongly attracted to the loop. We claim DP0(q) = 0. To see this, we look at behavior of orbits near p0. Choose local

coordinates in the neighborhood of p0 such that the stable direction is the x-axis and

the unstable direction is the y-axis. Denote the eigenvalues of Df0(p0) by α < 0 < β.

By assumption tr(Df0(p0)) = −α + β < 0 so α > β. To the extent that the linear

approximation is valid we have x(t) = x0e−αt ( ˙x = −αx) and y(t) = y0eβt ( ˙y = βy).

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it takes can be found by noting that y(t) = eβty(0) = ⇐⇒ t = β−1_{ln(/∆y) so}

that x(t) = e−αtx(0) = ∆x = (_∆y )

−α

β _{. This implies that} ∆x

∆y = ( ∆y

)

α

β−1 → 0 as

∆y → 0 since by assumption α_β > 1. Thus trajectories that start close to the x-axis approach much closer to the y-axis. This is where the proof is least rigorous (‘sketchy’): by assuming local linearity. By the (here not included) flow box theorem we see that the argument does not change much if we extend the trajectory backward and forward to crossing points on S. The calculation implies that a point x0 = P0(x) on S is much closer

to q then x was. Moreover, since (x_(x−q)0−q) → 0 as x → q we have that DP₀(q) = 0.

Figure 2.2: The occuring bifurcation for τ < 0 , ∆ > 0, with the construction of a periodic orbit. This picture is taken from [5], p. 311.

The theorem can also be formulated for τ > 0. Then there will be family γ(µ) of unstable periodic orbits that bifurcate from γ0 of which the periods are unbounded as

µ → 0. The proof is similar. The periodic orbit grows until it collides with the saddle point. At the bifurcation point the period of the periodic orbit has grown to infinity and it has become the homoclinic orbit. After the bifurcation there no longer is a periodic orbit.

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3 Melnikov’s method for

nonautonomous perturbations

We see that it is relatively easy to determine when a perturbation of the homoclinic orbit destroys it and gives birth to a nearby periodic orbit. This theorem requires knowledge of the whereabouts of s(µ) and u(µ) and more specifically requires the assumption

4 = d

dµ(s(µ) − u(µ))µ=06= 0.

We will now develop a method, (usually) attributed to Melnikov, for finding the lowest-order behavior of s(µ) and u(µ).

3.1 Derivation of the Melnikov function

We start with the somewhat general question: when can a system in the plane have an orbit that is a closed loop? Suppose that a system with a C1vector field f = (p, q)T has an invariant loop γ : S1 _{→ R}2_{. It could be periodic, homoclinic, or a family of heteroclinic}

trajectories that form a loop. Using the fact that ˙x − p(x, y) ≡ 0 ≡ ˙y − q(x, y) along any trajectory, consider the integral

0 = I γ (−( ˙y − q)dx + ( ˙x − p)dy) = I γ ( ˙xdy − ˙ydx) + I γ (qdx − pdy).

The first integrand can be written as ˙xdy − ˙ydx = ( ˙x ˙y − ˙y ˙x)dt ≡ 0 so that only the second term is possibly nonzero, giving

0 = I γ (qdx − pdy) = Z Int(γ) ∇ · f.

This last equality follows from (the here not included) Green’s theorem, using that γ is a positively oriented, piecewise smooth, simple closed curve in a plane. Here Int(γ) is the region bounded by γ (without the boundary) and q and p are functions of (x, y) defined on an open region containing Int(γ) that have continuous partial derivatives there. This hands us the lemma:

Lemma 2. If ˙x = f (x) has an invariant loop γ thenR

Int(γ)∇ · f = 0.

Consider for example the perturbed Hamiltonian system ˙

x = f1(x, y) + g1(x, y)

˙

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for some Hamiltonian H and f = (_∂y∂ H, −_∂x∂ H)T. Suppose that this system has an invariant closed loop γ for some value of . Since the inner product ∇ · f always equals

0 for a Hamiltonian system (like this one) and by applying the just introduced lemma we find that 0 = I γ (g2dx − g1dy) = Z Int(γ) ∇ · g.

