Dynamical Systems Generated by ODEs and Maps:

Final Examination Project

Jasmine Nirody

26 January, 2010

Contents

1 Introduction and Background 1

2 Equilibria 1

3 Attracting Domain 2

4 Overview of Behavior Under Changes in r 4

5 Bifurcations of Equilibria 5

6 The Strange Invariant Set 11

7 The Strange Attractor 14

8 Geometric Models 18

9 Period Doubling and Intermittent Chaos 21

10 Summary 23

1 Introduction and Background

In the following we consider the Lorenz system:

x = −σx+ σy

y = rx− y − xz

z = −bz + xy

Unless otherwise mentioned, the parameter values considered will be σ = 10, b = 83,

r > 0.

The system arose from a model proposed by Lord Rayleigh for thermal convection:

δ∇2ψ

δt= −δ(ψ,∇

2ψ)

δ(ζ, η)+ ν∇4ψ + gα

δΘ

δζ

δΘ

δt= −δ(ψ,Θ

δ(ζ, η)+δT

H+δψ

δζ+ κ∇2Θ.

In this model, ψ is a stream function for two dimensional motion, Θ is the functiondescribing the temperature variation when there is no convection, ζ, η are spatialcoordinates, ∆T the imposed temperature difference, g is the acceleration due togravity, α bouyancy, κ thermal diffusivity, ν viscosity, and t is time [7].

Using a Fourier series in the spatial coordinates to expand ψ and Θ, with functionsof t as coefficients, and then reducing the infinite system by setting all modes butthree to zero [10], we obtain a finite system of ordinary differential equations, whichare the basis for the system shown above [7].

It is important to note that the coordinates in the Lorenz system are not spatial coor-dinates. Rather, x is proportional to the speed of motion of the air due to convection,y is a measure of temperature difference between the rising (warm) and falling (cool)air, and z is the vertical temperature difference as we move through the system.The parameters also have physical meaning: σ is called the Prandtl number whichis related to the nature of the air involved, b is representative of the size of the areaconsidered, and r is called the Rayleigh number which says at which point convectionwill begin.

2 Equilibria

We now wish to solve for the stationary points or equilibrium points of the system,˙x = f(x), where f(x) : R

3 → R3 is defined as:

f(x) =

x = −σx+ σy

y = rx− y − xz

z = −bz + xy.

1

We define stationary points as those values at which f(x) = 0. It is easy to see thatthe origin, (x, y, z) = (0, 0, 0), is such a point, but the system also has two otherequilibria,

(x, y, z) = (±b√r − 1,±b

√r − 1, r − 1)

which we refer to from now on as C1(+) and C2(−). Note that C1, C2 are real-valuedfor r > 1.

At this point, let us quickly note that there is a natural symmetry defined by thetransformation:

(x, y, z) 7→ (−x,−y, z).This property is seen in the equilibria C1 and C2, but we wish to verify this symmetry.In order to show symmetry, we must find a 3 × 3 matrix, R, which commutes withthe system. That is:

Rf(x) = f(Rx).

We choose R based on the transformation mentioned above:

R =

−1 0 00 −1 00 0 1

.

We check the commutative relationship:

Rf(x) =

σ(x− y)x(r − z) − y

xy − bz

,

f(Rx) = f

−x−yz

=

σ(x− y)x(r − z) − y

xy − bz

= Rf(x).

Thus, we have shown that this relationship corresponds to a natural symmetry in thesystem.

3 Attracting Domain

Let us consider a general 3D system. The time evolution of a volume, V (t), enclosedby a surface, S(t), is given by:

V =

∫

S

f ·NdA,

where N is the normal vector to S. Then, by Green’s Theorem, this is equivalent to:

V =

∫

V

∇ · fdV.

2

The divergence of the Lorenz system is given as:

∇ · f =d

dx[σ(y − x)] +

d

dy(rx− y − xz) +

d

dz(xy − bz),

∇ · f = −(σ + 1 + b).

And so:V = −(σ + 1 + b)V

⇒ V (t) = V (0)e−(σ+1+b)t.

We see that the exponent is negative, and therefore volumes under the action of theLorenz system shrink exponentially fast, showing that there exists an attracting set

of zero volume.

A set is considered to be a positively-invariant domain of a system if, for any startingpoint x0 ∈ S, under the action of the system, x(t, x0) ∈ S for all t > 0.

We now show that there is a bounded sphere, S, into which all trajectories enterand upon entering, never leave (i.e. S is a positively invariant domain of the Lorenzsystem). First, let us define a Lyapunov function. A scalar function, V , is considereda Lyapunov function on a region D if it is continuous, positive definite, and hascontinuous first order derivitives on all ofD. Now, we consider a specific such function:

V (x, y, z) = rx2 + σy2 + σ(z − 2r)2.

The derivitive of a Lyapunov function, with respect to the ODE system ˙x = f(x), isgiven to be:

V = ∇V (x) · f(x).

So, considering the Lorenz system and Lyapunov function we have introduced:

V = −2σ(rx2 + y2 + bz2 − 2brz).

