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Bifurcations in Maps Juan Alejandro Valdivia Hepp [email protected] . . . in dedication to Carolina y Nicolas Contents 1 Concepts in MAPS 3 1.1 Bifurcation diagram .................................... 3 1.2 The attractor ....................................... 3 1.2.1 Lebesgue measure ................................. 6 1.3 Basin of attraction .................................... 6 1.3.1 invariant measure ................................. 7 1.3.2 Path integrals ................................... 8 1.4 Periodic Orbits and Period doubling bifurcation .................... 9 1.5 Renormalization group .................................. 12 1.5.1 General Approach to RGT ............................ 15 1.6 Path integrals ....................................... 17 2 Other situations of interest 17 2.1 Symbolic dynamics .................................... 17 2.2 More than one attractor ................................. 17 3 Higher Dimensions 18 3.1 Map examples ....................................... 21 3.1.1 Hennon Map ................................... 21 3.1.2 prey-predator model ............................... 22 3.1.3 Baker’s Map .................................... 22 3.1.4 Horse shoe map and non-attracting chaotic sets ................ 23 3.1.5 The cat map .................................... 23 3.1.6 Hamiltonian systems ............................... 25 3.1.7 City Traffic .................................... 26 3.2 Stable and unstable Manifolds .............................. 26 1
Transcript
Page 1: Bifurcations Maps

Bifurcations in MapsJuan Alejandro Valdivia [email protected]

. . . in dedication to Carolina y Nicolas

Contents

1 Concepts in MAPS 31.1 Bifurcation diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Lebesgue measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Basin of attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 invariant measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.2 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Periodic Orbits and Period doubling bifurcation . . . . . . . . . . . . . . . . . . . . 91.5 Renormalization group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5.1 General Approach to RGT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.6 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Other situations of interest 172.1 Symbolic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 More than one attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Higher Dimensions 183.1 Map examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 Hennon Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1.2 prey-predator model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.3 Baker’s Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.4 Horse shoe map and non-attracting chaotic sets . . . . . . . . . . . . . . . . 233.1.5 The cat map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.6 Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.7 City Traffic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Stable and unstable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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4 Generic Bifurcations 314.1 Generic Bifurcations 1-D and normal forms . . . . . . . . . . . . . . . . . . . . . . . 314.2 Hysteresis as a non-local bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.3 Non-generic local bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.4 Generic local Bifurcations N-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.5 Global bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.6 Discontinuous transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.6.1 Intermittent transitions to chaos . . . . . . . . . . . . . . . . . . . . . . . . . 374.6.2 Interior and exterior crisis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

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1 Concepts in MAPS

There are a lot of models in physics that are based in discrete maps,

xn+1 = M [xn, p]

with p as a controlling parameter. Similarly, there is a straight forward way to describe a set ofdifferential equations with discrete maps by using the Poincare surface of section.

For a given initial condition x0 the limit of xn as n → ∞ is described as the ω-set of x. Inmany situations characterizing the behavior of this system involves determining the possible limitsets (or asymptotic sets) to which the dynamics can evolve.

1.1 Bifurcation diagram

By changing the parameters p we can observe different behavior through a Bifurcation diagraminitiated at x0 that display the behavior of the limit set of a given initial condition as a functionof the parameter. In general, since in many situations this approach is not uniformly continuous,interesting things can happen such as discontinuous transitions in the attractor features (size,disappearance, etc).

One of the most studied equations is the logistic equation that represents the reproduction of acertain specie for a given fixed carrying capacity of the food source

xn+1 = rxn(1− xn)

with r as the controlling parameter. As we will see this system shows chaotic behavior withsensitivity to initial conditions. First, notice that the points in (∞, 0)U(1, +∞) map to the negativevalues for r > 0 and to −∞ for (r > 1). The set [0, 1] maps to itself for (r > 0). We will see thatthe range of interest (for 1 ≤ r ≤ 4) is for x in [0, 1].

The asymptotic state of the system can be computed numerically by starting with an initialcondition x0 = 0.1 and after removing the transient (say n < 1000) we plot for a given value ofr, the following iterations of the map (say n < 2000). The bifurcation diagram is constructed byrepeating the same procedure for different r. For the case in Fig. 1a-b we can see that the limitset for the same initial conditions converges to a fixed point of period one and two respectively.Figures 1e-f show the bifurcation diagram, and we see in Fig. 1d that the trajectory seems tobecome chaotic for r ∼ 3.8. This is called a period doubling bifurcation route to chaos.

1.2 The attractor

An attractor A intuitively is a set to which all neighborhood trajectories converge. In general forthe case of maps these sets are (a) fixed points, and (b) strange attractors. In the case of differentialequations, we will see that we can also have limit cycles.

Mathematically, A is the minimal (no smaller set satisfy these conditions) closed invariant set(if x0 ∈ A xn ∈ A ∀n > 0) that attracts an open set U of initial conditions. The largest set Uis the basin of attraction of A.

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Figure 1: (a) The logistic map for r=2.5 and the zig-zig line represent the trajectory of the x0 = 0.1.(b) The same but for r = 3.2 representing a orbit of period 2. (c) The trajectory of the map f 2 forr = 3.2. (d) The logistic map for r = 3.5 a chaotic case. (e) The bifurcation diagram. (e)A zoomof the bifurcation diagram.

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Figure 2: (a) The bifurcation diagram

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Numerically, we start with a set of initial conditions and see numerically whether they convergesto a fixed point for a given number of iterations. If there is a set of points that do not convergeto the already observed fixed points, we initially assume that we have a strange attractor SA. Ofcourse this is not necessarily true, for we may have a nonatracting set and those points just take along time to go to one of the previous attractors, so we must be sure that the number of points thatstay in the SA does not goes down in time. Different attractors have different basin of attractors.

Strange attractors are attractors in which there is sensitive dependence on initial conditions.Sometimes are referred as fractal attractors or chaotic attractors.

In Fig. 1 the attractor is obtained from the initial condition x0 = 0.1, but in principle therecould be more attractors than the one described in the bifurcation diagram, since in principle westarted with one initial condition, and other initial conditions may converge to other attractors.In the bifurcation diagram displayed above, we see two possible attractors, the one at −∞ andanother one in [0, 1] for r > 0.

Numerically, in the case of multiple attractors we may have to take an ensemble of initialconditions and construct a bifurcation diagram for each. If the attractors are well separated, thenthe bifurcations diagram can be superposed, but if the attractors are somehow interwoven, then itmay be more difficult to separate.

1.2.1 Lebesgue measure

For the case shown in Fig. 1, almost all (in the Lebesgue sense) initial conditions converge tothe attractor shown. The concept of the Lebesgue is important and necessary, since it is obviousthat if we start with an initial condition at an unstable periodic orbit, then the trajectory does notevolve towards the attractor. But this set is countable and has Lebesgue measure zero, and doesnot affect the statement above.

HOMEWORK PROBLEM: show numerically that almost all initial conditions con-verge to the same attractor

1.3 Basin of attraction

The closure (see comment on Lebesgue measure below) of the set of initial conditions that convergeto a given attractor is defined as its basin of attraction. In the case of the logistic map, we havethe trivial attractor at −∞ with basin [∞, 0]U [1, +∞]. The other attractor is in [0,1] with basingof attraction [0, 1].

Numerically we start with an ensemble of initial conditions and follow their evolution. Inprinciple some initial conditions will converge to a given set of fixed points (i.e., trajectories thatrepeat themselves after some period. If after a certain number of iterations, the trajectory doesnot converge to a fixed point, we define it initially as part of a strange attractor. The set of initialconditions that don’t converge to a fixed point form the strange attractor. First, there may bemore that one, and second, care must be taken that the set is not a non-attracting chaotic set byobserving if the number of points in the strange attractors does not go down in time.