Now suppose that we do have a closed trajectory γ0 at = 0. This implies in particular

that 0 = ∂ ∂ I γ (g2dx − g1dy) =0= I γ0 (g2dx − g1dy) = I Int(γ0) ∇ · g =: M, where the integrals are now taken along the ‘unperturbed’ orbit γ0. When γ0is a periodic

orbit, γ0(t) = γ0(t + T ), we can use dx = ˙xdt and dy = ˙ydt to convert this last integral

into a time integral over one period of the orbit. On the = 0 orbit f = (f1, f2), so the

last integral becomes 0 =

Z T

0

(g2(x(t), y(t)) ˙x − g1(x(t), y(t)) ˙y)dt =

Z T 0 (g2f1− g1f2)γ0(t)dt ≡ Z T 0 f ∧ g |γ0(t) dt,

where we have defined the wedge product : f ∧ g ≡ f1g2− f2g1. When y0 is a homoclinic

orbit, the period must be taken to infinity and x(t), y(t) are functions that limit to the saddle point in both directions of time. Since f (x(t), y(t)) → 0 as the orbit approaches the equilibrium and does so exponentially fast with time, the integral converges as t → ∞. We find the following expression for M :

M =

Z ∞

−∞

f ∧ g |_γ₀_(t) dt. (3.1)

This integral is known as the Melnikov integral. The vanishing of this integral is a necessary condition for the existence of a closed orbit γ near the original homoclinic orbit.

Lemma 3. Suppose that a perturbed Hamiltonian system has a closed loop γ0 when

= 0. Then a necessary condition for the existence of an invariant loop when is small is that the Melnikov integral vanishes.

Example 4. Consider the non-Hamiltonian perturbation of the Hamiltonian: H(x, y) =

1

2(y2− x2) + ax3.

˙

x = y + x y = x − 3ax˙ 2+ bxy,

so that g(x, y) = (x, bxy). To apply the Melnikov criterion, we only need an expression for the unperturbed solution, which is y± = ±x

√

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becomes M = I γ0 g2dx − g1dy = Z γ+ g2dx − g1dy + Z γ− g2dx − g1dy = Z 1 2a 0 (bxy+(x)dx − xdy+(x)) − Z 1 2a 0 (bxy−(x)dx − xdy−(x)) = Z 1 2a 0 (bxy+− x dy+ dx )dx − Z 1 2a 0 (bxy−− x dy− dx )dx.

The two integrals can be combined since y− = −y+ to give (with the substitution

u =√1 − 2ax) M = 2 Z 1 2a 0 x(by+− dy+ dx )dx = 2 2b + 7a 105a2 ,

which is nonzero unless b = −7a₂ . Therefore, except on this curve there are no nearby closed loop orbits. This result applies when 1. 4

3.2 Melnikov’s method for nonautonomous perturbations

Nonautonomous perturbations to a planar system can also be treated by similar methods. Begin as before with a Hamiltonian vector field in the plane that has a homoclinic loop. Upon adding a perturbation that could be non-Hamiltonian and periodically time dependent, the loop will typically be destroyed. We will compute the distance between the stable and unstable manifolds of the perturbed fixed point.

Letting z = (x, y), consider the system ˙

z = f (z) + g(z, t) ∇ · f = 0 g(z, t + T ) = g(z, t) (3.2)

where f and g are C2. Upon introducing a phace θ = ωt with ω = 2π/T , this system becomes autonomous on the extended phase space R2 × S1_:

˙

z = f (z) + g(z, θ) ˙

θ = ω

with g(z, θ + 2π) = g(z, θ). The solution (flow) for this (now autonomous) system with = 0 is given by:

(z(t), θ(t)) = (ϕt(z), θ + ωt)

where ϕt(z) is the flow for f . Therefore any equilibrium p0of f becomes a periodic orbit

of the extended system given by the closed loop γ0(t) = (p0, ωt mod 2π). Moreover, if

p0 is a hyperbolic equilibrium of f , then the implicit function theorem implies that in

the extended phase space, the periodic orbit persists for > 0. We state this in the following theorem.

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Theorem 5 (Persistence of hyperbolic periodic orbits). If γ0(t) = (p0, ωt) is a hyperbolic

periodic orbit of the just introduced autonomous system at = 0, then there is an 0> 0 such that for any |e∗| < 0 there is a unique periodic orbit γ∗(t) of period T that

continues from γ0(t).

This can be proven by applying the implicit function theorem to the Poincar´e map. Now suppose that when = 0 the equilibrium p0 has a homoclinic loop,

Γ ⊂ Ws(p0) ∪ Wu(p0).

Then the corresponding periodic orbit γ0 of p0 has a two-dimensional manifold

H(γ0) = {(z, θ) : z ∈ Γ0, θ ∈ S1}.

Every orbit on this manifold is both backward and forward asymptotic (in time) to γ0.

We now wish to study the effect of the perturbations on H(γ0). To do this we develop an

expression for the rate of change of the manifolds with . Note that for any (q, θ) ∈ H(γ0)

the mapping (q, θ, t) → (φt(q), θ + ωt) uniquely represents a point on H(γ0).

To measure the distance between the perturbed manifolds, we use the perpendicular

Figure 3.1: Flow of (3.2) for 6= 0 and the perturbed stable and unstable manifolds of γ. See [5], p. 317.