Let us call D the bounded region in which V > 0, and let c be the maximum of Vin D. Specifically, we can define D as the ellipsoid which consists of the points forwhich:

rx2 + y2 + bz2 = 2brz.

Consider now a bounded region, S consisting of the points which satisfy:

x2 + y2 + (z − r − σ)2 = R2,

and let us choose R so that D is completely contained within S. Then, if a pointx lies outside of S, then it must also lie outside of D, and by definition, V (x) ≤ 0.Rather, we can also say that V (x) ≤ −δ for some δ > 0. Now, if a trajectory be-gins with initial condition x outside of S, then V (x) will decrease with time and willeventually (within finite time for δ > 0) enter S. Additionally, since on the surfaceof S, V ≤ 0, trajectories may only cross inwards. Therefore, once trajectories passinwards through the boundary of S, they will never leave it, and S is therefore apositvely-invariant domain into which all trajectories will enter.

3

4 Overview of Behavior Under Changes in r

In the following sections, we wish to observe changes in the dynamics of the Lorenzsystem as we vary one parameter, namely r. In this section, we provide a quickoverview of the changes we see with this varying, with only a few proofs. However,most of the statements made in this section will be discussed in detail in subsequentsections.

• In the interval 0 < r < 1, the origin is globally stable and all trajectories areattracted to it.

We now show the global stability of the origin. Consider the Lyapunov function:

V (x, y, z) =1

σx2 + y2 + z2

with:

V = 2[(r + 1)xy − x2 − y2 − bz2]

= −2[x− r + 1

2y]2 − 2[1 − (

r + 1

2)2]y2 − 2bz2

From this we see that for 0 < r < 1, V < 0 if (x, y, z) 6= (0, 0, 0), and we can concludethat as t → ∞, V → 0. So we can conclude that any point (x, y, z) → (0, 0, 0); thatis, all points will tend to the origin.

• For 1 < r, the origin is unstable.

• In the interval 1 < r < 24.74.., C1,2 are stable.

The first statement can be shown by a simple analysis of the eigenvalues of thesystem evaluated at the origin. To analyse stability of equilibria, we first consider theJacobian:

J =

−σ σ 0r − z −1 −xy x −b

.

At the origin, this reduces to:

J(0,0,0) =

−σ σ 0r −1 00 0 −b

.

The eigenvalues of this matrix are:

λ1,2 =1

2

(

−1 − σ ±√

1 − 2σ + 4rσ + σ2)

,

4

λ3 = −b.For r > 1, λ1 > 0 and the origin becomes unstable.

A discussion of the stability of C1,2 will be given in a later section.

• For 24.74.. < r, both C1 and C2 are unstable. At this point, there are no stablefixed points in the system.

A discussion of this loss of stability will be provided in a later section. Our interestlies in what happens for r > 24.74....

We know there are no more stable fixed points, and we will see in later sections thatthere do not exist any limit cycles around C1,2 either. We also know that volumescontract exponentially fast, and so trajectories cannot escape to infinity.

We have shown that there exists an attracting set to which all trajectories tend. Inthis region, however, the attractor does not contain any critical points (as they arenot stable) and is unlikely to contain any limit cycles (as we will see, the Hopf bi-furcation which occurs nearby is subcritical, so the limit cycles caused by it will beunstable as well). A discussion about the “shape” of the attractor will be providedlater, as will a general discussion of techniques used to analyse such a question.

However, before we consider this region, we first quickly discuss the region r < 24.74..,which also has some interesting events.

5 Bifurcations of Equilibria

A bifurcation occurs when a small change in parameters causes a qualitative change inthe behavior of the system. We will first be concerned with local bifurcations, whichcan be analysed by observing changes in stability of equilibria, periodic orbits, orother invariant sets as parameters cross what we deem to be critical values. In a latersection, we discuss one kind of global bifurcation, the homoclinic bifurcation. In thissection, we focus on bifurcations concerned with equilibria, primarily the pitchfork

and Hopf bifurcations.

We consider again the Jacobian of the system:

J =

−σ σ 0r − z −1 −xy x −b

,

and evaluated at the origin:

5

J(0,0,0) =

−σ σ 0r −1 00 0 −b

.

A one parameter system ˙x = f(x, λ) with a pitchfork bifurcation is defined as one hashaving the following necessary conditions:

• fx(0, 0) = 0 (bifurcation condition),

• f(0, λ) = 0 (bifurcation condition),

• fxx(0, 0) = 0 (suggests local symmetry).

We wish to find such a point in the Lorenz system. We know the second bifurcationcondition must hold for any equilibrium point with any parameter values. At apitchfork point, we know that rank(J) is not maximal as per the first condition, thatis, rank(J) < 3. So the determinant of this matrix, J , must be zero. The determinantis given by:

det(J) = 0 =−80

3+

80

3r

⇒ rBP = 1.

Note that we have considered the parameter values for σ and b given in the firstsection.

Alternatively, we can also show that J(0,0,0) does not have 3 linearly independentcolumns. Because of the 0 in the right upper column, we will consider the first twocolumns.

A(−σ) = σ

A(r) = −1

A(y) = x.

