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Period Doubling Cascades
Jim Yorke
Joint Work with
Evelyn Sander
George Mason Univ.
Extending earlier work by Alligood, SN Chow,
Mallet-Paret, & Franks
Period-doubling cascades
If this picture were infinitely detailed, it would show infinitely
many period-doubling cascades, each with an infinite number
of period doublings. My goal is to explain this phenomenon
And give examples in 1 and n dimensions.
some period doubling cascades
Period 1
cascade
Period 3 & 5
cascades
cascade
Period-doubling cascades were first reported by Myrberg in
1962, and popularized by May using the logistic map in
the 1970’s.
For maps depending on a parameter, a cascade is an
infinite sequence of period doubling bifurcations in a
connected family of periodic orbits.
The periods in the cascade are k, 2k, 4k, 8k,… for some k.
• Feigenbaum’s rigorous methods suggest that when
period-doubling cascades exist, there is a regular behavior
in the sequence of period-doubling values.
cascade
Period-doubling cascades were first reported by Myrberg in
1962, and popularized by May using the logistic map in
the 1970’s.
For maps depending on a parameter, a cascade is an
infinite sequence of period doubling bifurcations in a
connected family of periodic orbits.
The periods in the cascade are k, 2k, 4k, 8k,… for some k.
• Feigenbaum’s rigorous methods suggest that when
period-doubling cascades exist, there is a regular behavior
in the sequence of period-doubling values.
Needed: new examples
• Maps like
α - x2
have played a prominent role in the history
of cascades. What is so special about
these maps? If anything?
The topological view for problems
depending on a parameter
Example of a geometric theorem.
Theorem. Assume
• g is continuous on [α0 , α1] and
• g(α0 ) < 0 and g(α1) > 0.
• Then
g(x) = 0 for some x between α0 & α1.
We find an analogous approach for
cascades
The topological view for problems
depending on a parameter
Example of a geometric theorem.
Theorem. Assume
• g is continuous on [α0 , α1] and
• g(α0 ) < 0 and g(α1) > 0.
• Then
g(x) = 0 for some x between α0 & α1.
We find an analogous theorems for
cascades
A snake is a (non-branching) path of periodic orbits
The topological view for cascades
Let F: [α0 , α1] X Rn → Rn be differentiable.
Theorem (terms explained later) Assume
1. there are no periodic orbits at α0 ; and
2. at α1 the dynamics are horse-shoe-like; and
3. On [α0 , α1] the set of periodic points is bounded in x.
4. F has generic orbit behavior;
Then if (α1, x1) is periodic and has no eigenvalues < -1,
it is on a connected family of orbits which includes a
cascade.
Distinct such orbits yield distinct cascades.
The topological view for cascades
Let F: [α0 , α1] X Rn → Rn be differentiable.
Theorem (terms explained later) Assume
1. there are no periodic orbits at α0 ; and
2. at α1 the dynamics are horse-shoe-like; and
3. On [α0 , α1] the set of periodic points is bounded in x.
4. F has generic orbit behavior;
Then if (α1, x1) is periodic and has no eigenvalues < -1,
it is on a connected family of orbits which includes a
cascade.
Distinct such orbits yield distinct cascades.
A new example
Let F(α; x) = α - x2 + g(α ,x)
Assume g(α, x) is a real valued function,
differentiable and bounded for α,x in R2, and
so are its first partial derivatives.
For example g = finite sum of fourier series
terms in α,x plus terms like tanh(α+x)
Let F(α; x) = α - x2 + g(α ,x)
A new example
Assume g(α ,x) is differentiable and bounded over all α ,x and
so are its first partial derivatives.
Let F(α; x) = α - x2 + g(α , x) Then
1. for α0 sufficiently small, there are no periodic orbits at α0 ;
and
2. for α1 sufficiently large, the dynamics are horse-shoe-like,
and
3. for “almost every” g, F has generic orbit behavior
4. the set of all periodic orbits in [α0 , α1] is bounded, and
Theorem. For such generic g,
if (α1, x1) is periodic and its derivative is > +1,
Then it is on a connected family of orbits which includes a
cascade.
Corollary: the map has infinitely many disjoint cascades.
A new logistic example
α x(1-x)g(α, x) for some α
•
A new logistic example
We require that g(α, x) is differentiable and positive
for x in [0,1], and bounded:
For some B1 & B2, 0 < B1 < g(α, x) < B2
and the partial derivatives fo g are also bounded.
a
Then
αx(1-x)g(α, x)
has cascades of period doublings as the
parameter α is varied (for typical g).
In fact we show the map has infinitely many
disjoint cascades as a is varied.