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vector to the unperturbed manifolds: f⊥ = (−f2, f1) is the two-dimensional vector

perpendicular to the unperturbed flow since (f1, f2) · (−f2, f1)T = 0. At (φt(q), θ + ωt)

the three-dimensional vector perpendicular to the manifold (with this point as its origin) is (f⊥(φt(q)), 0). Note that the dot product of a vector with f⊥ is equal to the wedge

product with f :

f⊥· v = −f2v1+ f1v2= f ∧ v.

Define a local section at a point (q, θ) ∈ H(γ0) by

S = {(z, θ) : z = q + σf⊥(q), σ ∈ (−δ, δ), θ ∈ [0, 2π)}

for some δ > 0. Since S is transverse to the flow at = 0 it is still transverse to the perturbed flow for small enough . We denote the perturbed flow of the system (3.2) ( 6= 0) by

(z(t), θ(t)) = (ψt(z, θ), θ + ωt).

The intersections of the manifolds with S are denoted by

{(s(θ), θ)} = Ws(γ) ∩ S, {(u(θ), θ)} = Wu(γ) ∩ S.

These sets continue from the original unperturbed intersections (so that s0(θ) = u0(θ) =

q for all θ ∈ [0, 2π) ). In figure 3.1 we see that Ws(γ) ∩ S = {(s(θ), θ) : θ ∈ [0, 2π)}.

In this figure the notation u(θ) = (u(θ), θ) is used and likewise for the intersection

of the stable manifold with S. Later on we will see that if the manifolds intersect transversely at some point (r, θ∗), Smale’s horseshoe theorem will imply there must be

infinitely many more transversal intersections of the stable and unstable manifold on the horizontal (“cross”) section S at height θ∗, and hence at every such section (at

all possible heights). So, right now we are only interested in finding one transversal intersection on S. For this we need to find a stronger expression for the perturbed flow.

3.2.1 Bounding and expanding the perturbed flow

The (here not formally proven but nonetheless given) smoothness of the flow with respect to parameters and with respect to initial conditions implies that we can find the solution ψt as a power series expansion away from ϕt. For example, the solution on the stable

manifold that starts at (s(θ), θ) can be expanded as

ψt(s(θ), θ) = ϕt(q) + ξts(q, θ) + O(2).

The difference ξs_t is bounded for any finite time, which can be shown by varying both the initial condition and the parameter and applying the (here not included) Gr¨onwall inequality. Since ψt(s(θ), θ) → γ as t → ∞ there is some finite time Tρsuch that ψt is

within ρ of γ for any ρ > 0. Moreover, the implicit function theorem implies γ is O()

close to γ0. Since the Gr¨onwall inequality implies that up to Tρ the deviation is O(),

we have that ψt= ϕt+ O() for all t > 0. A similar argument leads to the conclusion

that for points on the unstable manifold, the deviation is bounded for all t < 0. 1

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3.3 Expressions for the Melnikov function

While having the above consideration in mind we can now define the measure of the distance between the manifolds as the dot product of the difference between these points with the perpendicular vector field f⊥, or equivalently the wedge product with f itself:

∆(t, θ) : = f⊥(ϕt(q)) · (ψt(u(θ), θ) − ψt(s(θ), θ))

= −f2(ϕt(q)) · v1+ f1(ϕt(q)) · v2

= f (ϕt(q)) ∧ (ψt(u(θ), θ) − ψt(s(θ), θ)) = M (t, θ) + O(2),

where (ψt(u(θ), θ) − ψt(s(θ), θ)) = (v1, v2). We see that the above expression is

some-what implicitly dependent on q. To get the actual distance we could divide by the norm of f, but we are interested in only a measure of the distance and in particular whether the distance is zero. Note that ∆0(t, θ) ≡ 0 since the manifolds coincide at = 0. We

have hereby defined the Melnikov function as the rate of change of this distance with : M (t, θ) ≡ ∂ ∂∆(t, θ) =0= f (ϕ(q)) ∧ (ξ u t(q, θ) − ξts(q, θ)) = Mu(t, θ) − Ms(t, θ).

Lemma 4. On the section S itself (when t = 0), the deviation is given by

∆(θ) ≡ ∆(0, θ) = M (0, θ) + O(2). (3.3)

To compute the terms Mu(t, θ) and Ms(t, θ) we derive a differential equation for M . We consider the stable and unstable term separately. For example, the stable part has derivative ∂ ∂tM s_{(t, θ) = Df (ϕ} t(q)) ∂ ∂tϕt(q) ∧ ξ s t(q, θ) + f (ϕt(q)) ∧ ∂ ∂tξ s t(q, θ). (3.4)

To evaluate this we need the differential equation for ξ_ts(q, θ) = _∂∂ψt(s(θ), θ)

=0. This

is obtained by differentiation of (3.2) with respect to : ∂ ∂tξ s t(q, θ) = ∂ ∂ ∂ ∂tψt(s(θ), θ) =0 = ∂ ∂(f (ψt(s(θ), θ)) + g(ψt(s(θ), θ), θ + ωt))=0 = Df (ϕt(q))ξts(q, θ) + g(ϕt(q), θ + ωt).

