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Cours : Dynamique Non-Lineaire

Laurette TUCKERMAN

laurette@pmmh.espci.fr

Maps, Period Doubling and Floquet Theory

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1 Discrete Dynamical Systems or Mappings

A discrete dynamical system is of the form

yn+1= g(yn) (1)

wherey and g are real vectors of the same dimension. A fixed point of (1) is a solution to

y= g(y) (2)

In one dimension, linear stability analysis ofyis carried out by writing

yn = y+ n

yn+1 = g(yn)

y+ n+1 = g(y+ n)

= g(y) + g(y)n+1

2g(y)2n

n+1 g(y)n (3)

We see that|| decreases, i.e. y is a stable fixed point, if|g(y)| < 1 and that|| grows, i.e. y is anunstable fixed point, if|g(y)| > 1. In a multidimensional system, g(y) is replaced by the JacobianDg(y)and yis a stable fixed point if all of the eigenvaluesofDg(y)satisfy ||

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Figure 2: Graphical construction for iterating maps. f(x) =cx with various slopesc.

line across toy = x, then a vertical line down to thex-axis atx1and then repeat this procedure to iteratethe map. (We can combine the two consecutive vertical lines, i.e. drawing the vertical line directly to the

f(x)curve instead of to thex-axis.)

(x0, 0) (x0, x1 f(x0)) (x1, x1) (x1, x2 f(x1)) (4)

Figure 2 shows trajectories resulting from iterating linear mapsf(x) = cx

for values ofc

which are

positive and negative, and with absolute value greater and less than one. There is a fixed point at x= 0.This fixed point is stable if|c| 1. Trajectories proceed monotonically ifc >0 andoscillate between values to the right and left ofxifc

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Fixed points satisfyingf(x) = xexist for >0. Their stability is calculated via

f(x) = x +

x2

f(x) = 1 2x (6)f() = 1 2 1for >0

Pitchfork bifurcation:

x xn+1 xn= xn x3n = xn+1= f(xn) =xn+ xn x3n (7)

The fixed points0, satisfyf(x) = x. Their stability is calculated via

f(x) = x + x x3

f

(x) = 1 + 2x2

f(0) = 1 + 1for 0 (8)

f() = 1 0 (9)

Subcritical pitchfork bifurcations and transcritical bifurcations can also occur. Saddle-node and pitchfork

(super and subcritical) bifurcations are illustrated in figure 3.

Case (ii), when eigenvalues cross at ei, leads to a secondary Hopf, or Neimark-Sacker, bifurcation to atorus. We will discuss this in the next chapter. Case (iii), when eigenvalues cross at 1leads to a flip, ora period-doubling bifurcation, and is a phenomenon that cannot occur for continuous dynamical systems.

We now discuss this case in the context of the logistic map.

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Figure 3: Steady bifurcations for discrete dynamical systems.

Top row: Saddle-node bifurcation. f(x) =x + x2 for = 0.2(left) and for= 0.2(middle).Middle row: Supercritical pitchfork. f(x) =x + x x3 for = 0.2(left) and for= 0.4(middle).Bottom row: Subcritical pitchfork. f(x) =x + x + x3 for = 0.4(left) and for= 0.2(middle).Right: corresponding bifurcation diagrams.

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2 Logistic Map

The logistic map was proposed in the 1800s and popularized in the 1970s as a model for populationbiology. Population growth is geometric (the next value is a multiple of the current value) when the

population is small, but is reduced when the population is too large. It is with this map that the famous

period-doubling cascade was discovered, also in the 1970s, by Feigenbaum in Los Alamos, U.S. and,

almost simultaneously, by Coullet and Tresser in Nice, France.

2.1 Fixed points and period doubling

Figure 4: Left: the logistic mapf(x) =ax(1 x)fora= 0.4, 1.2, 2.0, 2.8, 3.6. Fixed points (dots) arelocated at x= 0and x= 1 1/a. Short dashed line segments have slopef(x).Right: fixed pointsx= 0and x= 1 1/aas a function ofa.

The logistic map is defined by:

xn+1= f(xn) axn(1 xn) forxn [0, 1], 0< a < 4 (10)

fis a quadratic function mapping[0, 1]into itself, with minima at the two endpointsf(0) =f(1) = 0and a maximum at the midpoint f(1/2) =a/4. Its fixed points are easily calculated:

x= ax(1 x) = x= 0 or1 =a(1 x) = 1 x= 1/a= x= 1 1/a (11)

These are shown in figure 4.

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The stability of these fixed points is also easily determined:

f(x) = ax(1

x) (12)

f(x) = a(1 x) ax= a(1 2x) (13)f(0) = a= |f(0)|

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Figure 5: Behavior of the logistic map. Fora = 0.8,xn 0. Fora = 2.0 and a = 2.6, xn x =1 1/a. Fora = 3.04,xnconverges to a two-cycle.