Periodic orbits of F(α,x)
We say (α,x) is p-periodic if Fp(α,x) = x.
If (α,x) is p-periodic, its “eigenvalues” are those of
its derivative DFp(α,x).
If x is one-dimensional, its “eigenvalue” is the
derivative (d/dx)Fp(α,x).
An orbit with no eigenvalues on the unit circle is
called “hyperbolic”; these include attractors.
Periodic orbits of F(α,x)
We say (α,x) is p-periodic if Fp(α,x) = x.
If (α,x) is p-periodic, its “eigenvalues” are those of
its derivative DFp(α,x).
If x is one-dimensional, its “eigenvalue” is the
derivative (d/dx)Fp(α,x).
An orbit with no eigenvalues on the unit circle is
called “hyperbolic”; these include attractors.
Types of hyperbolic orbits
Let (α,x) be a hyperbolic periodic point.
It is a flip saddle orbit or point if it has an odd
number of eigenvalues < -1.
If (α,x) is NOT a flip saddle orbit and the number of
eigenvalues with λ > 1 = n or n-2 or n-4 etc, then
it is a left orbit;
otherwise it is a right orbit.
For n=1, right orbits are attractors and
left orbits are orbits with derivative > +1.
A snake is a (non-branching) path of periodic orbits
Following segments of orbits
Follow a segment of left orbits to the left
(decreasing parameter direction)
Follow a segment of right orbits to the right.
(increasing parameter direction)
Never follow segments of flip orbits.
Generic Bifurcations of a path
For a family of period k orbits x(α) in Rn,
bifurcations can occur when
DFk(x) has eigenvalue(s) crossing the unit
circle. Generically they are simple.
• A Saddle node occurs when an e.v. λ = +1
• A Period doubling
. . . λ = -1
• Generically complex pairs cross the unit
circle at irrational multiples of angle 2π
Possible bifurcations affecting paths
Bifurcations for 1 dim x or more
Possible bifurcations affecting paths
Bifurcations for 1 dim x or more
Other Bifurcations only in dim x > 1
In addition each
period-doubling
bifurcation can
have both arrows
reversed
All low-period segments
are “right” segments
All new low-period segments
are “left” segments
Possible bifurcations affecting paths
Bifurcations for 1 dim x or more
Other Bifurcations only in dim x > 1
All S-N & P-D bifurcation points have one segment
approaching and one departing (except the upper-right one).
In addition each
period-doubling
bifurcation can
have both arrows
reversed
Coupling n 1-D maps
Coupling n 1-D maps. x = (x1, …,xn)
Let F(α; x) =
(αa1 - x1 2 + g1 (α, x1,…,xn),
...
αan - xn 2 + gn (α, x1,…,xn))
where each gj is bounded and so are its partial
derivatives;
Assume aj > 0 for each j = 1,…,n.
A new n-Dim example
Assume gm : RxRn → R for each m is differentiable and
bounded, and so are its first partial derivatives. Then
1. for α0 sufficiently small, there are no periodic orbits at α0 ;
and
2. for α1 sufficiently large, the dynamics are the horse-shoelike behavior of the uncoupled system (i.e. g=0), and
3. for “almost every” g = (gm), F has generic orbit behavior
4. the set of all periodic orbits in [α0 , α1] is bounded, and
Theorem. For such generic g
If (α1, x1) is periodic and
has an even number of eigenvalues < -1, (possibly none),
Then it is on a connected family of orbits which includes a
cascade.
Corollary: the map has infinitely many disjoint cascades.
A new n-Dim example
Assume gm : RxRn → R for each m is differentiable and
bounded, and so are its first partial derivatives. Then
1. for α0 sufficiently small, there are no periodic orbits at α0 ;
and
2. for α1 sufficiently large, the dynamics are the horse-shoelike behavior of the uncoupled system (i.e. g=0), and
3. for “almost every” g = (gm), F has generic orbit behavior
4. the set of all periodic orbits in [α0 , α1] is bounded, and
Theorem. For such generic g
if (α1, x1) is periodic and
has an even number of eigenvalues < -1, (possibly none),
Then it is on a connected family of orbits which includes a
cascade.
Corollary: the map has infinitely many disjoint cascades.
A new n-Dim example
Assume gm : RxRn → R for each m is differentiable and
bounded, and so are its first partial derivatives. Then
1. for α0 sufficiently small, there are no periodic orbits at α0 ;
and
2. for α1 sufficiently large, the dynamics are the horse-shoelike behavior of the uncoupled system (i.e. g=0), and
3. for “almost every” g = (gm), F has generic orbit behavior
4. the set of all periodic orbits in [α0 , α1] is bounded, and
Theorem. For such generic g
If (α1, x1) is periodic and
has an even number of eigenvalues < -1, (possibly none),
Then it is on a connected family of orbits which includes a
cascade.