This linear equation is to be solved with the initial condition ξ₀s(q, θ) = ∂

∂s(θ)

=0.

Substituting ˙ϕt= f (ϕt) and our expression for _∂t∂ξts into (3.4) gives

d dtM

s_{= Df (ϕ}

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While using simplified notation, we notice that (Df (ϕt)f (ϕt)) ∧ ξ + f (ϕt) ∧ (Df (ϕt)ξ) = (Df1jfj)ξ2− (Df2jfj)ξ1 + f1(Df2jξj) − f2(Df1jξj) = (−Df21f1− Df22f2+ f1Df21− f2Df11)ξ1 + (Df11f1+ Df12f2+ f1Df22− f2Df12)ξ2 = −(Df22+ Df11)f2ξ1+ (Df11+ Df22)f1ξ2 = − Tr(Df )f ∧ ξ. Since f ∧ ξ = Ms we obtain d dtM s_{(t, θ) = − Tr(Df (ϕ} t(q)))Ms(t, θ) + f (ϕt(q) ∧ g(ϕt(q), θ + ωt).

Since f is by assumption a Hamiltonian vector field, Tr(Df ) = 0. This implies that the ordinary differential equation is now trivial,

d dtM

s_{(t, θ) = f (ϕ}

t(q)) ∧ g(ϕt(q), θ + ωt),

since everything on the right hand side is now known. Thus we can simply integrate the equation to obtain Ms. Note that Ms(t, θ) vanishes exponentially fast as t → ∞ because ϕt(q) → p0 and f (p0) = 0. Therefore if we integrate from t to ∞ we get

Ms(t, θ) = − Z ∞

t

f (ϕτ(q)) ∧ g(ϕτ(q), θ + ωτ )dτ.

A similar calculation gives Mu(t, θ) =R_−∞t f (ϕτ(q)) ∧ g(ϕτ(q), θ + ωτ )dτ . Putting these

together yields the Melnikov function expressed as: M (t, θ) = Mu(t, θ) − Ms(t, θ) =

Z ∞

−∞

f (ϕτ(q)) ∧ g(ϕτ(q), θ + ωτ )dτ = M (θ), (3.5)

which is independent of t. This expression is almost identical to (3.1) for the autonomous case. Note that

M (θ + 2π) = M (θ) since g is a periodic function.

3.4 Finding transverse homoclinic points with the

Melnikov function

By construction, when M (θ) = 0, the deviation in the manifolds is zero to first order in . This means that there is a transverse crossing of the manifolds nearby, provided that a nondegeneracy condition is satisfied:

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Theorem 6 (Melnikov ). Suppose there is a point θ0 on the saddle connection such that

M (θ0) = 0 and DθM (θ0) 6= 0. Then when is sufficiently small, Ws(γ) and Wu(γ)

intersect transversely at a point within O() of (q, θ0).

Proof. This proof follows from the implicit function theorem. The separation between the manifolds on the section S is measured by the formal expression (3.3). By definition ∆0(θ) = 0 and _∂∂∆(θ)

=0 = M (θ). The implicit function theorem cannot be applied

to ∆(θ). However, the new function

F (θ, ) := ∆(θ)

= M (θ) + O()

does satisfy the required conditions at the point θ0 : F (θ0, 0) = 0 and DθF (θ0, 0) 6= 0.

Thus there is a unique curve θ() such that F (θ(), ) = 0 or equivalently ∆(θ()) = 0.

Since M changes sign as θ traverses θ0, the separation must change sign upon crossing θ0.

So the assumptions M (θ0) = 0 and DθM (θ0) 6= 0 result in a transverse intersection of the

manifolds (as visualized in figure 3.1) at θ0, instead of having the manifolds intersecting

tangently.

3.5 Homoclinic tangle

When the manifolds (as visualized in figure 3.1) cross at a point s(θ) = u(θ), they cross

on the orbit of this point as well. Since this orbit (ψt(s(θ)), θ + ωt) moves periodically

in θ, as it approaches the equilibrium, the crossing point will intersect each section ¯

S = {(x, θ) : θ = θ0} infinitely many times. We will explain this concept furthermore

and pose it as a more general phenomenon.