Figure 6: Graph off2(x), where f(x) = 1ax(1x), and ofy= x. For a= 1.6, f2 has the same fixedpoints asf, namelyx = 0 (unstable, hollow dot) and x = 1 1/(a) (stable, filled dot). Fora = 3.4,the fixed pointx = 1 1/ahas become unstable and two new fixed points x1,2 (filled triangles) haveappeared, which together comprise a two-cycle forf.

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We will need:

x1+ x2 = 1 +

1

a =

a + 1

a (22)

x1x2 = a + 1

a2 (23)

As you might be beginning to guess, we can continue this process and check for stability of the two-cycle.

We have:d

dxf2(x) =f(f(x))f(x) (24)

For an element of a two-cycled

dxf2(x1) =f

(x2)f(x1) (25)

For the two-cycle (21), we have

f(x1)f(x2) = a(1 2x1) a(1 2x2)

= a2(1 2(x1+ x2) + 4x1x2)= a2

1 2

a + 1

a

+ 4

a + 1

a2

= a2 2a(a + 1) + 4(a + 1)= a2 + 2a + 4 (26)

0 = f(x1)f(x2)

1 =

a2 + 2a + 4

1 =

a2 + 2a + 3 (27)

a = 2 4 + 12

2 =

2 162

=2 4

2 = 3 (28)

0 = f(x1)f(x2) + 1 = a2 + 2a + 4 + 1 = a2 + 2a + 5 (29)

a = 2 4 + 20

2 =

2 +

24

2 =

2 + 2

6

2 = 1 +

6 = 3.44948 . . . (30)

where we have discarded the negative value ofr in (30). Equation (28) confirms that the two-cycle off,a fixed point off2, is created via a steady-state (pitchfork) bifurcation at a= 3. Equation (30) states thatthis two-cycle loses stability via a flip bifurcation at a = 3.44948 . . ., leading to a four-cycle.

The successive period-doubling bifurcations occur at successively smaller intervals inr and accumulate

ata = 3.569945672 . . ., as shown in table 1. This is called the period-doubling cascade.

2.2 Renormalization

The behavior of the logistic map is actually universal. The qualitative and quantitative properties above

(specifically, the value of in Table 1) hold for any family of maps with a quadratic maximum, suchasr sin x Why should this be? Feigenbaum and Coullet and Tresser provided an explanation of theperiod-doubling cascade in terms ofrenormalization. The idea is that the interior portion off2 resemblesa scaled version off. To facilitate the calculations, we use the map

f(x) = 1

ax2 (31)

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n an n an an1 n n1/n1 1

2 3 24 2.44948 0.228 4.46

8 3.54408 0.0948 4.747

16 3.56872 0.0244 4.640

32 3.5698912 0.00116 4.662...

......

...

3.569945672 0 4.669

Table 1: Period-doubling cascade for logistic map.

on the interval[1, 1], shown in figure 7. Figure 7 also showsf2(x)and a box surrounding the centralminimum off2. The idea is that the portion off2 inside the box, when scaled horizontally and vertically,resemblesf. Specifically, figure 7 also shows

T f(x) 1

f2(x) (32)

The resemblance is quantified as follows:

f(x) T f(x) (33)1 ax2 1

f2(x)

= 1

f(1 a(x)2)= 1

(1 a(1 a(x)2)2)

= 1

(1 a(1 2a(x)2 + a2(x)4))

= 1

(1 a + 2a2(x)2 a3(x)4) (34)

For the constant and quadratic terms in (34) to agree, we require

a 1

= 1 2a22

=a

a 1 = 2a= 12a(a 1) = 1

= 0.366 a=1 +

3

2 = 1.366 (35)

The dashed curve in figure 7 is T f(x) with the parameters of (35). To go from f2 to f4, or moregenerally fromf2

ntof2

n+1, we would repeat this procedure. At each stage the scaling factor n is set

ton= f2n(1)andanis determined by an equation like (33). T f is not a quadratic polynomial likef(31), but a quartic polynomial; T2fis an eighth-order polynomial, and so on. It has been shown thatthis procedure converges to a function

(x) = 1 1.528 . . . x2

+ 0.105 . . . x4

+ 0.0267 . . . x6

+ . . . (36)

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Figure 7: Left: Graph off(x) = 1ax2, off2(x), and ofT f(x) 1

f2(x)(dashed curve).T f(x)consists of the portion off2 inside the box, rescaled by . Fixed points off2 andT fare indicated bysolid dots and hollow dots, respectively. Parameters area = 1.366 and = 0.366. Right: Schematicdiagram of action ofTon space of mappings. Each sheet consists of mappings that are just undergoing aperiod-doubling bifurcation from2n to2n+1. The fixed pointis a mapping with one unstable direction

under the action ofT.

Figure 8: Tent map (37) for r = 0.4 and r = 0.85. The origin is a fixed point, stable forr < 0.5 andunstable forr >0.5. Forr >0.5, there is another fixed point, x= 2r/(1 + 2r).

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We view T as itself a mapping on mappings, with a fixed point ofT. The fixed point has a singleunstable directio