Corollary: the map has infinitely many disjoint cascades.
Following families of period p
points
Let F : R X Rn → Rn be differentiable.
Assume Fp(α0 ,x0) = x0
When does there exist a continuous path
(α, x(α)) of period-p points through (α0 ,x0) for
α in some neighborhood (α0 -ε,α0 +ε) of α0?
This can answered by trying to compute the
path x(α) as the sol’n of an ODE..
A p-period Orbit (α0 ,x0) can be
continued if +1 is not an eigenvalue
If
Fp(α, x(α)) - x(α) = 0, then
(d/dα) {Fp (α, x(α)) - x(α)} = 0 (*)
i.e., Fpα, +Fpx dx/dα – Id dx/dα = 0
If Fpx – Id is invertible, then x(α) satisfies
dx/dα = [Fpx – Id]-1 Fpα (**)
It is easy to check (*) is satisfied by any solution
of (**).
If (α0 ,x0) is periodic and +1 is not an eigenvalue,
then (α,x(α)) can be continued, ending only
when +1 is an eigenvalue.
A p-period Orbit (α0 ,x0) can be
continued if +1 is not an eigenvalue
If
Fp(α, x(α)) - x(α) = 0, then
(d/dα) {Fp (α, x(α)) - x(α)} = 0 (*)
i.e., Fpα, +Fpx dx/dα – Id dx/dα = 0
If Fpx – Id is invertible, then x(α) satisfies
dx/dα = [Fpx – Id]-1 Fpα (**)
It is easy to check (*) is satisfied by any solution
of (**).
If (α0 ,x0) is periodic and +1 is not an eigenvalue,
then (α,x(α)) can be continued, ending only
when +1 is an eigenvalue.
A p-period Orbit (α0 ,x0) can be
continued if +1 is not an eigenvalue
If
Fp(α, x(α)) - x(α) = 0, then
(d/dα) {Fp (α, x(α)) - x(α)} = 0 (*)
i.e., Fpα, +Fpx dx/dα – Id dx/dα = 0
If Fpx – Id is invertible, then x(α) satisfies
dx/dα = [Fpx – Id]-1 Fpα (**)
It is easy to check (*) is satisfied by any solution
of (**).
If (α0 ,x0) is periodic and +1 is not an eigenvalue,
then (α,x(α)) can be continued, ending only
when +1 is an eigenvalue.
Snakes of periodic orbits
• A snake is a connected directed path of
periodic orbits.
• Following the “path” allows no choices
because it does not branch.
A snake is a (non-branching) path of periodic orbits
Generic Behavior of F(α,x)
In a bounded region of (α,x) space, for each
period p,
• there are finitely many p-periodic (α,x) having +1
as an eigenvalue and all such are generic
saddle-node bifurcation orbits.
• there are finitely many p-periodic (α,x) having -1
as an eigenvalue and all such are generic period
doubling orbits.
• If (α,x) has complex eigenvalues on the unit
circle, they are irrational multiples of 2π.
Generic Behavior of F(α,x)
In a bounded region of (α,x) space, for each
period p,
• there are finitely many p-periodic (α,x) having +1
as an eigenvalue and all such are generic
saddle-node bifurcation orbits.
• there are finitely many p-periodic (α,x) having -1
as an eigenvalue and all such are generic period
doubling orbits.
• If (α,x) has complex eigenvalues on the unit
circle, they are irrational multiples of 2π.
Generic Behavior of F(α,x)
In a bounded region of (α,x) space, for each
period p,
• there are finitely many p-periodic (α,x) having +1
as an eigenvalue and all such are generic
saddle-node bifurcation orbits.
• there are finitely many p-periodic (α,x) having -1
as an eigenvalue and all such are generic period
doubling orbits.
• If (α,x) has complex eigenvalues on the unit
circle, they are irrational multiples of 2π.
Generic maps
• Almost every (in the sense of prevalence)
map is generic.
The reason why cascades occur
• Each left segment must terminate (at a SN or PD
bifurcation) because there are no orbits at α0.
• Each right segment must terminate (at a SN or
PD bifurcation) because there are no right orbits
at α1.
• The family then continues onto a new segment.
This leads to an infinite sequence of segments
and corresponding periods (pk).
• Each period can occur at most finitely many
times, so pk →∞. So it includes ∞-many PDs.
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