Let p be a hyperbolic fixed point for f . If q is a transverse homoclinic point for f , then the whole orbit of q is formed by transverse homoclinic points. The iterations of q tends to p for t → ±∞. Thus Wu(p) and Ws(p) must be intersecting on the orbit of q. The created folds grow larger and thinner, creating new homoclinic points when intersecting. The result is a so called homoclinic tangle. See figure 3.2 for a impression of the arising complexity (while imagening the red-blue-green patch to not be there). This phenomenon paves the way to chaos, with its formation of horseshoes.

3.6 Smale horseshoe map

Smale provided an elegant topological description of the chaotic nature of the trajectories in the homoclinic tangle. First, consider what happens to a rectangular patch on the surface of section S in the vicinity of the hyperbolic fixed point under the iterated application of the map. In figure 3.2, a rectangular patch, consisting of three strips colored red, blue, and green, is mapped forward around the tangle (three iterates in (a) and another three in (b)). In the immediate vicinity of the fixed point, the linearized flow dominates, and we see that the patch is squeezed in one direction and lengthened in the other. Under repeated action of the map, the patch is eventually brought again into

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the region near the fixed point. Thus for some positive integer m, the map Q := Pm is a map from the patch into itself. A single iteration of this map yields an interesting outcome. First we notice that the middle blue third of the patch has been removed

Figure 3.2: figure (a) contains three iterates of the map, after which, figure (b) illustrates an additional three iterates of the map. This picture is taken from [7], p. 19. entirely. The portion of the patch that remains inside the tangle will be mapped again under the action of the dynamics. The crux of this exercise is to carefully observe which points remain inside the tangle. Carefully keeping track of the red and green strips during the mapping around the tangle, allows us to determine their orientation once they return to the patch. Figure 3.2 (c) shows the initial horizontal orientation of the strips, overlaid with the final vertical orientation of the strips. Not only have the strips been squeezed and stretched: the red strip has been turned upside down! The action of the homoclinic tangle just described is abstracted as the horseshoe map Q : S → R2

acting on the square [0, 1] × [0, 1] ⊂ R2 and is shown graphically in figure 3.3. Again, we consider three horizontal strips of the square: the first labelled A, the third C, while the middle strip B does not return to the square under the action of the map. We may also consider the backward iteration of the map (figure 3.4). In this case, a vertical third of the patch is removed upon each iteration.

3.7 Cantor set of transverse homoclinic points

The invariant set of points that remain under infinite backward and forward iteration, denoted by Λ := k=∞ \ k=−∞ Qk(S), is a Cantor set.

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Figure 3.3: (a) Forward iteration of the Smale horseshoe map. (b) The set that remains in the square after two iterations of the forward map is colored blue. [7], p. 20.

Figure 3.4: (a) Backward iteration of the Smale Horseshoe map. (b) The set that remains in the square after two iterations of the backward map is colored blue. [7], p. 20.

Definition 13. A set Λ ⊂ R2 is a Cantor set if it is (1) closed (2) perfect (each point in Λ is a limit point) and (3) totally disconnected.

The set Λ contains uncountably infinite points, but has measure zero.

We have come across a more general phenomenon and therefore include it as well. Lemma 5. Let p is a hyperbolic fixed point for the dynamical system given by f on R2. If non-empty, the set of all the transverse homoclinic points for p is a Cantor set.

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Figure 3.5: On the left the set that remains invariant under one iteration of both forward and backward maps and on the right the invariant set after two iterations of both the forward and backward maps. [7], p. 21.

3.8 Symbolic dynamics

We assign to each point in Λ a symbolic representation of its entire bi-infinite trajectory. We begin with the map ϕ : Λ → {0, 1} that assigns to a point q ∈ Λ a symbol 0 or 1 according to the rule

ϕ(q) = 0 if q ∈ A and ϕ(q) = 1 if q ∈ C.

Next, using the symbol assignment map ϕ we define the map φ : Λ → Σ that maps a point q ∈ Λ to the space of all possible bi-infinite sequences of the symbols 0 and 1 according to:

φ(q) = [ϕ(Pk(q)]k=∞_k=−∞.

By this construction, we now have a representation for every trajectory under P as a bi-infinite ordered sequence of symbols that describes precisely how the trajectory meanders back and forth between A and C. The action of the map results in a simple shift on the sequence of symbols. That is,

φ(P (q)) = σ(φ(q)),

where σ : Σ → Σ simply shifts the list of symbols one place to the left. Furthermore, the fact that both the images and pre-images of A lie in both A and C (and vice versa), implies that to every bi-infinite sequence of symbols, we may assign a trajectory in Λ. Thus, φ : Λ → Σ is a homeomorphism (continuous, invertible mapping) between Λ and Σ.

