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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

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Period doubling cascades
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 cascades1
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
some period doubling cascades

Period 1 cascade

Period 3 & 5 cascades


Cascade
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.


Cascade1
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
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
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 parameter1
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
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 cascades1
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
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 example1
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 example x 1 x g x for some
A new logistic exampleα x(1-x)g(α, x) for some α


A new logistic example
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 x
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 x1
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
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
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
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
Possible bifurcations affecting paths

Bifurcations for 1 dim x or more


Possible bifurcations affecting paths1
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 paths2
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

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
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 example1
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 example2
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
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 0 x 0 can be continued if 1 is not 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 0 x 0 can be continued if 1 is not an eigenvalue1
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 0 x 0 can be continued if 1 is not an eigenvalue2
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
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 x
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 x1
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 x2
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
Generic maps

  • Almost every (in the sense of prevalence) map is generic.


The reason why cascades occur
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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