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1.3.1 invariant measure

It is possible to assign a “weight”, or density (in the Lebesgue sense), to the points in phase space.Suppose we assign a certain initial density ρ0(x) to each point in phase space, with∫

ρn(x)dx = 1

then we can evolve that density using

ρn+1(x) =

∫ρn(y)δ(x−M [y])dy =

∑i

ρn(y(i))

|M ′(y(i))|(1)

where M(y(i)) = x. An invariant density is one that satisfies ρn+1(x) = ρn(x) = ρ(x). Inpractice ρ(x) is usually a very discontinuous function, and in many situations is very difficult todefine. For that it is more customary to define a measure µ. The measure of a set S is describedby µ(s) =

∫S

ρ(x)dx and the average of a function by∫f(x)ρ(x)dx =

∫f(x)dµ(x)

The measure µ is smoother that the density ρ, so it is more manageable. It is possible toestimate the relevance of each attractor in phase space by measuring the measure of its basin ofattraction (starting with ρ0(x) = 1 in phase space).

0 0.2 0.4 0.6 0.8 1x

0.0001

0.001

0.01

0.1

0 0.2 0.4 0.6 0.8 1x

0.0001

0.001

0.01

0.1

Figure 3: The analytical density for r=3.25 and r=3.8 using Eq. 1 for n = 1 (red line), n = 5 (blueline), and n = 10 (green line), assuming ρ0 = 1. The black line corresponds to the the densityconstructed in time after n = 106 iterations using x0 = 0.1. It is possible to show that this is indeedthe natural invariant density.

Notice, that we have looked from the point of view of phase space, we will now look from thepoint of view of time. Let’s start with an initial condition x0 = 0.1 in the basin of attraction ofan attractor, and compute the fraction of time the trajectory spends in an interval S, denoted byµ(S, xo). In Fig. 3 we display the density ρ(x, x0) that is generated by the fraction of time thetrajectory spends around x. If µ(S, xo) is the same for almost all the points in x0 in the basin ofattraction then we denote it by the natural measure µ(S). For smooth 1-D maps natural measuresusually exists, but in higher dimensional systems is an open question. This implies that the measureis ergodic since it satisfies (only in the basin of attraction B)

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< f >t= limN→∞

1

N

N∑n=0

f(Mn[x0]) =

∫B

f(x)ρ(x)dx =< f >x

for any function f(x) and almost any point x0εB.

1.3.2 Path integrals

Suppose we start with an initial density ρ0(x0) in phase space (essentially it accounts for the factthat we cannot determine our initial conditions with infinite precision). Then starting from

sn+1 = M(sn) + ηn

with ηn as a random noise, then the probability of a particular sequence [x0, x1, x2, . . . , xn−1] canbe written as

P [x0, x1, . . . , xn] = 〈δ[x1 − s1] . . . δ[xn − sn]〉

from an initial condition x0, where the average is over the noise sequence ηn . Using

δ[x] =1

∫dkeikx

we can rewrite

P [x0, x1, . . . , xn] =(

12π

)n ∫[Πn

m=1dkm]⟨ei

Pnm=1 km(xm−sm)

⟩=

(12π

)n ∫[Πn

m=1dkm]⟨ei

Pnm=1 km(xm−M [xm−1]−ηm−1)

⟩given that xm = M(xm−1) + ηm−1 and xm−1 = sm−1.

Usually the differential DK = Πnm=1dkm describe all possible paths connecting this specific

sequence from x0 and xn. It is interesting to compare this formalism with the propagator inquantum mechanics. We see that ρn(xn) is just the phase space average of this probability over allintermediate sequences [x0, x1, x2, . . . , xn−1] with initial probability ρ0(x0).

The probability that connects an initial condition x0 with a final condition xn is

P [xn] =

∫Πn−1

m=0ρ0(x0)P [x0, x1, . . . , xn]

and the phase space density at the nth step is then

ρn(xn) =1

Zn

∫dx0ρ0(x0)

[Πn−1

m=1dxmdkm

]dkn

⟨ei

Pnm=1 km(xm−M [xm−1]−ηm−1)

⟩where we have defined the partition function at the nth step as

Zn =

∫[Πn

m=1dxmdkm] ρ0(x0)⟨ei

Pnm=1 km(xm−M [xm−1]−ηm−1)

⟩This partition function can be use to describe any system, even away from equilibrium. An equi-librium is reached if an invariant density is obtained in some basin. From the partition function,

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using functional derivation, dynamical variables for the system can be obtained. But that is a topicfor another book.

The probability of a phase space variable of function as a function of time can be calculatednow as

< f >n=

∫f(x)ρn(x)dx

Let’s note that this equations can be derived from Eq. 1 using

δ(x−M [yn]) =

∫eikn(x−M [yn])dkn

This path integral formalism is useful when the noise distribution is know, so that⟨eikn+1(xn+1−M(xn)−ηn)

⟩can be evaluated. For example

P [η] =1

πσ2e−

„η2

σ2

«→

⟨eb+aη

⟩= e

b−“

aσ2

4

HOMEWORK PROBLEM: For the logistic map for r = 3.8 start with ρ0(x) = 1, whathappens to ρn as n → ∞?. Is this an invariant measure? Is it equal to the invariantnatural measure?

HOMEWORK PROBLEM: Evolve ρn with ρ0(x) = 1 using Eq. 1 analytically

HOMEWORK PROBLEM: For the map zn+1 = z2n +(0.341, 0.341) construct the basin of

attraction of infinity in the complex plane. The basin boundary should look fractal.Estimate the dimension of the basin boundary using boxes of size ε. Help: the numberof boxes that cover the basin boundary diverges as N ∼ ε−D as ε → 0 (see below).

HOMEWORK PROBLEM: For the identity map, and a Gaussian independent noise,calculate the partition function and the probability density for a few n. Repeat thesame analysis for the logistic map for a few values of r and n

1.4 Periodic Orbits and Period doubling bifurcation

As seen in the figures above, the dynamics seems to be controlled to a certain extent by the stabilityproperties of the periodic orbits of the map. The periodic orbits of period m can be calculatedfrom the fixed points of Mm given by

Mm(x∗) = x∗

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Let’s take the logistic map and study the details of the bifurcation diagram. The period doublingbifurcation to chaos seems to be a continuous bifurcation. In the bifurcation diagram we observewindows of periodicity inside the chaotic range, where the transition to the periodic windows seemsto occur discontinuously. These are the discontinuous transitions mentioned above and that willbe studied in detail below.

0.5 1 1.5 2 2.5 3 3.5 4r

0.2

0.4

0.6

0.8

1x

0.5 1 1.5 2 2.5 3 3.5 4r

0.2

0.4

0.6

0.8

1x

3.2 3.4 3.6 3.8 4r

0.2

0.4

0.6

0.8

1x

3.82 3.84 3.86 3.88 3.9r

0.2

0.4

0.6

0.8

1x

Figure 4: The stability of the period (a) m=1, (b) m=4, (c) zoom of m=4 and (d) m=3 orbits.

The period m = 1 orbits are defined by the solutions of

M(x∗) = x∗ →

x

(1)0 = 0

x(2)0 = 1− 1

r

and we still need to clarify the stability of them. The stability of a periodic m = 1 orbit x∗ can beconstructed from all possible perturbations xn = x∗ + δn, then

xn+1 = x∗ + δn+1 = M(x∗ + δn) ≈ M(x∗) +dM(x∗)

dxδn

δn+1 ≈dM(x∗)

dxδn

therefore, the periodic orbit is attractive if |δn+1| < |δn|. Furthermore, the trajectory will convergeat a rate

δn ∼ eλnδ0

with

λ ≈ ln|d(M(x∗)

dx|

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In this particular case, we have

λ0 = →

λ

(1)0 = ln|r|

λ(2)0 = ln|2− r|

respectively. Therefore, the x1 is an attractive fixed point for r < 1 while x2 is an attractive fixedpoint for 1 < r < 3. Otherwise they are repulsive.

The stability of a periodic m orbit x0, . . . , xm−1 can be constructed from all possible perturba-tions xn = x∗ + δn, then

xn+m = x∗ + δn+m = Mm(x∗ + δn) ≈ Mm(x∗) +d(Mm)

dxδn

δn+m ≈ d(Mm)

dxδn =

d(M(x0)

dx...

d(M(xm−1)

dxδn

therefore, the periodic orbit is attractive if the product in absolute value is less than one. Further-more, the trajectory will converge at a rate

δn ∼ eλnδ0

with

λ ≈ 1

m

m−1∑n=0

ln|d(M(xn)

dx|

Figure 4 shows the behavior of a few of these orbits. We see that in the period doublingbifurcation at the point at which the periodic m-orbit looses stability, an stable and unstable period2m-orbit appears. Notice that the other 2m-orbits are still there but are unstable. Obviously theM2m also describe the Mm orbits as it should. This is a very important point. Take notice.

