3.6 Bifurcation of M-Cycles
73
As was shown by Bountis (1981), the first few period-doubling bifurcations
can be computed analytically. The period 1 fixed points of this map are given by
y (1)
x (1)
= Q a
y (1)
x (1)
. It is easy to show that this equation has two solutions
(we write them as row vectors for convenience), (x
(1)
+ , y
(1)
+ ) and (x
(1)
− , y
(1)
− ), where
x
(1)
± = y
(1)
± =
1
a (−1 ±
√
1 + a). The stability of each of these fixed points can be
determined from the tangent map
∇Q ± =
0
1
−1 −2ax
(1)
±
,
(3.58)
which is obtained by linearizing Q a about the fixed points x
(1)
± . As discussed in
Sect. 3.2, the fixed point will be elliptic if |Tr(∇Q ± )| < 2. We find that the
fixed point (x
(1)
− , y
(1)
− ) is always hyperbolic, while the fixed point (x
(1)
+ , y
(1)
+ ) is
elliptic for −1 < a < 3 and hyperbolic otherwise. When a = 3, ∇Q + has the
doubly degenerate eigenvalue λ = −1. Thus, we expect that the tangent map for
the mapping Q 2
a will have a doubly degenerate eigenvalue λ = 1, and a perioddoubling bifurcation can occur. This bifurcation is shown in Fig. 3.15, where the
neighborhood of the fixed point (x
(1)
+ , y
(1)
+ ) is shown for a = 0.95, a = 2.98, and
a = 3.02.
The mapping Q 2
a is defined as
y n+2
x n+2
= Q
2
a
y n
x n
=
1 − y n − ax 2
n
1 − x n − a(1 − y n − ax 2
n ) 2
.
(3.59)
This mapping has fixed points at (x
(2)
1
= A + , y
(2)
1
= A − ), (x
(2)
2
= A − , y
(2)
2
=
A + ),(x
(2)
3 = −B + , y
(2)
3 = −B + ), and (x
(2)
4 = −B − , y
(2)
4 = −B − ), where A ± =
1
a (1±
√
a − 3) and B ± =
1
a (1±
√
a + 1). For a < 3, the fixed points (x
(2)
1 , y
(2)
1 ) and
(x
(2)
2 , y
(2)
2 ) don’t exist, and the fixed points (x
(2)
3 , y
(2)
3 ) and (x
(2)
4 , y
(2)
4 ) correspond
to the hyperbolic and elliptic fixed points of Q a . However, for a ≥ 3, all four fixed
points of Q 2
a exist and the original orbit has bifurcated and period-doubled. It is
easy to check that when a = 3, the tangent map for Q 2
a has a doubly degenerate
eigenvalue λ = +1. Bountis found that the symmetry curve plays a dominant role
in determining the location of the orbits in the period-doubling sequence. If he let
(y m , x m ) denote the location of the mth member of 2 k periodic points obtained after
k period-doublings of the original orbit, then he found that the orbits m = 1 and
m = 2 k−1 will lie on the symmetry line 2x − 1 + ay 2 = 0. He called this symmetry
line the “symmetry road” of the period-doubling sequence.
73
As was shown by Bountis (1981), the first few period-doubling bifurcations
can be computed analytically. The period 1 fixed points of this map are given by
y (1)
x (1)
= Q a
y (1)
x (1)
. It is easy to show that this equation has two solutions
(we write them as row vectors for convenience), (x
(1)
+ , y
(1)
+ ) and (x
(1)
− , y
(1)
− ), where
x
(1)
± = y
(1)
± =
1
a (−1 ±
√
1 + a). The stability of each of these fixed points can be
determined from the tangent map
∇Q ± =
0
1
−1 −2ax
(1)
±
,
(3.58)
which is obtained by linearizing Q a about the fixed points x
(1)
± . As discussed in
Sect. 3.2, the fixed point will be elliptic if |Tr(∇Q ± )| < 2. We find that the
fixed point (x
(1)
− , y
(1)
− ) is always hyperbolic, while the fixed point (x
(1)
+ , y
(1)
+ ) is
elliptic for −1 < a < 3 and hyperbolic otherwise. When a = 3, ∇Q + has the
doubly degenerate eigenvalue λ = −1. Thus, we expect that the tangent map for
the mapping Q 2
a will have a doubly degenerate eigenvalue λ = 1, and a perioddoubling bifurcation can occur. This bifurcation is shown in Fig. 3.15, where the
neighborhood of the fixed point (x
(1)
+ , y
(1)
+ ) is shown for a = 0.95, a = 2.98, and
a = 3.02.
The mapping Q 2
a is defined as
y n+2
x n+2
= Q
2
a
y n
x n
=
1 − y n − ax 2
n
1 − x n − a(1 − y n − ax 2
n ) 2
.
(3.59)
This mapping has fixed points at (x
(2)
1
= A + , y
(2)
1
= A − ), (x
(2)
2
= A − , y
(2)
2
=
A + ),(x
(2)
3 = −B + , y
(2)
3 = −B + ), and (x
(2)
4 = −B − , y
(2)
4 = −B − ), where A ± =
1
a (1±
√
a − 3) and B ± =
1
a (1±
√
a + 1). For a < 3, the fixed points (x
(2)
1 , y
(2)
1 ) and
(x
(2)
2 , y
(2)
2 ) don’t exist, and the fixed points (x
(2)
3 , y
(2)
3 ) and (x
(2)
4 , y
(2)
4 ) correspond
to the hyperbolic and elliptic fixed points of Q a . However, for a ≥ 3, all four fixed
points of Q 2
a exist and the original orbit has bifurcated and period-doubled. It is
easy to check that when a = 3, the tangent map for Q 2
a has a doubly degenerate
eigenvalue λ = +1. Bountis found that the symmetry curve plays a dominant role
in determining the location of the orbits in the period-doubling sequence. If he let
(y m , x m ) denote the location of the mth member of 2 k periodic points obtained after
k period-doublings of the original orbit, then he found that the orbits m = 1 and
m = 2 k−1 will lie on the symmetry line 2x − 1 + ay 2 = 0. He called this symmetry
line the “symmetry road” of the period-doubling sequence.