From this, it is apparent that these columns are linearly dependent when x = y =0, A = −1, rBP = 1.

The local symmetry condition holds (as expected, given the natural symmetry of thesystem), apparent from the fact that there are no quadratic terms in the system.

We wish to analyse the direction of this bifurcation; that is, whether it is consideredsubcritical or supercritical. In order to do this, we begin by introducing the paramterρ = r − 1 so that the system now becomes:

6

x = −σx+ σy

y = ρx+ x− y − xz

z = −bz + xy

Now, we realise that the pitchfork bifurcation will occur at ρ = 0 and so we analysethe Jacobian at this state:

J =

−σ σ 01 −1 00 0 −b

.

We see that J has eigenvalues λ1,2,3 = 0,−(σ+ 1),−b. We now rewrite our system inthe form:

f(u) = T−1JT [u] + T−1R(T (u)),

where:

R(x) = f(x) − J

and

T−1JT =

(

Ac

As

)

,

where Ac and As are block matrices whose diagonals contain the eigenvalues withRe(λ) = 0 and Re(λ) < 0 respectively. Note that we have accounted for the fact thatRe(λ) ≤ 0 for all eigenvalues in this system.

Using T as the matrix consisting of the eigenvectors:

T =

1 σ 01 −1 00 0 1

,

we arrive at the extended system which takes into account the derivitive of ρ:

f(u) =

u = σ1+σ

(ρ− w)(u+ σv)

v = −(1 + σ)v + 11+σ

(−(ρ− w)(u+ σv))

w = −bw + 11+σ

(1 + σ)(u+ σv)(u− v)

ρ = 0.

The central manifold will be of the form:

W c = {(u, v, w, ρ) : v = h1(u, ρ), w = h2(u, ρ), hi(0, 0) = 0, Jh(0, 0) = 0}.

7

We substitute these values for v and w and consider the power series:

h1(u, ρ) = a1u2 + a2uρ+ a3ρ

2 + higher order terms

h2(u, ρ) = b1u2 + b2uρ+ b3ρ

2 + higher order terms.

After comparing coefficients for u2 and uρ, we find:

v = − 1

(1 + σ)2

w =1

bu2,

leaving us with:

u = − 1

1 + σ(ρu− u3

b)

ρ = 0.

Therfore, the equilibrium is stable for ρ ≤ 0 and unstable for ρ > 0, correspondingto r ≤ 1 and r > 1, respectively, as we found previously. Furthermore, since thesign in front of the cubic term is negative, we say we have a supercritical pitchforkbifurcation, which we already suspect from the fact that the bifurcation results in theappearance of two stable fixed points C1,2.

We also find this bifurcation numerically:

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

r

X BPBPBP

Figure 1: Numerical continuation in MATCONT showing branching point and re-sulting “pitchfork” appearance.

A Hopf bifurcation occurs when a pair of complex conjugate eigenvalues cross theimaginary axis, that is when Re(λx,y) = 0 and the eigenvalues are purely imaginary.Specifically, the necessary conditions for a Hopf bifurcation are:

8

• fx(0, 0) = 0 (bifurcation condition),

• Re(µ1,2) ∈ σ(J) = 0 and Im(µ1,2) ∈ σ(J) 6= 0, where sigma is the spectrum ofJ . That is, the Jacobian has a pair of purely imaginary eigenvalues.

Genericity conditions usually seen are:

• There are no other eigenvalues ∈ σ(J) on the imaginary axis

• ddλRe(µ1,2) 6= 0 for all µ ∈ σ(J).

The genericity conditions assure that the bifurcation is topologically equivalent to thenormal form.

We wish to find the value at which a Hopf bifurcation may occur in the system. Fora Hopf point in this system, we realise that there are three eigenvalues, two of whichwill be purely imaginary conjugates. Therefore, at the Hopf point, tr(J) will be equalto the third eigenvalue.

We see that tr(J) = −σ − 1 − b = λ3 (which is not a critical value, as per the firstgenericity condition), where λ1,2 are purely imaginary. Remembering that det(J −λ3I) = 0 (because λ3 is an eigenvalue of the system), we call A = J − λ3I:

A =

1 + b σ 0r − z σ + b −xy x σ + 1

.

We calculate:

det(A) = 0 = x(x+ bx− σy) + (1 + σ)(b+ b2 + σ + bσ − rσ + σz).

Using the equations for the stationary points, C1,2, of the system:

(x, y, z) = (±b√r − 1,±b

√r − 1, r − 1),

we arrive at:

rH =σ(3 + b+ σ)

σ − b− 1.

We now check the stability of C1,2. The Jacobian at C1 is shown as:

J =

−σ σ 0

1 −1 −√

b(r − 1)√

b(r − 1)√

b(r − 1) −b

,

and at C2:

J =

−σ σ 0

1 −1√

b(r − 1)

−√

b(r − 1) −√

b(r − 1) −b

.

9

The eigenvalues of these matrices are the roots of:

f(λ) = λ3 + λ2(σ + b+ 1) + λb(σ + r) + 2σb(r − 1).