It is interesting to note that we could define the stability criteria of a m →∞ periodic orbit, i.e.,something that is non-periodic. In essence all the periodic orbits are there, but are unstable. Thechaotic attractor include the set of all unstable periodic orbits. We define the Lyapunov exponentas the infinite limit

λ ≈ LimN→∞1

N

N∑n=0

ln|d(M(xn)

dx|

and due to the ergodicity of the system, it can also be computed from

λ ≈∫

ln|d(M(x)

dx|dµ(x) =

∫ln|r(1− 2x)|dµ(x)

If the orbit is periodic, we naturally obtain the previous result for the stability of the orbit. Figure5 shows the Lyapunov exponent for the logistic map.

Note that in the period m=4, the other period 4 orbit becomes stable at r a little less that r,but through a bifurcation that has a different nature from the period doubling. We will study thesebifurcations below.

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HOMEWORK PROBLEM: repeat the whole analysis for the tent map xn+1 = r ∗ (0.5−|x− 0.5|), including attractors, bifurcation diagrams, Lyapunov exponents.

HOMEWORK PROBLEM: repeat the whole analysis for the tent map xn+1 = r∗(1−x2),including attractors, bifurcation diagrams, Lyapunov exponents.

HOMEWORK PROBLEM:For the logistic map show that the two methods for com-puting the Lyapunov exponents give the same results, hence the system is ergodic.

1.5 Renormalization group

From the map of the logistic map we can characterize the universal behavior of the period doublingbifurcation. This universal behavior occurs for all maps with a single quadratic maximum. Let’sdo the logistic map and then we can generalize. From the Fig 6 we can observe that the m and(m + 1) period doubling bifurcations look the same except for a zooming factor and a shifting.

Let’s define the m-super-stable fixed point at rm when dM2m(x∗, rm)/dx = 0. It is found that

δm =rm − rm−1

rm+1 − rm

→ δ ∼= 4.669201 . . .

is universal for all maps with a single quadratic maximum. This implies that |rm − rc| ∼ δ−ν .Numerically it is found that rc

∼= 3.57 . . .. There are other universal quantities. When the stablefixed point of Mm(x∗) become super-stable (dMm(x∗, rm)/dx = 0), we can estimate the value

∆m = M2m−1

(1

2, rm

)− 1

2

αm = − ∆m

∆m+1

→ α ∼= 2.50280 . . .

We will use a renormalization group transformation (RGT) to estimate these values (seeFig. 7). The RGT is a coarse graining transformation that in essence transform the map M2m

(x)to the map M2m+1

(x). Notice from Fig. 6 that the map close to the supercritical fixed points of M2

at r2 looks exactly the same as the map M at r1, except for a coordinate transformation (actuallya flip and a zooming) which can be described by

y − 1

2= α

(x− 1

2

)and we expect α to be larger than one and negative. We therefore, expect

xn+2 = M2(xn, rm+1)yn+1 = M(yn, rm)

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Figure 5: (a) The Lyapunov exponent of the logistic map.

0.2 0.4 0.6 0.8 1x

0.2

0.4

0.6

0.8

1f r1=2

0.2 0.4 0.6 0.8 1x

0.2

0.4

0.6

0.8

1f r1=2

Figure 6: (a) The super-stable fixed points for M and M2

α0

δ0

δ−1

α−1

Figure 7: (a) Grafical description of the parameters

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which can be written in the general case as

M [y, rn]− 1

2= α

(M2

[1

2+

(y − 1

2

, rn+1

]− 1

2

)The can construct the transformation rm = R(rm+1) and α = G(rm+1) which are called the RGTwhich do the zooming and shifting, and permits to transform from the rm to the rm+1 super-stablepoints down the period doubling bifurcation. This transform from the m period doubling to them + 1 period doubling, and following this way we approach to chaos at m → ∞. This coarsegraining transformation is enough to determine the behavior of the period doubling bifurcation.

In general is usually very difficult to carry a RGT exactly, which means that we have to resortto some approximation. Notice that this transformation only makes sense close to the super-stablefixed points which permits us to do the analysis close to x ∼ 1/2. We rewrite the logistic map as

M(x, r) = r

[1

4−(

x− 1

2

)2]

(2)

It is left to the reader to work out the transformation of other similar maps proving that the resultsare in essence universal. We then have using Eq. 2 that

M2(x, r) =r

4−(

r

4− 1

2

)2

+ 2r2

(r

4− 1

2

)(x− 1

2

)2

+ O(

(x− 1

2

)4

)

Therefore, following the above prescription we have obtained 3 solutions for rm = R(rm+1) andα = G(rm). These solutions give a nontrivial fixed point (of the Rr transformation) only in onecase, namely

R(r) = 1 +1

8

√64 + 8r2(r − 2)(8− 4r2 + r2)

where the fixed point of this new transformation corresponds to the saturation point rc = 1 +√3 + 2

√3 ∼= 3.54 . . . which is close to the numerical value. This values is not universal, and

depend on the particular system. At the same time, α = G(rc) → −2.732.. corresponds to theFeigenbaum number.

Now, let’s estimate δ. First notice that close to the fixed point of the RGT (i.e., close to thesaturation point of the period doubling bifurcation) the variable εm = rm − rc represent the onlyrelevant variable, and we can conclude that

εm = R(rm+1)− rc = R(εm+1 + rc)− rc∼= (4 +

√3)εm+1 + O(ε2

m+1)

hence δm → (4+√

3) = 5.73, which is ok due to the approximation we made initially. Furthermore,let’s define N(ε) = N(r − rc) as the number of period doubling at r (notice that N depends onlyon r − rc). Clearly, we have

N(εm) = 2N(εm−1) = 2N(δεm) = 2nN(δnεm)

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from this relationship we estimate N since it applies for all n. Then chose an n, such that δnεm ∼ 1(n = −ln2εm/ln2δ), then

N(εm) = 2nN(1) = 2−ln22/ln2δ → N(r) ∼ 1

|r − rc|ν

with ν = 1/ln2δ = 0.396 . . ..

1.5.1 General Approach to RGT

We discuss rapidly the general RGT, which has a much more general applicability.The renormalization group (RG) approach is a method to describe the behavior of scaling in

a system that has some inherent symmetry. Furthermore, within the RG approach the presenceof scaling usually represents some kind of critical behavior as we will see below. The RG methodusually starts with a set of equations, that can include dynamical equations,

f(x, a)

described by a set of variables x and a set of parameters a. We make explicit use of the symmetryto make a scaling transformation that transform the variables and parameters as

x′ ∼= Rx(x)a′ ∼= Ra(a)

so that the equation is left (almost) invariant

f(x, a) → ≈ f(x′, a′)

The interesting point, is that the scaling properties are completely described by the transfor-mation equation of the parameters, as it should be, and the critical behavior can be completelycharacterized.

Suppose that the transformation a′ = Ra(a) is repeated many times, then the system willapproach a fixed point in the parameter space defined by

ac = Ra(ac)

If the initial system start with those parameters it becomes invariant under those transforma-tions and we say that the system is at the critical state. Of course there are trivial fixed pointslike at infinity or zero. But in general if there is critical behavior there will be finite critical points(or fixed points). Furthermore, such a fixed point in general have stable and unstable manifolds.The stable manifold described the criticality of such a system, which means that a system onthe stable manifold approaches the critical point under the RG transformation. In general, manycritical points will describe different critical or scaling behaviors of the system.

Around a fixed point ac we can linearize the transformation Ra → La to characterize thestable (λi < 0) and unstable (λi > 0) manifolds by the respective eigenvectors Vi with eigenvalueλi. Since any a =

∑aiVithen we note that there are only a few relevant directions described by

the dimension of the unstable manifold.