Definition 14. Two differentiable dynamical systems (R, M, η), (R, N, ψ) are topolog-ically conjugate if there exists a homeomorphism h : M → N such that h ◦ ηt= ψt◦ h for all t ∈ R. This implies that h takes the orbits of (R, M, η) to orbits of (R, N, ψ).

The commutation diagram that illustrates the conjugacy between the map P : Λ → Λ and σ : Σ → Σ through φ : Λ → Σ is shown in figure 3.6. The upshot of this construction

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Figure 3.6: commutation-diagram showing the topological conjugation. [7], p. 22.

is that the morass of complicated trajectories in the homoclinic tangle envisioned by Poincar´e is completely and elegantly described by a consideration of the shift map on the space of bi-infinite sequences of two symbols. Specifically, Smale was then able to use this conjugacy to describe the possible orbits in Λ and by considering all possible sequences of symbols concluded that Λ has a countable infinity of periodic orbits with arbitrarily high period, an countable infinity of periodic orbits, and a dense orbit. The dense orbit is obtained by considering the concatenation of all possible finite sequences of one fixed and then increased length. Moreover, given any finite trajectory, it is easy to see how repeating the sequence produces a periodic orbit that is arbitrarily close. Theorem 7 (Poincar´e-Smale-Birkhoff Homoclinic Theorem). Let p be a hyperbolic fixed point of P , where P is a diffeomorphism (a differentiable map with a differentiable inverse) on S and suppose that q is a transversal homoclinic point of P (where Ws_(p)

and Wu(p) intersect transversely). Then there is a Cantor set Λ ⊂ S and N ∈ Z+ such that PN(Λ) = Λ and PN(·) restricted to Λ is topologically a shift automorphism.

The value of N is chosen large enough, precisely to ensure that the square patch in S is returned by the map P to S after its journey around the homoclinic tangle. In sum, the conjugacy of the horseshoe map with a shift automorphism provides a succint framework in which to understand homoclinic dynamics. For instance, we see the existence of sensitivity to initial conditions: there exist two sequences with identical histories, but with very different futures.

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4 Application and conclusion

With the knowledge we have gained we can now illustrate the usefulness of the Melnikov method.

Example 5. Consider the time-periodic Hamiltonian H(q, p, t) =1

2(q

2_{+ p}2_{) −}1

3q

3_{− q sin(t)}

where is a small parameter. We can sketch the phase diagram for the corresponding Hamiltonian system ˙q = _∂p∂ H , ˙p = −_∂q∂ H and determine the equilibria together with their type, and possible homoclinic or heteroclinic orbits. See figure 4.1 for a streamplot. Then we can determine an expression in terms of p and q for all homoclinic and

hete-Figure 4.1: Streamplot of the Hamiltonian system derived from the Hamiltonian H(q, p, t) = 1₂(q2+ p2) −1₃q3− q sin(t), drawn by Wolfram Alpha.

roclinic solutions that exist for = 0. For this we use the Hamiltonian. By assuming = 0 we have that ˙q = q − q2 = 0 ⇐⇒ q = 0 or q = 1, and ˙p = p ⇐⇒ p = 0. We find by linearization of f a clockwise-turning centre around (0, 0) and a saddle point at (1, 0). The Hamiltonian is constant on the homoclinic orbit, with the same value as it has on the saddle point: H(1, 0) = 1₆. Notice that

1 2q 2₊1 2p 2₋1 3q 3 ₌ 1 6 ⇐⇒ r 1 3 − q 2₊2 3q 3_{= ±p.}

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Since ˙q = p we get ±t + c = Z ∞ −∞ 1 q 1 3 − q2+ 2 3q3 dq,

where we take c = 0 as the initial condition. By using the substitution u = √2q + 1 (such that du = √ 1 2q+1 dq) we get Z ∞ −∞ 1 q 1 3− q2+ 2 3q3 dq = √ 3 Z ∞ −∞ 1 (q − 1)√2q + 1dq = √ 3 · −2 Z ∞ −∞ 1 3 − u2du =√3 · −2 tanh−1( √ 2q+1 √ 3 ) √ 3 ,

of which the last equality can be derived with the use of some more (somewhat) advanced calculus. More specifically: the substitution v = √u

3 and the definition of tanh

−1 _would

do the trick. This implies

q(t) = q =3 2tanh 2₍₋t 2) − 1 2. Also, we can apply the fact that tanh2(₂t) = tanh2 −t

2 . Now we have almost arrived

at our solution for the unperturbed homoclinic orbit. We find p(t) = d dtq(t) = 3 2tanh( t 2) sech 2₍t 2) so that (q(t), p(t)) is the solution. By definition (3.5) we have that

M (θ) =

Z ∞

−∞

f (φτ(q)) ∧ g(φτ(q), θ + ωt)dτ

for some q on the homoclinic orbit and φτ being the flow of the solution. Evaluation of

this above exprssion requires finding f along the orbit, but since f (z) = ˙z with z = (q, p) this is equivalent to taking time derivatives so we can rewrite to find

M (θ) = Z ∞ −∞ ˙ q(t)g2(q(t), p(t), ωt + θ) − ˙p(t)g1(q(t), p(t), ωt + θ)dt = 3 2 Z ∞ −∞ tanh(t 2) sech 2₍t 2) sin(ωt + θ)dt,

using g1(t) = 0 and g2(t) = sin(t). To solve this we will first apply partial integration.