All three roots are real when r ≈ 1, but when r > 1.436.. (with σ, b as usual), wehave a pair of complex conjugate roots and one real root. All three roots will havea negative real part for r < rH , while the two complex conjugate roots will havepositive real part for r > rH , implying they have non-zero velocity as they cross theimaginary axis, as per the the genericity condition. This means that the equilibriumpoints C1,2 are stable for 1 < r < rH ≈ 24.74.., and lose stability after. [8].

We find the Hopf points numerically as well:

0 5 10 15 20 25−10

−8

−6

−4

−2

0

2

4

6

8

10

r

X BPBPBPBP

H H

BPBP

H H

Figure 2: Numerical continuation in MATCONT showing Hopf bifurcation points.

From numerical calculations, we find that the first Lyapunov coefficient, l1(0), is1.333832 × 10−3. A Hopf bifurcation is considered subcritical (and will result in astable equilibrium point losing stability by absorbing an unstable periodic orbit) ifl1(0) > 0 and supercritical (and will result in a stable equilibrium point losing stabilityby emitting a stable periodic orbit) if l1(0) < 0. Our Hopf bifurcation is subcriticalfor the values we have chosen.

It appears that the bifurcation is subcritical for all values σ and b for which it occursas long as r > 0 [3]. Because this implies that the Hopf point will absorb periodicorbits rather than emit them, we must investigate where these periodic orbits origi-nate.

10

1015

2025

30

−10−5

05

1015

20−10

−5

0

5

10

15

20

r

LPCLPC

X

Y

Figure 3: A family of limit cycles going into the Hopf bifurcation point for C1.

6 The Strange Invariant Set

For r > 1, trajectories tend to have a certain behavior. Trajectories that begin on oneside of R

3 spiral into C1 while those that begin on the other side spiral to C2. Thesetwo halves are divided by the stable manifold of the origin [11]. The trajectorieswhich begin on the stable manifold, of course, tend towards the origin. However as rgets larger, and crosses some value r∗, trajectories on each side of the stable manifoldmake bigger and bigger spirals until finally, at r > r∗ ≈ 13.926.., trajectories on oneside of the stable manifold of the origin “cross over” and become attracted to thefixed point on the other side.

We show this in a MATCONT simulation:

−10 −5 0 5 100

5

10

15

20

25

x

z

(a) Trajectory for r = 10 < r∗

−10 −5 0 5 100

5

10

15

20

25

x

z

(b) Trajectory for r = 15 > r∗

Figure 4: Trajectories beginning from the same point in space (x, y, z) = (2, 3, 0) fordifferent values of r.

We know that the stable manifold includes the entire z axis, since it is invariant andall points on it will tend to the origin, and we can explain this change of behavior byexpecting a homoclinic orbit at some value r = r′, meaning there will be a tangency

11

between the stable and unstable manifolds of the origin. This means that, in bothforward and backward time, trajectories will tend towards the origin. Formally, anorbit, φ(t), is called homoclinic if φ(t) → x0 as t → ±∞, where x0 is an equilibriumof the system.

We see numerical evidence of a homoclinic orbit if we plot the period of the limitcycles, T , vs. r:

10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

16

18

20

r

Per

iod

LPCLPC

Figure 5: Numerics suggest the period of limit cycles tends to ∞ as r → r∗.

It is difficult to find this orbit numerically, because once approximations jump off thestable manifold, they will move away from it very quickly. However, we can convinceourselves of its existence in another way: if a trajectory beginning at x1 with a cer-tain r reaches x2 in a certain time, then a trajectory beginning at a point x1 usingr′ near r will reach a point near x2 in the same time. But, if we choose some ǫ > 0,r = r∗ − ǫ

2, the trajectory will spiral in towards a fixed point, whereas for r′ = r∗ + ǫ

2

it will spiral into the other. The existence of a homoclinic orbit at r = r∗ explainsthis change, since at this time the trajectory will spend “infinite time” at the origin.

Let us choose a surface Σ which contains the equilibria C1,2 and their 1D stablemanifolds such that Σ intersects the 2D stable manifold of the origin along a curveD (shown as the midline in Figure 6). In addition, we choose Σ so that any orbitstarted on it returns to it after some time, so that we have now defined a Poincaremap, P , on Σ\D. We can put this more intuitively: consider a small box around theorigin with two tubes entering the top face and exiting the sides and turning aroundthe branches of the unstable origin. Then, we call F = P (Σ)

⋂

Σ a return map, whichanalyses where a trajectory entering in the top face of the box will hit the top faceonce again (rather, “return”) after travelling through the tubes. We show the returnmap presented by Sparrow in Figure 6.

The line D is mapped to two points, denoted by the tips of the shaded triangles,H±. Let us make note that these points are where the unstable manifold of the origin

12

Figure 6: Return maps for the Lorenz system given by Sparrow [11].

hits Σ for the first time. Near the origin, the flow contracts (because λ1 < 0) in thedirection of the eigenvector corresponding to the largest eigenvalue λ1. Using somecoordinates (u, v), where u = 0 defines D, we write F as:

F (u, v) = (f(u), g(u, v)),

where0 < gv(u, v) < c < 1 for u 6= 0,

andg → 0 as u→ 0.