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Any physical “macroscopic” parameter of the system P(a) can now be described in this scalingregion by the behavior of the relevant directions, i.e., (like generalized coordinates)

P (a) ∼= P (V1, V2, ...VN)

with N as the dimension of the unstable manifold. It is important to stress that this physicalvariable transform under the RG transformation and it depends on the system under considerationand the type of the transformation. In equilibrium statistical mechanics we are usually interestedrescaling the variables by a factor β with the correlation length ζ and the free energy F transformingas

F (V1, V2, ...VN) ∼ β−ndF (V1λn1 , V2λ

n2 , ...VNλn

N)ξ(V1, V2, ...VN) ∼ βnξ(V1λ

n1 , V2λ

n2 , ...VNλn

N)(3)

since F is an extensive variable (Careful if we have non-extensive entropy) and ζ clearly decreases aswe do the transformation of the d variables in x. Both F and ζgre invariant under the transformationand the transformed variables x represent a smaller system. From this relation we can get rid ofone variable and obtain the scaling relations since n is arbitrary, i.e.,

V1λn1 = 1 → n = − ln V1

ln λ1

F (V1, V2, ...VN) ∼ Vd ln β

ln λ11 F (1, V2

V

ln λ2ln λ1

1

, ... VN

V

ln λNln λ1

1

)

ξ(V1, V2, ...VN) ∼ V− ln β

ln λ11 ξ(1, V2

V

ln λ2ln λ1

1

, ... VN

V

ln λNln λ1

1

)

which reduces to the scaling since what it inside the function are all constant but in general describemacroscopic variables like temperature T and magnetic fields H as in the case of spin systems.

Figure 8: The RG transformation in parameter space

In Fig. 8 we show the RG transformation in parameter space, with multiple fixed points. Usuallysymmetry breaking occurs as we change from the neighborhood of one fixed point to another.

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1.6 Path integrals

It is interesting to note that this RG approach can be done directly using the path integral formalismdescribed above for P [x0, x1, . . . , xn] by

• integrating every other term in the sequence

• rewriting the result as it look before the summation, but with a renormalized function for r.

2 Other situations of interest

2.1 Symbolic dynamics

Take the map

xn+1 = 2xn mod2

we can define a point by its binary representation

x0 = 0.a1a2a3 . . . =∞∑

j=1

2−jaj

with ai = 0, 1 which applied to the map gives

x1 = 0.a2a3 . . . xn = 0.an+1an+2 . . .

and it is called a binary shifting. It is then very easy to study periodic orbits in this system. Noticethat the fixed points are all unstable since λ = ln2 and in fact they are dense in the interval [0, 1].We then clearly can find that the natural invariant density must be

ρ(x) = 1 xε[0, 1]

since the unstable periodic orbits have measure zero.

2.2 More than one attractor

In many situations we have more than one attractor. Let’s study the map shown in Fig. 9c.This is constructed in Mathematica by connecting points, namely connecting crossing points atx = 0, 1/5, 2/5, 3/5, 4/5, 1 with lines. There are two clear attractors, namely x = 2 and x = −1,which are attractive. The basin of attraction for each attractor is clearly fractal defining somethingsimilar to a cantor set with finite structure at all scales. The basin of attraction is not a cantor set,for it has dimension 1/2. But the basin boundary has a fractional dimension.

HOMEWORK PROBLEM: Estimate δ and α and ν and rc numerically for the logisticmap.

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-1 1 2

-1

-0.5

0.5

1

1.5

2

0.2 0.4 0.6 0.8 1x

0.2

0.4

0.6

0.8

1

Figure 9: (a) The map and (b) and fractal basin boundary where the color determines whichattractor it belongs to.

HOMEWORK PROBLEM: Repeat the RGT for the logistic map, but by mappingM2 → M4, and show you obtain similar results.

HOMEWORK PROBLEM: Estimate δ and α and ν and rc numerically for the mapr(1− x2) and check the universality.

HOMEWORK PROBLEM: word out the RGT for the map M = r ∗ (1− x2) and find δand α. What are the values of δ for the map r ∗ (1− x2p)?

HOMEWORK PROBLEM: Try to do the RGT for the logistic map but using the pathintegral approach

HOMEWORK PROBLEM: Estimate the dimension of the fractal basin boundary ofthe map shown in Fig. 9

HOMEWORK PROBLEM: By mapping the logistic map for r = 4 to the map usedfor symbolic dynamics, construct the density ρ(x) in analytical form.

HOMEWORK PROBLEM: Repeat Fig 9 but including also the time it takes to ap-proach the attractor. Construct the probability function for the time it takes toapproach the attractor. Calculate the dimension of the boundary of the basin.

3 Higher Dimensions

The stability of a fixed point x∗ can be constructed from all possible perturbations xn = x∗ + δn,then

xn+1 = x∗ + δn+1 = M(x∗ + δn) ≈ M(x∗) + D(M)δn

δn+1 ≈ DM(x∗)δn

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therefore, the fixed point is attractive if all the eigenvalues |wi| < 1. Notice that in this casethe eigenvalues can have an imaginary value, and therefore the stability criteria must be that theabsolute value must be less than one, in amplitude. For imaginary eigenvalues the orbits spiral inor out of the fixed point in the direction of the stable or unstable manifold perpendicular to thetwo eigendirections corresponding to the complex eigenvalues. In the case of continuous systems,it becomes extremely relevant the fact that we have complex eigenvalues, for it usually signal theappearance of a quasiperiodic orbit when the fixed point losses stability.

We can see that points that start close-by, begin separating at the rate of the largest eigenvalue.

δn ∼ wni δ0 = eλnδ0

If we are interested in the looking at the stability of a fixed point, we are only required to followany perturbation, which will eventually converge to the most unstable direction, or largest λ.

The stability of a periodic m-orbit x0, . . . , xm−1 can be constructed from all possible perturba-tions xn = x∗ + δn, then

xn+m = x∗ + δn+m = Mm(x∗ + δn) ≈ Mm(x∗) + D(Mm)δn

δn+m ≈ D(Mm(xn))δn = DM(xm−1)..DM(x0)δn

therefore, the periodic orbit is attractive if all the eigenvalues |wi| < 1. Notice that λ includes thefactor m−1.

It is interesting to note that we could define the stability criteria of a m → ∞ periodic orbit,i.e., something that is non-periodic. Again we could start with a perturbation in any direction andthe evolution will take it towards the most unstable direction. But if we are interested in computingthe whole spectrum of Lyapunov exponents, as they are called, we have to resolve to a trick sincediagonalization of a large number of matrices is very unstable numerically. Also the stable andunstable directions are changing as the trajectory progresses. Instead we do a QR decomposition

DM(xn)Qn−1 = QnRn

with Q0 = I, Q as an orthogonal matrix, and R as an upper triangular matrix. This impliesnumerically

D(Mn) = QnRnRn−1 . . . R1

hence

λi = limn→∞1

N

N∑n

ln|(Rn)i,i|

because the Rn are upper triangular matrices. In essence, the QR decomposition accounts for therotation of eigendirections and the numerical instability. Note that in the case of a periodic orbit,this Lyapunov spectrum is exactly equal to the one computed above due to the periodicity in xn.Note also that the convergence of the above spectrum is guaranteed by the Ergodic theorem forvery general systems.

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It is very easy to discover that the QR decomposition is in general correcting for the rotation ofthe manifolds. For example, let’s take a 2-D system and we evolve two unitary δ vectors that areinitially perpendicular. Eventually these two vectors become v1 and v2, which are not perpendicular,and have grown in size if one of the Lyapunov exponents is positive. At this point it is convenientto renormalize to two unitary vectors that are perpendicular to each other, namely,

v1 = v1

|v1|v2 = v2−v2·v1v1

|v2−v2·v1v1|

or

v1 = |v1|v1 + 0v2

v2 = v2·v1

|v1| v1 +∣∣∣v2 − (v2·v1)v1

|v1|2

∣∣∣ v2

The transformation can be rewritten in matrix form as

(v1, v2) = (v1, v2)T

[|v1| v2·v1

|v1|

0∣∣∣v2 − (v2·v1)v1

|v1|2

∣∣∣]

which is precisely what we have above DM(xn)Qn−1 = QnRn. We see that |v1| corresponds to the

maximum Lyapunov exponents, and that∣∣∣v2 − v2·v1

|v1|2

∣∣∣ corresponds to the 2nd Lyapunov exponents.