We use that d dtsech 2₍t 2) = 2 sech( t 2)∗ d dtsech( t 2) = 2 sech( t 2)∗− 1 2tanh( t 2) sech( t 2) = − sech 2₍t 2)∗tanh( t 2) so that M (θ) = 3 2 ∗ ([sech 2₍t 2) sin(ωt + θ)] − − Z ∞ −∞ sech2(t 2) cos(ωt + θ) ∗ ω dt) = 3 2ω Z ∞ −∞ sech2(t 2) cos(ωt + θ) dt,

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since sech(₂t) goes to zero for t → ±∞. Also we have that

cos(ωt + θ) = e

i(ωt+θ)_{+ e}−i(ωt+θ)

2 ,

so that we can split our integral into two parts 3 2ω Z ∞ −∞ sech2(t 2) cos(ωt+θ) dt = 3 4ω∗( Z ∞ −∞ sech2(t 2)e i(ωt+θ)_dt+ Z ∞ −∞ sech2(t 2)e −i(ωt+θ)_dt).

Since sech is an even function we can substitute −t for t in the second integral, resulting in 3 2ω ∗ eiθ_{+ e}−iθ 2 Z ∞ −∞ sech2(t 2)e iωt _{dt =} 3 2ω cos(θ) Z ∞ −∞ sech2(t 2)e iωt _dt = 3 2ω cos(θ) Z ∞ −∞ sech2(τ )e2iωτ dτ.

By substituting τ = ₂t. We’ll now apply the (here not included) Cauchy residue theorem, by noting that sech2(τ ) = (cos(iτ ))−2 has double poles at the zeros of the cosine, at τn= iπ(2n + 1)/2. Make a contour in the upper half plane with positive orientation and

radius R = π(2m + 2)/2 for m ∈ Z and let m → ∞ (with steps such that the boundary of the contour does not intersect a pole). We get

M (θ) = 3 2ω cos(θ)2πi n=∞ X n=0 Resτn(sech 2_{(τ )e}2iωτ_).

Near a pole, cos(iτ ) = 0 − i sin(iτn)(τ − τn) + O(τ − τn)3 = i(−1)n(τ − τn) + O(τ − τn)3.

The redisue at τn is given by the O(τ − τn)−1 term in the expansion of sech2(τ )e2iωτ.

Thus we find

Resτn(sech

2_{(τ )e}2iωτ_{) = −2iωe}2iωτn _,

giving us the answer we were looking for: M (θ) = 3

2ω cos(θ)2πi

n=∞

X

n=0

−2iωe2iωτn _{= 6πω}2_cos(θ)

n=∞ X n=0 e−2πω(2n+1)2 = 6πω2cos(θ)e−πω n=∞ X n=0 e−2nπω= 6πω2cos(θ)e−πω e −πω 1 − e−2πω = 6πω2cos(θ) csch(πω).

Note that M is periodic in θ and vanishes for θk= π₂ + kπ with k ∈ Z but is otherwise

nonzero. Also, DθM (θk) 6= 0. Therefore we may apply Melnikovs theorem and conclude

that the stable and unstable manifolds intersect transversely. At last, the Smale-Birkhoff theorem now tells us that we have found the chaotic behavior that we were looking for. 4

Likewise, our simple perturbed planar pendulum exhibits this route to chaos. This method introduced by Poincar´e and developed by Melnikov is still used today to prove the existence of chaotic dynamics in various fields analysis (and physics).

