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Period Doubling Cascades

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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.

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

- there are no periodic orbits at α0 ; and
- at α1 the dynamics are horse-shoe-like; and
- On [α0 , α1] the set of periodic points is bounded in x.
- 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

- there are no periodic orbits at α0 ; and
- at α1 the dynamics are horse-shoe-like; and
- On [α0 , α1] the set of periodic points is bounded in x.
- 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

- for α0 sufficiently small, there are no periodic orbits at α0 ; and
- for α1 sufficiently large, the dynamics are horse-shoe-like,and
- for “almost every” g, F has generic orbit behavior
- 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 exampleα 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.

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.

a

a

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 - x12 + g1 (α, x1,…,xn),

. . .

αan - xn2 + 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

- for α0 sufficiently small, there are no periodic orbits at α0 ; and
- for α1 sufficiently large, the dynamics are the horse-shoe-like behavior of the uncoupled system (i.e. g=0), and
- for “almost every” g = (gm), F has generic orbit behavior
- 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

- for α0 sufficiently small, there are no periodic orbits at α0 ; and
- for α1 sufficiently large, the dynamics are the horse-shoe-like behavior of the uncoupled system (i.e. g=0), and
- for “almost every” g = (gm), F has generic orbit behavior
- 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

- for α0 sufficiently small, there are no periodic orbits at α0 ; and
- for α1 sufficiently large, the dynamics are the horse-shoe-like behavior of the uncoupled system (i.e. g=0), and
- for “almost every” g = (gm), F has generic orbit behavior
- 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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