For the general case, the QR decomposition is precisely this gram-Schmitt diagonalization in ndimensions.

We could also construct the maximum Lyapunov exponents numerically, by following 2 trajec-tories v1 = xn and v2 = xn + δn separated by a small initial condition δ0. But we now that astime increases, the separation of the two trajectories also increases up to the saturation size of theattractor. Hence, every m steps we must re-normalize the separation of the two trajectories to

v2 = xm + δ0δm

|δm|→ αi =

|δm|δ0

and continue integrating v1 and v2. The Lyapunov exponents is then the average of this quantity

λ1 =1

Nm

N∑i=1

log αi

If we want to compute the 2nd Lyapunov exponents this way, then we must follow 3 trajectoriesv1, v2, v3 such that the separation between v1 and v2 and the separation between v1 and v3 areperpendicular. Every m steps, we 1st Lyapunov exponent is obtained the same manner as abovefrom v1 and v2. The 2nd Lyapunov exponent is obtained by looking at the growth of the projectionof the vector v3 − v1 in the direction perpendicular to v2 − v1. This is equivalent to looking at thesize of the area Am spanned by the vectors v2− v1 and v3− v1 (see Fig. 10) which is related to the2nd Lyapunov exponent by

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λ1 + λ2 =1

Nm

N∑i=1

log Am

But this is precisely the QR decomposition of the vectors v2− v1 and v3− v1. The extension to theother exponents and more dimensions is now straight forward. For example the renormalization ofvolumes Vm give

λ1 + λ2 + λ3 =1

Nm

N∑i=1

log Vm

and so on.To do this in a general manner, the vector product and the determinant of a matrix may be usefulfor calculations here.

Figure 10: Evolution of a small volume in 2-D space

Note that these formulations allow us to construct locally (at each point in phase space) stableand unstable directions with local Lyapunov exponents. It is interesting to note that each localLyapunov exponent does not in principle has to converge to the global exponent for two reasons:(a) the system may not be ergodic, (b) the orientation of the stable-unstable directions changesover the attractor.

3.1 Map examples

Notice that the attractor usually has a volume (or a Lebesgue measure) smaller than its basinof attraction, this gives rise to the concept of fractals, with dimension smaller that the originalvolume.

3.1.1 Hennon Map

A typical 2-D map is the Hennon map, given by

xn+1 = A− x2n + Byn

yn+1 = xn

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Note that this map is in general dissipative for a range of parameters, hence there must be somenegative Lyapunov exponents. We can visualize this map with a bifurcation diagram as shown inFig. 11. Even though the attractor seems to be broken (composed of many parts), this is an effectof the numerical procedure. We could take a square of initial conditions and iterate forward intime, and because the map is continuous, the attractor should be connected. In essence, we cansee that the orientation of the stable and unstable directions changes along the attractor (in phasespace) forcing the system to have the shape shown in the picture (with one stable and one unstabledirection).

Figure 11: (a) A bifurcation diagram for the Hennon map for B = 0.3. (b) The attractor forA = 1.4.

3.1.2 prey-predator model

Let’s take a prey-predator model using a generalization of the logistic map

xn+1 = raxn(1− xn)(1− Aayn)yn+1 = rbyn(1− yn)(1− Ab(1− x))

with the obvious parameters. This map will be useful later on.A variant of the predator-pray model is to take the logistic map but with r changing from

period to period. For example, rn = A and rn+1 = B (period two situation). The results areshown in Fig. 12 for the Lyapunov exponent in the A−B parameter space. We see that by givingdynamics (period 2) to the parameter in a system we can change considerably the result. This iswhy predator-pray models are so interesting.

3.1.3 Baker’s Map

Another well know map is the Baker map, namely

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3 3.2 3.4 3.6 3.8 4a

3

3.2

3.4

3.6

3.8

4

b

Figure 12: The Lyapunov exponent of the variant of the logistic map in the A−B space.

xn+1 =

λaxn yn < α

(1− λb) + λbxn yn > α

yn+1 =

yn/α yn < α

(yn − α)/β yn > α

with β + α = 1 and λa + λb ≤ 1. This is dissipative system that is very useful to study.

3.1.4 Horse shoe map and non-attracting chaotic sets

The complex horse shoe dynamics are useful to study the chaotic dynamics in non-attractingsets, usually with the use of a symbolic description (see an example above). We show in Fig.13 one iteration of the map, which corresponds to the set M(S) = V1 ∪ V2. The inverse setM−1[M(Vi] = Hi is also shown. In the same manner we can construct the sets Mm(S) (verticallines) and M−m[Mm(S)] (horizontal lines). If we continue the iterating the map, we can clearlysee that the set that stays forever has measure zero. But there is an invariant set Λ that staysforever, and this set is a chaotic non-attracting set. This set can be obtained by the successiveintersections of

∩∞m=1M−m[Mm(S)] ∩ ∩∞m=1M

m(S)

Therefore, we can have chaos in sets that are not attractors.

3.1.5 The cat map

Sometimes it is useful to work with the following map(xn+1

yn+1

)=

(1 11 2

)mod 1

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V1V2

H 2

H 1

H 2

H 1

V1V2

Figure 13: (a) Horse shoe map, and its inverse. (b) The sequence to construct the invariant set.(c) The equivalent dynamics by extending stable and unstable manifold in a Homoclinic Tangency.

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3.1.6 Hamiltonian systems

In a conservative map we have volume preservation, which requires that the Jacobian

Det[DF [x]] = 1

If the map is volume preserving, it does not have an attractor since volume cannot decrease inphase space. Hence the sum of the Lyapunov exponents must be zero,

∑i λi = 0.

An example is a discretization of a Hamiltonian system with a Poincare surface of section(or stroboscopic map). Hamiltonian systems have also the symplectic property that forces theLyapunov exponents to occur in pairs that cancel each other. A good example is the circle map(Fig. 14) generated by a kicked pendulum without gravity generated by the Hamiltonian system.

H =p2

2+∑

n

εCosθδ(t− nT ) → pn+1 = pn + εSinθn+1

θn+1 = θn + pn ∗ T

Figure 14: (a) The pendulum of the kicked rotator.

We can find the fixed points of the M map

p = 2nπθ = mπ

→ λ± =1

2

(2 + ε(−1)m+2n ±

√ε√

ε + 4(−1)m+2n)

hence in the range 0 < ε < 4 we have an elliptic point at θ = nπ and a hyperbolic fixed point atθ = 0. For ε > 4 the elliptic point becomes an hyperbolic point. Because of the elliptic nature of afixed point, the orbits must spend a long time close to it, generating what is called an island. It islike an attractor, but since there are not attractors, |λ| = 1, the trajectory eventually must moveon. This is shown in Fig. 15b, where this particular island is clearly defined. Note that for ε = 2it is still alive. Therefore, an orbit spends a long time close to these islands, with sporadic motionbetween the islands. People have develop theories based on fractal integrals and derivatives for thedynamics on these islands.

Hamiltonian systems will be studied in more detail later on.

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3.1.7 City Traffic

A single car moving through a sequence of N traffic lights. The velocity dynamics between trafficlights is

dv

dt=

[a+θ(vmax − v) accelerate−a−θ(v) brake

The states between traffic lights are:

• accelerate with a+ until reaching cruising speed vmax

• constant cruising speed vmax until decision point xn = Ln − v2max

2a−

sin(ωntn − φn) =

[> 0 green< 0 red

• a− braking while red

It generates a 2D map from light to light (Tn+1, vn+1) = M [Tn, vn]. Sometimes, it is possible tonormalize with respect to cruising time Tc = L

vmax

A+ =a+L

v2max

= 10 A− =a−L

v2max

= 30 Ω = ωTc Ω = Ω/2π

3.2 Stable and unstable Manifolds

For a fixed point, the eigendirections Ei defined by DM(x∗) define a tangent space Tx∗ at x∗. Thistangent space can be decomposed in stable E− (|wi| < 1), unstable E+ (|wi| > 1) and centeredE0 (|wi| = 1) subspaces or directions. If we continue the stable directions we obtain the stablemanifold W s(x∗) of the fixed point x∗

W s(x∗) = xεU |Limn→∞Mn(x) → x∗, with xnεU ∀nεZ+

Similarly if we continue the unstable directions we construct the unstable manifold of the periodicorbit. Continuing the centered directions we obtained the center manifolds of the periodic orbit.The manifold have the same dimension as the linear subspace, and are tangent to them. Thereis a small issue related to the degeneracy of the eigenvalues, in which case the contraction (orexpansion) may not be exponential.