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5 Summary

In general1: this thesis reviews the use of the Melnikov method in the detection of chaotic dynamics. It is a bulky subject and to get to understanding this method I will sketch some foundations of it. There are multiple mathematical objects (theory and definitions) introduced to start with and to be able to discuss the central phenomenon: the route to chaos of the perturbed simple pendulum. The unperturbed pendulum has a fixed length and point mass and is acted on by a constant gravitational field. The phase space consists of possible values of the pendulum’s position, represented by an angle and its angular velocity. The motion only takes place in two dimensions. The pendulum can be described by differential equations (which can be time-dependent, non-autonomous). We can study it with the use of linearizations thanks to the Hartman - Grobman theorem which gives information about the flow (also called solution curve, trajectory, orbit ) of the system. But we would not have gotten there without the also very basic local existence and uniqueness theorem. Knowledge about manifolds and vector fields would help as well. The same holds for comprehension of fundamental concepts used in (this) analysis such as continuity and differentiability. Topology and linear algebra are also part of the toolkit. There is simply too much to get you up to speed (if you did not make it through the thesis itself yet). Scrolling this paperwork to have a look at the images and figures could also support a good impression. Moreover, we encounter fixed points (which are also called equilibria) which are defined as constant solutions for the differential equation. Also important here are the homoclinic orbits on which the flow ‘moves’ to a hyperbolic saddle equilibrium (which exists when the pendulum is hanging motionless upside down). We unfold a vector field and use theorems such as the implicit function theorem and the stable manifold theorem. These will guarantee lots of things such as the preservation of nondegenerate equilibria. Of great importance are the first return map and the concept of transversal intersection, to be able to study homoclinic bifurcation. For the corresponding (homoclinic bifurcation) theorem we make assumptions such as the requirement that the sum of the eigenvalues of the total derivative of the vector field f evaluated in the saddle equilibrium needs to be unequal to 0 (it is a non-Hamiltonian system). The period of the periodic orbit grows to infinity and it collides with the saddle point to become a homoclinic orbit and after the bifurcation there is no longer a periodic orbit (or this just explained process can occur the other way around, depending on the conditions). So, with this theorem it becomes relatively easy to determine whether a perturbation of a homoclinic orbit destroys it and gives birth to a nearby periodic orbit. With all this information acknowledged we have gotten much closer to the actual Melnikov theory. Yet, there is more insight needed. When we take a section at a

1

This chapter is written for high school students with affinity for mathematics and first year bachelor mathematics students

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point in the homoclinic loop, we need to know how the intersections of the perturbed stable and unstable manifold that continue from this point behave on this section. For this Green’s theorem is applied, as well as the Gr¨onwall inequality. Again, one of the main tools for determining the existence of (or nonexistence of) chaos in a perturbed Hamiltonian system is Melnikov theory. In this theory, the distance between the stable and unstable manifolds of the perturbed system is calculated up to the first-order term. We start with considering a smooth (sufficiently differentiable) dynamical system ˙z = f (z) + g(z, t) with g(z, t) = g(z, t + T ), which means that g is periodic with period T . Suppose for = 0 the system has a hyperbolic fixed point p0 and a homoclinic orbit

γ corresponding to this fixed point. Then for sufficiently small 6= 0 there exists a T-periodic hyperbolic solution (as stated in the persistence of hyperbolic periodic orbits theorem). We find the solution of the perturbed flow as power series expansian away from the unperturbed flow. The stable and unstable manifolds of this periodic solution intersect at some height θ0 when the Melnikov integral vanishes. This integral measures

the distance between these manifolds along a direction that is perpendicular to the unperturbed homoclinic orbit γ, by using the famous wedge product. The Melnikov function is defined as the rate of change of this distance with . When DθM (θ0) 6= 0 we

find a transversal intersection. This leads to the formation of a homoclinic tangle (and transversal intersection at every section). Within this tangle the formation of Smale’s horseshoes can be found. This horseshoe map is topologically conjugated to a shift map on the space of bi-infinite sequences of two symbols. The upshot of this construction is that the complicated trajectories in the homoclinic tangle can be completely and elegantly described. The set of transverse homoclinic points form a Cantor set. This comes together in the Poincar´e-Smale-Birkhoff theorem. At last, we may call a dynamical system chaotic on a given invariant set X for a flow φ, when this flow is (1) topologically mixing (2) its periodic points are dense and (3) it has sensitive dependence on initial conditions. All of the above can of course be reviewed more extensively in the thesis itself, which includes clarifying examples. The also beautiful Cauchy residue theorem is part of a somewhat voluminous illustration of the usefulness of the Melnikov method that can be found at the end of my thesis. Thanks for reading!

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Bibliography

[1] Azzari, P. (2016) “Homoclinic chaos and the Poincar´e-Melnikov method.” [2] Devaney, Robert “An Introduction to Chaotic Dynamical Systems.” [3] Francis, Matthew “Physics Quanta: Pendulums Revisited”.

Re-trieved from https://galileospendulum.org/2011/05/31/physics-quanta-pendulums-revisited/.

[4] Guckenheimer, John “Nonlinear oscillations, dynamical systems, and bi-furcations of vector fields.”

[5] Meiss, James D. “Differential Dynamical Systems.”

[6] Robinson, R. Clark “An Introduction to Dynamical Systems.”

[7] du Toit, Philip C. (2010) “Transport and separatrices in time-dependent flows.”

The perturbed simple pendulums route to chaos