This assumes the existence of a global contracting foliation, a concept which we willrevisit when discussing the geometric Lorenz attractor. Now, we wish to see what willhappen to F as we increase r. The left panel of Figure 6 shows F for r < r∗. Here,the fixed points C1,2 at the left and right sides of the triangles are global attractors.As we increase r to r = r∗, we see that the points of the triangle (which we rememberrepresent the first intersection of the unstable manifold of the origin with the returnsurface) hits D, thus suggesting that the unstable manifold of the origin is containedin the stable manifold, and a homoclinic orbit occurs (as we had expected).

Beyond r = r∗, we can only calculate return maps numerically, and so the followingresults are based on numerical, rather than analytical, evidence. For r > r∗, F hastwo new fixed points within the triangles, which we refer to as P±. These pointsare from the limit cycles which arise at r∗ from the homoclinic orbits. Now, thereexists an invariant set, Λ, of Cantor structure between P+ and P−. So, points thatenter this region stay forever within this region. Furthermore, the dynamics on Λ isconjugate to the shift map, s, of two symbols: [4, 11, 15]

s(w) = θ, θj = wj+1, j ∈ Z

where w is a list of symbols

w = w1w2w3... , wi ∈ 0, 1

We now show that this implies that there exist within Λ a countable infinity of peri-odic orbits and an uncountable infinity of aperiodic orbits and orbits that tend to theorigin. Any periodic sequence with block length n repeating k ∈ Z amount of times(in this case n ≤ 2) is called a n-periodic orbit. This set of all periodic sequences

13

is countable. Since we can choose any k, the orbits are dense as well. Now supposethat all non-periodic orbits (consisting of the set of aperiodic orbits and orbits thattend to the origin) are listed. Then, we can construct a sequence with w1 differentthan in the first sequence, w2 different from the second sequence, etc, and thereforeconstruct some sequence w not listed, which is a contradiction. Therefore, the set ofall non-periodic orbits is uncountable.

On this set, the dynamics also show sensitive dependence on initial conditions, mean-ing that trajectories which begin arbitrarily close to each other move apart withincreasing time. Let us consider a map f t with state space M . Then f t shows sensi-tive dependence for initial conditions if for every point x ∈M , there exists a point yin a neighborhood N of x such that at time t = τ :

d(f τ (x), f τ (y)) > δ

for some δ > 0.

Due to the appearance of this strange invariant set, we arrive at two possible con-clusions. We see that the two simplest periodic orbits (one whose period consists ofone journey through the right tube, and the other its symmetric counterpart in theleft tube) are probable candidates for the periodic orbits which the subcritical Hopfbifurcation at rH will absorb. Other periodic orbits cannot be considered because,as the z axis is invariant, an orbit which winds around the axis will never be able toseparate itself from it, since it cannot cross it. Finally, we also see that this invariantset could possibly become the attractor after the three critical points become unstable(since we know there always exists an attractor in the system).

7 The Strange Attractor

Formally, we define an attractor as a region in space that is invariant under the evo-lution of time and attracts most, if not all, nearby trajectories. An attractor is calledstrange if it has a non-integer dimension or if the trajectories within it display chaotic

behavior (that is, display high sensitivity to initial conditions).

As we increase r, both P± and H± move towards C1,2, but numerical experimentsshow that H± move more slowly. A trajectory here eventually settles to C1 or C2 butwill wander about the invariant set for some time, τ , before it does so. Finally, atr = rA ≈ 24.06.., P± hits H±. This implies that the unstable manifold of the originintersects the stable manifolds of the periodic orbits. Here, we will see heteroclinic

orbits which connect C1,2 at τ = ∞. A trajectory that passes through the invari-ant set remains wandering aperiodically in the invariant set forever and it becomes astrange attractor [15]. Figure 7 shows a typical trajectory for r∗ < r = 22 < rA.

We show some numerical experiments that highlight the sensitive dependence oninitial conditions observed:

14

0 20 40 60 80 100 120 1400

5

10

15

20

25

30

35

40

t

z

Figure 7: A typical trajectory with finite but high τ , r = 22.

−25 −20 −15 −10 −5 0 5 10 15 20 250

5

10

15

20

25

30

35

40

45

50

x

z

(a) Trajectory for z = 10

−25 −20 −15 −10 −5 0 5 10 15 20 250

5

10

15

20

25

30

35

40

45

50

x

z

(b) Trajectory for z = 9.9

Figure 8: Trajectories beginning from the nearby points in space (x, y, z) = (0, 2, z)for parameter values, σ = 10, b = 8

3, r = 28.

0 5 10 15 20 25 30 35 40 450

5

10

15

20

25

30

35

40

45

t

z

Figure 9: Trajectory shown with respect to time for z = 10 (red) and z = 9.9 (blue).All other values as above.

Note that rA < rH ≈ 24.74... Therefore, at rA, C1,2 are still stable, and some trajec-tories will be attracted to these even though the strange attractor exists.