Notice that these sets are by construction invariant in the sense that M(W s(x∗)) = W s(x∗).Numerically is easier to construct the unstable manifold of a fixed point than stable manifolds,specially in higher dimensions. Also, it may be difficult to construct the stable manifold if the mapis not invertible.

We can think of a number of ways to compute the manifolds. First, the unstable manifoldscan be trivially computed by iterating forward an initial condition close to the fixed point but inthe unstable direction. If the map in invertible, we can compute the stable manifold by iterating

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1 2 3 4 5 6q

-6

-4

-2

2

4

6

P

1 2 3 4 5 6q

-6

-4

-2

2

4

6

P

1 2 3 4 5 6q

-6

-4

-2

2

4

6

P

1 2 3 4 5 6q

-6

-4

-2

2

4

6

P

Figure 15: Trajectories in the phase space for (a) ε = 0, (b) ε = 0.5, (c) ε = 1, (d) ε = 2 usingmany initial conditions.

Figure 16: Possible trajectories between traffic lights

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Figure 17: The bifurcation diagram, associated Lyapunov exponent.

backwards along the stable directions. If it is non-invertible, then we could iterate many initialconditions forward in time. The initial conditions that stay in the initial volume should approximatethe stable manifold.

Mathematically, a n-dimensional manifold is a space which looks locally like Rn, but it is notnecessarily Rn. On each point in the manifold, there is a local coordinate neighborhood that canbe described by n coordinates (Rn). This allow us to define functions, differential forms, etc. onthe manifold. The simpler example is the 2-sphere S2 which is clearly not R2 (non-Euclidean) butthat a neighborhood close to any point in the sphere can be considered as part of R2 (Euclidean)represented by two coordinates. Locally we cannot distinguish between this and a small domain ofR2.

Figure 18: (a) The stable and unstable manifold of a fixed point. (b) Evolution of a square closeto the fixed point.

The linear spaces, and the invariant (stable, unstable, center) manifolds of periodic m-orbitscan be defined similarly. In particular, the stable manifold applies to all the points in the period

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m-orbit x0, x1, . . . , xm−1. Visually and numerically, it may be easier to treat the periodic x-orbitas a fixed point of Mm and study the local subspaces, with their invariant manifolds, of this fixedpoint with the understanding that the invariant manifold apply to all the elements of the fixedpoint.

We define a hyperbolic fixed point (or a periodic m-orbit) if it has no center subspace. Thetangent subspace can then be uniquely specified as the direct sum of Tx∗ = E+ ⊕ E−.

For example for a Hamiltonian map in 2D, we have that around a fixed point

w1 =1

w2

→ λ1 = −λ2

and we usually define two situations

w > 0 Hyperbolicw > e±iθ Elliptic

Let’s note that we cannot have an attractor (periodic orbit, etc) because they do not guarantee theconservation of the volume.

The Baker map yields an example of an hyperbolic system, since

DM(x) =

[wx(y) 00 wy(y)

]with λx(y) < 1 and λy(y) > 1. The unstable manifold are vertical lines, and the stable manifoldsare horizontal lines. This is in essence one of the reasons for creating this map. The Horse shoemap also generates horizontal and vertical manifolds. In fact they are cantor sets of horizontal andvertical lines whose intersection is the invariant set.

The concept of hyperbolicity not only applies to fixed points, but also to more general invariantsets of a map, such as strange attractors, or nonattracting chaotic sets. We say that an invariantset Λ is hyperbolic if there is a direct sum decomposition of Tx = E+ ⊕ E− for all xεΛ, and thatthe splitting varies continuously with x and is invariant DM(E±

x ) = E±M(x). We obtain that

yεE−x |DMn(x)y| < Kρn|y|

yεE+x |DM−n(x)y| < Kρn|y|

for K > 0 and 0 < ρ < 1. The stable and unstable manifold can then be locally defined andthat points in the stable manifold approach each other exponentially, and points in the unstablemanifold separate exponentially (e.g., chaotic attractors).

Hyperbolic invariant sets are nice because a number of theorems can be proven, such as (a) theyare structurally stable, (b) there is shadowing, (c) a natural measure exists, (d) can be representedby symbolic dynamics, etc. It seems that the logistic map is not structurally stable since a smallperturbation in the parameter can shift from chaotic to periodic.

In general, many of the interesting chaotic phenomena seems to occur in systems that are nothyperbolic and not structurally stable. The Hennon map is non hyperbolic since there are pointson the attractor whose stable and unstable manifold at x are tangent, and therefore they cannotspan the whole space.

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The manifolds generated by fixed points and periodic orbits are extremely relevant in deter-mining the dynamics of the system. There are situations in which the dynamics of the stable andunstable manifolds of fixed points are of interest in understanding chaos. For example, in the Hen-non map, the strange attractor is the closure of the unstable manifold of a fixed point in its basinof attraction (see homework).

In the particular case of fixed points of invertible maps (for example maps derived fromdifferential equations) the stable manifold cannot intersect stable manifolds (in particular itself)and similarly unstable manifold cannot intersect unstable manifolds (in particular itself) (WHY?).But there can be intersections of stable with unstable manifolds (of itself or with other fixedpoints). In particular we have Homoclinic (intersections of the stable and unstable manifolds)and Heteroclinic (intersections of the manifolds of two periodic orbits) tangencies. This is alsorelevant in continuous systems as we will see later on.

In a Homoclinic intersection, chaotic behavior occurs because if there is one intersection, theremust be an infinite number of them. Note that since W± are invariant, then intersections mapinto intersections. This shows the complicated dynamics forming a chaotic or strange attractor.The dynamics are horseshoe type for a high enough iteration of the map, and hence generatenon-attracting chaotic sets.

Figure 19: (a) Homoclinic Tangency, (b) Heteroclinic tangency

HOMEWORK PROBLEM:For the predator-pray model with Aa, Ab fixed, draw themaximum Lyapunov exponent in the ra − rb plane.

HOMEWORK PROBLEM: For the Hennon Map in a chaotic regime, show that thetwo Lyapunov exponents can be obtained by (a) using the QR decomposition of theinfinitesimal vectors evolved by DM , and (b) by following trajectories with δ0 = 10−10

HOMEWORK PROBLEM:Let’s take the logistic map, but one in which rn = A andrn+1 = B. In the A−B space paint the value of the Lyapunov exponent for that system.Use a color function that makes the picture interesting

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HOMEWORK PROBLEM: Sketch the dynamics of the Baker map (λa +λb = 1). Takea square in [0, 1]× [0, 1] and follow what happens to it. Is this map conservative? Is themap chaotic? Calculate numerically the spectrum of Lyapunov exponents?

HOMEWORK PROBLEM: Calculate the stable and unstable manifolds for the fixedpoint of the Hennon map in the chaotic case by iterating the map forward and back-ward. Show that the second method to compute the stable manifold also works.Repeat for a period 2 orbit.

HOMEWORK PROBLEM: Show that the unstable manifold of a point in the attractoris the chaotic attractor and that the stable manifold is tangent to it in many places.This suggest that the system is not hyperbolic. Why?

HOMEWORK PROBLEM: Take the Horseshoe map. To visualize the complex dy-namics that can happen, consider the invariant set Λ. Follow the iterations of the 16elements of M2[M−2(S)] forward in time. Does it show chaos? Can you estimate theLyapunov exponent. Can you estimate the dimension of the invariant set?

HOMEWORK PROBLEM: For the city traffic problem. Construct the map, and abifurcation diagram for A+ = 10, A− = 30 with Ω.

