3.6 Bifurcation of M-Cycles
71
winding numbers whose continued fractions have random entries. Thus, scaling in
the neighborhood of these tori fluctuates as one goes from one level to another. This
has been called “renormalization chaos” by Chirikov and is discussed in (Chirikov
and Shepelyansky 1986).
3.6 Bifurcation of M-Cycles
We have seen from the Walker-Ford models (Sect. 2.5) that stable (elliptic) periodic
orbits can suddenly appear in the phase space as parameter values of the system
(in that case, the coupling constant) are varied. Existing periodic orbits may also
bifurcate and give rise to additional periodic orbits with periods that are some
integer multiple of the original period. A period-doubling bifurcation can be seen
in the strobe plots of the Henon-Heiles system in Fig. 2.3. In Fig. 2.3c, the elliptic
periodic orbit that is located at (p 2 = 0, q 2 ≈ 0.3) in Fig. 2.3b has bifurcated into
a hyperbolic periodic orbit, and two elliptic periodic orbits have been created along
the p 2 = 0 axis.
3.6.1 Some General Properties
There are a number of general statements that can be made about when a periodic
orbit can be created or destroyed or might collide with another periodic orbit. Most
of this can be determined from properties of the tangent map. Let us consider a twist
map, T K , and let us assume that it has a fixed point, x 0 . That is, x 0 = T K (x 0 ). We
can make the following statements about the neighborhood of x 0 .
1. This fixed point will be isolated (will have no other fixed points in its neighborhood) if the tangent map, ∇T K (x 0 ), has no eigenvalues λ = 1. To see
this, let F (x) = x − T K (x) so that F (x 0 ) = x 0 − T K (x 0 ) = 0. In the
neighborhood of x 0 , the tangent maps ∇F (x 0 ) and ∇T K (x 0 ) satisfy the equation
∇F (x 0 ) = 1 − ∇T K (x 0 ). If ∇F (x 0 ) =0, then ∇T K (x 0 ) has no eigenvalue, λ = 1.
Also, if ∇F (x 0 ) =0, the slope of F (x) at x 0 is finite and there can be no other
zeros of F (x) in the neighborhood of x 0 .
2. The fixed point, x 0 , is stable to small perturbations as long as ∇T K (x 0 ) has no
eigenvalues λ = 1. Consider a small perturbation, δT (x), to the map T K (x).
Assume the fixed point of the map, T K + δT , lies at x = x 0 + δx. Then x =
T K (x) + δT (x). If we expand to first order in δx and δT , we find
δx = (1 − ∇T K (x 0 ))
−1 δT (x 0 ).
Thus, if (1 − ∇T (x 0 )) has no zero eigenvalues, the fixed point is merely shifted
in position under the perturbation but is not destroyed.
71
winding numbers whose continued fractions have random entries. Thus, scaling in
the neighborhood of these tori fluctuates as one goes from one level to another. This
has been called “renormalization chaos” by Chirikov and is discussed in (Chirikov
and Shepelyansky 1986).
3.6 Bifurcation of M-Cycles
We have seen from the Walker-Ford models (Sect. 2.5) that stable (elliptic) periodic
orbits can suddenly appear in the phase space as parameter values of the system
(in that case, the coupling constant) are varied. Existing periodic orbits may also
bifurcate and give rise to additional periodic orbits with periods that are some
integer multiple of the original period. A period-doubling bifurcation can be seen
in the strobe plots of the Henon-Heiles system in Fig. 2.3. In Fig. 2.3c, the elliptic
periodic orbit that is located at (p 2 = 0, q 2 ≈ 0.3) in Fig. 2.3b has bifurcated into
a hyperbolic periodic orbit, and two elliptic periodic orbits have been created along
the p 2 = 0 axis.
3.6.1 Some General Properties
There are a number of general statements that can be made about when a periodic
orbit can be created or destroyed or might collide with another periodic orbit. Most
of this can be determined from properties of the tangent map. Let us consider a twist
map, T K , and let us assume that it has a fixed point, x 0 . That is, x 0 = T K (x 0 ). We
can make the following statements about the neighborhood of x 0 .
1. This fixed point will be isolated (will have no other fixed points in its neighborhood) if the tangent map, ∇T K (x 0 ), has no eigenvalues λ = 1. To see
this, let F (x) = x − T K (x) so that F (x 0 ) = x 0 − T K (x 0 ) = 0. In the
neighborhood of x 0 , the tangent maps ∇F (x 0 ) and ∇T K (x 0 ) satisfy the equation
∇F (x 0 ) = 1 − ∇T K (x 0 ). If ∇F (x 0 ) =0, then ∇T K (x 0 ) has no eigenvalue, λ = 1.
Also, if ∇F (x 0 ) =0, the slope of F (x) at x 0 is finite and there can be no other
zeros of F (x) in the neighborhood of x 0 .
2. The fixed point, x 0 , is stable to small perturbations as long as ∇T K (x 0 ) has no
eigenvalues λ = 1. Consider a small perturbation, δT (x), to the map T K (x).
Assume the fixed point of the map, T K + δT , lies at x = x 0 + δx. Then x =
T K (x) + δT (x). If we expand to first order in δx and δT , we find
δx = (1 − ∇T K (x 0 ))
−1 δT (x 0 ).
Thus, if (1 − ∇T (x 0 )) has no zero eigenvalues, the fixed point is merely shifted
in position under the perturbation but is not destroyed.