15

Because return maps can only be calculated numerically past r = r∗, complete mapscannot be computed since only a finite amount of points can be produced. However,using judgment and selecting a “sufficient” number of points, certain statementsabout the properties of the flow can be made. The following are assumptions andobservations from computed return maps by Sparrow [11].

• We assume that the map stretches almost all distances, implying that we stillobserve “sensitive dependence on initial conditions” because (almost) all tra-jectories will be divergent. The exceptions are those which begin “equidistant”from the intersection of the smooth manifold and the return plane, which wecan call RM . The trajectories of these “equidistant” points provide the returnmap with a chaotic appearance.

• The attractor has a Cantor-like structure. This partially explains why it iscalled a strange attractor. A strange attractor is one defined as having a non-integer dimension or one on which chaotic dynamics are observed. The Lorenzstrange attractor is observed to fit both these qualities.

• The attractor contains a countable infinity of non-stable periodic orbits and anuncountable infinity of aperiodic orbits and trajectories which terminate at theorigin.

• For r > rA, the simplest periodic orbits (each consisting of one turn aroundC1 or C2) are no longer part of the attracting set. This is expected becausethey are known to have been absorbed by the Hopf bifurcation, but it has beenshown that the same is true for rA < r < rH as well. We do not observe anyfixed points on the computed return map, which we would expect if there weresuch periodic orbits.

• Each periodic orbit is denoted by a series of x’s and y’s, which are arbitrarilychosen as opposite sides of the return plane, which we call RM . Choosing aninitial point R, each application of ψ(R) will bring it closer to RM , until finallyit appears on the other side. Therefore, there is a finite i∗ such that theredoes not exist a sequence with more than i∗ consecutive x’s or y’s. However,these orbits all existed in the original invariant set, as there was a one-to-onecorrespondence to each possible sequence. (Note, there is still only at most oneorbit corresponding to each sequence.)

• The explanation for the last point given is that periodic orbits, as well as someaperiodic trajectories, are removed from the attractor in homoclinic explosions.Consider, for example, r = 28, at which value i∗ = 26. Here, ψ26(R) lies exactlyon RM , and so corresponds to a homoclinic orbit.We must note that i∗ is a veryinaccurate measure of the “size” of the attractor, since we are only consideringthe first few entries in the symbolic sequence of x’s and y’s, k(r), which describesthe behavior of the right branch of the unstable manifold (an arbitrary choice).

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• Each homoclinic explosion creates a new strange invariant set, and so there willbe an uncountable number of topologically distinct attractors in any neighbor-hood of an arbitrary r [14].

• Numerical experiments suggest that while the attractor loses orbits (that is, i∗

decreases monotonically) for rA < r < 28, at some value slightly greater than28, i∗ begins to increase. At r ≈ 30, i∗ begins to decrease again, reaching 2 atr ≈ 45 and 1 at r ≈ 54.6. For r > 54.6, the original point R′ lies to the right ofRM for the first time.

• Since we consider the attractor to be monotonic on some interval, each periodicorbit can only be removed once from the attractor. Then, the set of periodicorbits in the original set can be divided into the following disjoint subsets:

1. The simplest periodic orbits absorbed by the Hopf bifurcation.

2. Orbits remaining in the strange attractor throughout.

3. A disjoint union of orbits removed by homoclinic explosions.

Using this criterion, we realise that only certain homoclinic explosions can be“allowed” so as to divide these orbits into completely disjoint seets. This result,which excludes certain homoclinic orbits from the Lorenz system, comes upagain important in geometric models.

8 Geometric Models

Several intuitive arguments based on numerical experiments and return maps can bemade. However, there has also been a considerable amount of work done on model

flows or geometric models that contain “Lorenz attractors”.

Many of the observations presented in the last section are based on the concept ofa contracting foliation. Let us consider two points, x and y, with distance betweenthem c such that at some nth iterate, d(ψn(x), ψn(y)) ≤ cλn, where 0 < λ < 1. Re-membering the observation from the previous section, we assume these two points tobe “equidistant” from RM , and so we then can assume there is a whole arc of pointsbetween x and y such that the above argument can be applied to any two trajectoriesstarted on this arc. If it is possible to fill the return plane with a continuum of sucharcs (i.e. that it is possible to begin a trajectory on one such arc that the return mapwill always hit a point on another such arc), then we can say there exists a contractingfoliation [11].

We now wish to present a model flow that is meant to mimic the Lorenz equations.But first, we present some basic concepts related to one-dimensional maps and knead-

ing sequences necessary for analysing these flows.

Let us consider a map, f , which satisfies the following conditions [11]:

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• f : I → I, I = [0, 1],

• f is continuous and differentiable except at c, 0 < c < 1,

• f is monotone increasing on [0, c) ∪ (c, 1],

• limx↑c f(x) = 1, limx↓c f(x) = 0, f(c) = c,

• For every interval J ⊂ I, there is some n such that fn(J) = I. This conditionwe can call “locally eventually onto” (LEO).

Now we define a kneading sequence, k(x), for every point in I. We define this as:

k(x) = s0s1s2s3... where si =

{

0 if x < c

1 if x > c

The sequence ends if f i(x) = c.