HOMEWORK PROBLEM: Take the kicked pendulum. Take many initial conditionsand calculate λ. Count how many have λ > 0 as a funcion of ε. This are called Arnoldtongues. Can you estimate the dimension of the chaotic set as a function of ε?

4 Generic Bifurcations

One of the most important topics here is the issue of the type of allowed bifurcations. We haveobserved a period doubling bifurcations in which an orbit of period 2m losses stability and an orbitof period 2m+1 appears and becomes stable.

4.1 Generic Bifurcations 1-D and normal forms

Define a local generic bifurcation if its basic character cannot be altered by an arbitrary smallperturbation M(x, r) + εg(x, r). There are 3 generic bifurcations in 1-D continuous maps. In factit is interesting to note that we can define normal forms that describe (with the minimum detailpossible) the local bifurcation (not its global form), namely

• Period doubling bifurcation,

f(x) = rx + x− x3

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• Tangent bifurcation

f(x) = −r + x− x2

• Inverse period doubling bifurcation

f(x) = −rx + x + x3

there are other bifurcations that are not generic, as we will see later. The 3 bifurcations are shownin Fig. 2 can also be understood in terms of the fix point solution and their stability.

S

S

US

Forward Backward

U

S

InversePeriod doubling

Tangent

DoublingPeriod

SS

U

U

Figure 20: The generic bifurcations

In the case of the logistic map the period doubling bifurcation has been clearly described.We can now understand that the periodic windows that appear in the chaotic see of the logisticmap occur because of a tangent bifurcation as described in Fig. 20. When we calculate thefixed point of a polynomial function, like the logistic map, in the tangent bifurcation 2 complexconjugate fixed points become real. In the case of the logistic map for the period 3 window, 3pairs of complex conjugate become real at the same time. The other two solutions of the degree 8polynomial corresponds to the 2 period one unstable fixed points.

Since it can be shown that the logistic does not have backward bifurcations not inverse perioddoubling, then we have explained the bifurcations diagram of the logistic map. There are a theoremby Sarkovskii (1964) that describe the order of the windows.

Notice that all the unstable fixed points created during the period doubling bifurcations (fromperiod one or from the periodic windows) are still present. In fact, they form what is called a

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Page 33: Bifurcations Maps

chaotic saddle (or a non-attracting chaotic set). Understanding these sets is one of thecrucial open problems in complex system theory.

It is important to notice that the by expanding the map M around a fixed point, we couldinvestigate the type of bifurcation that will happen.

M(x) = f(xo) + f ′(xo)(x− xo) + f ′′(xo)(x− xo)2 . . .

-1 -0.5 0.5 1x

-1

-0.5

0.5

1

f@xD

-1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1r

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1x

-1 -0.5 0.5 1x

-2.5

-2

-1.5

-1

-0.5

0.5

1

f@xD

-1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1r

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1x

-1 -0.5 0.5 1x

-2

-1

1

2

f@xD

-1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1r

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1x

Figure 21: The generic bifurcations with the normal forms.

4.2 Hysteresis as a non-local bifurcation

The bifurcations studied above, are local bifurcations. But we can add a few of these bifurcationsto form a global type of bifurcation. For example we can take the map

M(x, µ, r) = µx− x3 + r (4)

clearly it is not of the forms given above, and as such it will display nonlocal bifurcations. Thismap could have 3 fixed point solutions depending on r.

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Page 34: Bifurcations Maps

-1 -0.5 0.5 1x

-1

-0.5

0.5

1

f@xD

-0.4 -0.2 0.2 0.4r

-1

-0.5

0.5

1

x

Figure 22: The hysteresis generated by the collision of two tangent bifurcations, given by µ = 1.5.(a) the function, (b) the bifurcation diagram. We should study the phase r − µ diagram. But wewill leave it for a homework problem.

4.3 Non-generic local bifurcation

Notice that in the logistic map a non-generic bifurcation occurs at r = 1, which cannot be describedby the above 3 bifurcations. To see that it is non-generic, take the map

xn+1 = r ∗ xn(1− xn) + ε

then the fixed points no longer go through a bifurcation depending on the sign of ε.

0.25 0.5 0.75 1 1.25 1.5 1.75 2r

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1x

0.25 0.5 0.75 1 1.25 1.5 1.75 2r

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1x

Figure 23: Non generic bifurcations for ε > 0 and ε < 0.

4.4 Generic local Bifurcations N-D

In more dimension we can have the same type of transitions. There is the small complication thatthe eigenvalues can be complex.

Let’s take the predator-prey map and look at the bifurcation diagram, as shown in Fig. 24a,b,d.We clearly see that we have a bifurcation (inverse) that we have not see before at around Aa = 0.4.If we study what is going on, we realize that we have complex eigenvalues λ± = λr ± iλi when theeigenvalues go through|λ±| = 1, as shown in Fig. 24e. This is called a Hoft bifurcation and at thetransition we obtain a rotation that is quasiperiodic, which means that it rotates with frequencythat is an irrational number. A rational periodic orbit is equivalent to a m-periodic orbit. We canestimate the frequency of the orbit by using

34

Page 35: Bifurcations Maps

ω(ra) ∼1

N

N∑i=1

∣∣∣∣xi+1 − 2xi + xi−1

xi

∣∣∣∣which is plotted if Fig. 24c, which clearly has to go to zero at the bifurcation because the fixed pointis stable. We will study in more detail quasiperiodicity later on in the context of Hamiltonianchaos.

But what is going on here? Well, to understand it better we should look at the following mapin polar coordinates (x = rcosθ,y = rsinθ), namely

rn+1 = µrn(1− rn)θn+1 = θn + ω + br2

n

shown in Fig. 24f. Clearly, there is a period doubling bifurcation at (x = y = 0) for µc = 0in the radial variable with a stable solution at r(µ). On the other hand, θ rotates with angularfrequency which varies with the stable radius. This is called a supercritical Hoft bifurcation.The eigenvalues are

wi =1√2

[1 + r ±

√−1− 2r − r2

]therefore, the eigenvalues, which are complex become in magnitude greater than one at µc. Ifω + br(µ) = p/q is a rational number, then we have a periodic solution that repeats itself after qiterations. On the other hand if it is an irrational number then we have a quasiperiodic orbit,that never repeats itself. Since r(µ) is continuous, we will get a countable set of periodic solutionsembedded in the aperiodic solutions. But note that even though we have no periodic orbits, thisis not chaos because the Lyapunov exponent is clearly zero, i.e., there is no sensitivity to initialconditions.

There is the subcritical Hoft bifurcation in which we have a destabilizing cubic term, i.e.,f(r) = µr + r2 − r5.

The idea of a Hoft bifurcation makes also sense in a continuous dynamics system in which thereal part of the eigenvalue goes from negative to positive (we will see later). A fixed point lossesstability and a limit cycle becomes stable.

4.5 Global bifurcations

There are other bifurcations that are a little more obscure and deserve some time.First, we have the tangent bifurcations of cycles in which a tangent bifurcation occurs in

the radial variable and a stable radius appears.

rn+1 = µrn + r3n − r5

θn+1 = θn + ω + br2n

the r5 is to make the origin stable.Second, we have the infinite period bifurcation of cycles in which a cycle losses stability

and a fixed point appears at a particular angle,

35

Page 36: Bifurcations Maps

0.1 0.2 0.3 0.4 0.5r

0.2

0.4

0.6

0.8

1

x

0.1 0.2 0.3 0.4 0.5r

0.001

0.1

10

1000

Ω

0.2 0.3 0.4 0.5r

0.2

0.4

0.6

0.8

1

¨l¨

-0.4 -0.2 0.2 0.4r

-0.4

-0.2

0.2

0.4

x

Figure 24: (a) The bifurcation diagram for ra = 4, rb = 4, Ab = 1.0. (b) The attractor r = 0.4.,(c) The frequency as a function of r. (d) The attractor r = 0.35. (e) |λmax| as a function of r. (f)The attractor for the polar map.

