Period Doubling Cascades Jim Yorke Joint Work with Evelyn Sander George Mason Univ. Extending...
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Transcript of Period Doubling Cascades Jim Yorke Joint Work with Evelyn Sander George Mason Univ. Extending...
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.