We can see that a sequence k can be for at most one point x ∈ I due to the LEOcondition: if x1 6= x2 then for some n, fn(x1) and fn(x2) will be on opposite sides ofc. The ordering of sequences k is the binary ordering with the additional conditionthat the empty space is between 0 and 1. So: 0000... < 00 < 0011... . With thisordering we now have a relationship between k(x) and k(f(x)).

k(x) = s0s1s2s3... and k(f(x)) = s1s2s3s4...

So, generally: k(f(x)) = t0t1t2... where ti = si+1. Remember, we call this the shift

map.

We require that for a sequence to be a kneading sequence for some x ∈ I, k(0) ≤si(k) ≤ k(1). These kneading sequences give us a picture of the behavior of f .Aperiodic/periodic points have infinitely nonrepeating/repeating kneading sequences,whereas points that map to c have finite sequences. This condition also implies thatk(0) ≤ k(1). We define k′(x) as being the sequence derived from k(x) where all 0’shave been replaced by 1’s and vice versa.

Let us define the concept of topological equivalence for maps using kneading se-quences. Two maps f1 and f2 are considered topologically equivalent if and onlyif kf1

(0) = kf2(1), kf2

(0) = kf1(1) or if k′f1

(0) = kf2(0), k′f1

(1) = kf2(0). Let us call

(k(0), k(1)) the kneading invariant for a map f [11].

While these conditions might seem sufficient to define a kneading sequence for x ∈ I,we must think of the events that occur within the Lorenz system, specifically the ex-istence of only some certain homoclinic orbits. Some of those which are not allowedwill satisfy the above conditions. Let us consider a kneading invariant (000, 111).Then (000111, 111000) is a sequence which fits the first conditions. The map f givenby this kneading sequence will not be LEO (this can be seen by explicitly computingthe map [shown in Figure 4] and observing that an interval [A,B] ⊂ I is mapped to

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itself but does not fill I). We present, then, a final condition:

Suppose w1 and w2 are two 0-1 sequences such that w1 begins with 0 and w2 beginswith 1 and both sequences are of length at least two and at least one is finite. Then,k(0) and k(1) cannot be of the forms:

k(0) = w1(0w2)n1 (1w1)

n2 (0w1)

n3 ....

k(1) = w2(0w1)p1(1w2)

p2(0w1)

p3....

where, when w1 and w2 are finite, the sequences, k(0), k(1), are either finite or infinitewith n1, p1 > 0 and, when either w1 or w2 is infinite, k(0), k(1) are both infinite witheither n1 or p1 = 1 and the remaining ni, pi = 0 [14].

We can see that the sequences given in our example do not fulfill this condition(n1 = p1 = 1, n2 = p2 = 0...). It has not yet been proved that these conditions areboth necessary and sufficient, though they are generally accepted as such.

Note that the maps, f , are not invertible, so before we can construct model flows, wemust give a unique “past” to each point. For this, we construct a map f ∈ I suchthat every point x ∈ I consists of points ..., x−3, x−2, x−1 such that f(x−i) = x−i+1.For points x that we choose x−i = 0 or 1, we have a problem, since there is nox ∈ I such that f(x) = 0 or 1. So we say the history will end if x−i = 0 or 1. This iscalled the pinched inverse limit of f [11].

Now we can construct a 2D return map, F , of the unit square onto itself with thefollowing properties [11]:

• F (x, y) 7→ (f(x), g(x, y)).

• F is one-to-one everywhere except on a discontinuity c× I.

• dg

dy< 1

2(i.e there is a contraction in the y direction.)

• As x→ c±, g|x×I tend to constant functions b±.

Since F is contracting, it is logical to assume there is an attracting set Λ.

From this, we can construct a flow by placing the unit square with our map over anon-stable fixed point with 2D stable manifold and 1D unstable manifold, and thenembedding this cell into a vector field which takes trajectories that leave the area ofinterest back into the square as determined by the return map. (We can very roughlysay that this vector field acts as the “tubes” in our “box and tubes” model shownpreviously.)

Finally we present a 3D Lorenz flow. Geometric Lorenz models are vector fields in3D space (as mentioned above) with the following properties [5]:

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• There exists some stationary point O. O has a 1D unstable and a 2D stablemanifold.

• There exists some cross section to this vector field where the return map of thecross section has a contracting foliation.

• The 1D map induced by the factorization along the leaves of the stable foliationis uniformly expanding. This essentially implies that the 1D map of the returnmap will fit the conditions we outlined above.

It was shown that, not only do such vector fields exist and contain strange attractors,but even more so, under a small perturbation which produces a flow Φ′ with a returnmap F ′ and one-dimensional map f ′, Φ′ will also have a Lorenz attractor (that is, F ′

and f ′ will have the properties we have given above) [2].