36

Page 37: Bifurcations Maps

rn+1 = µ(1− r2n)

θn+1 = θn + µ− sin(θ)

which make an angle stable when µ < 1.There is also the possibility that a limit cycle can collide with the unstable manifold of periodic

orbit forming a Homoclinic orbit (at the bifurcation point). Beyond that the cycle will disappear.This is called the Homoclinic bifurcation. Here the key is the unstable manifold of the saddlewhich is capable of destroying the cycle. This is similar to a boundary crisis as we will see later.

4.6 Discontinuous transitions

We can see that the period-doubling bifurcations is in essence a continuous bifurcation, while thetangent bifurcations is a discontinuous transition. In the same way, the logistic map grows abruptlyagain at the end of the periodic window in another discontinuous transition of the attractor.

4.6.1 Intermittent transitions to chaos

See Fig. 26 for a zoom of the period 3 window. Observe that in the logistic map, the transition whenthe periodic windows appear is clearly discontinuous and the attractor changes size abruptly (notuniformly continuous) at r3c. We can appreciate from the figure that before the tangent bifurcationsthere is an intermittent behavior, in which the trajectory spends a long time between burst closeto the fixed point that is going to form (we could also find this by computing the invariant naturalmeasure). This are called intermittent transition to a chaotic attractor or inverse as in thecase of the logistic map (period 3 window). We can define PT as the mean time between burst, andof course PT diverges at the bifurcation. We will estimate the scaling of PT .

In Fig. 25a we see what occurs close to r3c, and the trajectory spends a long time around thetunnel. We can expand and form an equivalent map close to the fixed point yn = x − xp andε = r − r3c,

yn + 1 = ε + yn + y2n

which is shown in Fig 25b (this is the normal form described above). It is interesting to note thatthis is one of the normal forms for the transitions. Since we are very close to the transition, we canassume that in the tunnel we have

dx

dn∼ xn+1 − xn = x2 + ε T ∼

∫ ∞

xo

dx

x2 + ε=

∫ ∞

−∞

dx

x2 + ε∼ ε−1/2

hence PT ∼ (r − rT )−1/2.Pomeau and Manneville (1980) distinguish, besides the tangent bifurcation, 3 generic types of

intermittent transitions (or inverse)

• Tangent bifurcations to chaos, with PT ∼ (r − rT )−1/2, which we studied above.

• Hopf bifurcations to chaos, with PT ∼ (r− rT )−1 in which a quasiperiodic orbit (similar to atangent bifurcation) is formed (see latter).

37

Page 38: Bifurcations Maps

0.52 0.54 0.56 0.58 0.6

0.52

0.54

0.56

0.58

0.6r=2.5

-1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1x

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1f@xDr=2.5

Figure 25: (a) The iterated map f 3 close to the period 3 window and close to the formation of theperiod 3 orbit. (b) the equivalent map.

• Inverse period doubling, with PT ∼ (r− rT )−1 in which a periodic, but unstable, orbit in theattractor becomes stable by an inverse period doubling bifurcation.

where the other scalings can be obtained in a similar manner.

4.6.2 Interior and exterior crisis

In the same way, the logistic map grows abruptly again at the end of the periodic window, see Fig26 in another discontinuous transition, called a crisis, of the attractor.

An interior crisis, in which the attractor collides with a unstable orbit inside the basin ofattraction of the attractor. Therefore, once every so often the trajectory will spend time away fromthe initial attractor (before the transition) along the unstable manifold of the unstable periodicorbit. But since this orbit is in the basin or attraction of the attractor it will eventually come backto the place of the original attractor. of course now the attractor is the whole thing. In this waythe attractor growths in size at the crisis, showing an crisis induced intermittency as clearly inFig. 26 or Fig. 2.

It is possible to construct a scaling for the time between burst τ assuming that it has a distri-bution which seems random. Hence,

P (τ) ∼ e−τ/<τ>

This for is suggestive of the smoothness of the measure of the region of the attractor (beforecrossing) that cross the unstable manifold, i.e, the measure should not be too singular (see alsobelow the exterior crisis situation).

The average time between burst can be derived in a manner similar to the one above (see Grebogiet al., 1986, 1987). They determined that

< T >∼ (r − rc)−γ γ =

1

2+

ln|α1|ln|α2|

38

Page 39: Bifurcations Maps

with α1 and α2 are the expanding and contracting eigenvalues of the map at A (in the case of aperiod 3 orbit use the M3). The case of a 1-D map, we have that α2 = 0, i.e., only an expandingdirection. Then γ = 1/2.

3.82 3.83 3.84 3.85 3.86 3.87 3.88r

0.2

0.4

0.6

0.8

1x

3.82 3.83 3.84 3.85 3.86 3.87 3.88r

0.35

0.4

0.45

0.5

0.55

0.6

0.65

x

Figure 26: (a) The periodic 3 window with the created periodic orbits at the tangent bifurcation.(b) a zoom.

There are other types of crisis. In particular, there are boundary crisis in which the attractorhits an unstable manifold on its basin boundary. This transition destroys the attractor leaving onlya chaotic transient. The number of points that stay decays as

P ∼ exp(−τ/ < τ >)

Again, this for is suggestive of the smoothness of the measure of the region of the attractor (beforecrossing) that cross the unstable manifold, i.e, the measure should not be too singular.

A clear example in the logistic map happens when r > 4 in which case the only attractor is −∞.Again with ε = (r− 4) we can estimate that the set of points in which M > 1 is proportional to ε,and since this region has probability ε1/2, then < τ >∼ (r−4)−1/2. Remember that 1/ < τ > is theprobability that per iterate of falling in that area proportional to ε. The fact that this probabilitydensity ε1/2 can be found numerically, or can be determined from the fact that the invariant densityat r = 4 is smooth, and that the region that maps to |x− 1/2| ε is of the order of ε1/2 which givesthe above result. Notice the similarity with the above value.

In more dimensions we may distinguish between:

• Homoclinic tangency crisis: the collision of the stable and unstable manifolds of a unstableperiodic orbit A become tangent. Grebogi et al., [1986, 1987] construct the scaling

< T >∼ (r − rc)−γ γ =

1

2+

ln|α1|ln|α2|

39

Page 40: Bifurcations Maps

0 10 20 30 40 50n

1

10

100

1000ÈxÈ

0 0.2 0.4 0.6 0.8 1xo

2

5

10

20

50

100

200n

0 50 100 150 200n1

10

100

1000

10000

P@nD

0.0001 0.001 0.01 0.1Ε-41

5

10

50

100

<Τ>

Γ=0.524

Figure 27: (a) One trajectory for the logistic map with r = 4.02. (b) The time in takes for aninitial condition to leave the region [0, 1]. (c) The surviving probability. (d) The average lifetimeof as a function of ε− 4.

40

Page 41: Bifurcations Maps

with β1 and β2 are the expanding and contracting eigenvalues of the map at B.

• Heteroclinic tangency crisis: The stable manifold of an unstable periodic orbit B becomestangent with the unstable manifold of an unstable periodic orbit A. Before the crisis A was onthe attractor, and B was on the boundary. Grebogi et al., [1986, 1987] construct the scaling

< T >∼ (r − rc)−γ γ =

ln|β2|ln|β1β2|2

Of course we have not discussed the other relevant issues, for example of a boundary crisis inwhich an unstable manifold or a periodic orbit collides with a non attracting chaotic set.

Furthermore, there are other crisis of interest. For example, there are attractor mergingcrisis in which the two attractor collides generating a large attractor showing clear intermittency.Also, there are transitions in which the two states are for example low dimensional chaos to highdimensional chaos. We will see more on this later.

HOMEWORK PROBLEM: Show numerically that indeed the mean time betweenburst diverges as PT ∼ (r − rt)

−1/2 for the tangent bifurcation. Also check that for thecrisis we have P (τ) exp(−τ/ < τ >) with < τ >∼ (r − rt)

−1/2 for the crisis of the periodic3-window of the logistic map.

HOMEWORK PROBLEM: Repeat the analysis for the map M = r ∗ x2(1 − x) with rin [3.5, 6.74], and study the type of crisis that appear. Can you estimate when theyhappen?

HOMEWORK PROBLEM: Describe the phase diagram r− µ of the map given in Eq.4.

41


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