Period Doubling Cascades

Jim YorkeJoint Work with Evelyn SanderGeorge 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 numberof 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 • 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 exampleLet 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 exampleAssume 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-doublingbifurcation canhave 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

In addition each period-doublingbifurcation canhave both arrows reversed

All S-N & P-D bifurcation points have one segment approaching and one departing (except the upper-right one).

Coupling n 1-D mapsCoupling 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 exampleAssume 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-shoe-like

behavior of the uncoupled system (i.e. g=0), and 3. for “almost every” g = (gm), F has generic orbit behavior4. 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 exampleAssume 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-shoe-

like behavior of the uncoupled system (i.e. g=0), and 3. for “almost every” g = (gm), F has generic orbit behavior4. 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 exampleAssume 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-shoe-

like behavior of the uncoupled system (i.e. g=0), and 3. for “almost every” g = (gm), F has generic orbit behavior4. 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, +Fp

x dx/dα – Id dx/dα = 0

If Fpx – Id is invertible, then x(α) satisfies

dx/dα = [Fpx – Id]-1 Fp

α (**) 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, +Fp

x dx/dα – Id dx/dα = 0

If Fpx – Id is invertible, then x(α) satisfies

dx/dα = [Fpx – Id]-1 Fp

α (**) 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, +Fp

x dx/dα – Id dx/dα = 0

If Fpx – Id is invertible, then x(α) satisfies

dx/dα = [Fpx – Id]-1 Fp

α (**) 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.