Geometric models prove a useful tool to globally analyse the properties of the Lorenzattractor, but their relevance to the Lorenz equations could be completely invalidatedif it was shown that there does not exist a contracting foliation in the map of theLorenz equations, and therefore, there also does not exist a strange attractor in theequations. Fortunately, it was shown definitively through a computer-assisted proofthat there does exist a strange attractor in the Lorenz equations for the parametervalues Lorenz originally considered (σ = 10, b = 8

3, r = 28) [13].

9 Period Doubling and Intermittent Chaos

We now move away from geometric models back to the original equations for a lookat behavior for values of r > 28. In numerical simulations, stable periodic orbitsare observed in some intervals: 99.534 < r < 100.795, 145.96 < r < 166.07, and214.364 < r < ∞. In each of these intervals, we see a period doubling cascade, ora series of consecutive period-doubling bifurcations. Numerical simulations outsidethese intervals show more chaotic behavior [1, 9].

First, let us review some basic terminology and give an overview of three commontypes of bifurcations (which we say occur at some r = r∗) which may occur in periodicorbits [6]:

• On one side of r∗ there exists a periodic orbit which, after the crossing of r∗,gives birth to an invariant torus. This is called the Neimark-Sacker bifurcation

and occurs when a complex conjugate pair of eigenvalues of the monodromy

matrix, M , cross the unit circle in the complex plane at a point, |λ1,2| = 1. (Wecan also choose to consider the Poincare return map of the system and look atthe eigenvalues of the fixed point corresponding to the periodic cycle.) Notethat this bifurcation cannot occur in the Lorenz system, as an invariant toruswould not allow for the volume contraction observed in the equations.

• Suppose on one side of r∗ we have two periodic orbits (these can be non-symmetric or symmetric). As r → r∗, the orbits move very closely together

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and after r > r∗, both orbits disappear. This is referred to as the saddle-node

bifurcation. This bifurcation is characterised by an eigenvalue λ = +1 for M .

• On one side of r∗, we have a periodic orbit. As r → r∗, its period approachessome value T . On the other side of r∗, the original orbit continues to exist;however, there is, alongside it, another orbit of period 2T . This is referred toas a period-doubling bifurcation, and corresponds to an eigenvalue λ = −1 forM .

Actually, it is not perfectly accurate to say that we observe chaos in between pe-riod doubling windows. In fact, at the lower boundaries of the windows, such as197.6 < r < 214.364, we observe something called noisy periodicity. If on somereturn plane, we can find n non-overlapping, connected regions U1, U2, U3...Un suchthat all trajectories pass through these regions in a cyclic order, then we say that thesystem is semi-periodic with period n.

At the top of the in-between regions, we observe intermittent chaos. This is defined astrajectories which are “almost” periodic, and then they wander off and act chaoticallyfor some time, before returning to almost periodic behavior again, and so on infinitely.

We show some simulations of stable periodic orbits in the period doubling windows,as well as some trajectories in the in-between times:

0 5 10 15 20 25 30 35 40 45

−20

0

20

40

60

80

100

120

140

160

t

z

Figure 10: Intermittent chaos at r = 100.8.

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0 5 10 15 20 25 30 35 40 45 50100

120

140

160

180

200

220

240

260

280

300

t

z

Figure 11: Noisy periodicity at r = 200.

Figure 12: A stable periodic orbit at r = 400.

10 Summary

Although we have done so once before, we conclude with a summary of how thebehavior of the Lorenz system changes when one parameter, r, is varied:

• When 0 < r < 1, the origin, (0, 0, 0), is globally stable. All trajectories areattracted to it.

• When r = 1, a supercritical pitchfork bifurcation occurs. Two new stableequilibrium points C1 and C2 “appear”, while the origin loses stability.

• When 1 < r < rH ≈ 24.74.., C1,2 are stable fixed points.

• When r = r∗ ≈ 13.926..., a homoclinic bifurcation occurs. The simplest un-stable limit cycles created here are later absorbed by a Hopf point. At thispoint a “strange invariant set” is born. This set contains a countable infinity of

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periodic orbits and an uncountable infinity of aperiodic orbits and trajectorieswhich tend to the origin.

• When r = rA ≈ 24.06..., the invariant set becomes a strange invariant attractor–the “Lorenz attractor”.

• When 24.06.. < r < 24.74.., both stable fixed points (C1,2) and a strangeattractor co-exist. Depending on the initial conditions, the solution may tendtowards the fixed points (be non-chaotic) or enter the attractor (be chaotic).

• When r = 24.74.., a subcritical Hopf bifurcation occurs. C1,2 lose stability asthey absorb unstable limit cycles.

• When 24.74.. < r < 99.524.., the Lorenz attractor exists, and behavior is highlysensitive on initial conditions.

• When 99.524.. < r < 100.795.., the first period doubling window is observed.Stable periodic behavior occurs within this window.

• When 100.795.. < r < 145.96.. and 166.07.. < r < 214.364.., we are betweentwo period doubling windows. Here, again, trajectories are generally aperiodicand wandering. At the lower boundaries of the period doubling windows, noisyperiodicity is observed, while at the upper boundaries, intermittent chaos isobserved.

• When 145.96.. < r < 166.07.. and 214.364.. < r < ∞, the second and thirdperiod doubling windows occur.

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