1. Period Doubling Cascades Jim Yorke Joint Work with Evelyn Sander George Mason Univ. Extending earlier work by Alligood, SN Chow, Mallet-Paret, & Franks

2. 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.

3. some period doubling cascades Period 1 cascade Period 3 & 5 cascades

4. 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.

5. 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.

6. Needed: new examples Maps like α - x 2 have played a prominent role in the history of cascades. What is so special about these maps? If anything?

7. The topological view for problems depending on a parameter Example of a geometric theorem. Theorem. Assume g is continuous on [α 0, α 1 ] and g(α 0 ) 0. Then g(x) = 0 for some x between α 0 & α 1. We find an analogous approach for cascades

8. The topological view for problems depending on a parameter Example of a geometric theorem. Theorem. Assume g is continuous on [α 0, α 1 ] and g(α 0 ) 0. Then g(x) = 0 for some x between α 0 & α 1. We find an analogous theorems for cascades

9. A snake is a (non-branching) path of periodic orbits

11. The topological view for cascades Let F: [α 0, α 1 ] X R n → R n be differentiable. Theorem (terms explained later) Assume 1.there are no periodic orbits at α 0 ; and 2.at α 1 the dynamics are horse-shoe-like; and 3.On [α 0, α 1 ] the set of periodic points is bounded in x. 4.F has generic orbit behavior; Then if (α 1, x 1 ) 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.

12. The topological view for cascades Let F: [α 0, α 1 ] X R n → R n be differentiable. Theorem (terms explained later) Assume 1.there are no periodic orbits at α 0 ; and 2.at α 1 the dynamics are horse-shoe-like; and 3.On [α 0, α 1 ] the set of periodic points is bounded in x. 4.F has generic orbit behavior; Then if (α 1, x 1 ) 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.

13. A new example Let F(α; x) = α - x 2 + g(α,x) Assume g(α, x) is a real valued function, differentiable and bounded for α,x in R 2, 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 - x 2 + g(α,x)

14. 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 - x 2 + g(α, x) Then 1.for α 0 sufficiently small, there are no periodic orbits at α 0 ; and 2.for α 1 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 [α 0, α 1 ] is bounded, and Theorem. For such generic g, if (α 1, x 1 ) 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.

15. A new logistic example α x(1-x)g(α, x) for some α

16. A new logistic example We require that g(α, x) is differentiable and positive for x in [0,1], and bounded: For some B 1 & B 2, 0 < B 1 < g(α, x) < B 2 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

17. Periodic orbits of F(α,x) We say (α,x) is p-periodic if F p (α,x) = x. If (α,x) is p-periodic, its “eigenvalues” are those of its derivative DF p (α,x). If x is one-dimensional, its “eigenvalue” is the derivative (d/dx)F p (α,x). An orbit with no eigenvalues on the unit circle is called “hyperbolic”; these include attractors.

18. Periodic orbits of F(α,x) We say (α,x) is p-periodic if F p (α,x) = x. If (α,x) is p-periodic, its “eigenvalues” are those of its derivative DF p (α,x). If x is one-dimensional, its “eigenvalue” is the derivative (d/dx)F p (α,x). An orbit with no eigenvalues on the unit circle is called “hyperbolic”; these include attractors.

19. 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.

20. A snake is a (non-branching) path of periodic orbits

21. 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.

22. Generic Bifurcations of a path For a family of period k orbits x(α) in R n, bifurcations can occur when DF k (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

23. Possible bifurcations affecting paths Bifurcations for 1 dim x or more

24. 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

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

26. Coupling n 1-D maps Coupling n 1-D maps. x = (x 1, …,x n ) Let F(α; x) = (αa 1 - x 1 2 + g 1 (α, x 1,…,x n ),... αa n - x n 2 + g n (α, x 1,…,x n )) where each g j is bounded and so are its partial derivatives; Assume a j > 0 for each j = 1,…,n.

27. A new n-Dim example Assume g m : RxR n → R for each m is differentiable and bounded, and so are its first partial derivatives. Then 1.for α 0 sufficiently small, there are no periodic orbits at α 0 ; and 2.for α 1 sufficiently large, the dynamics are the horse-shoe- like behavior of the uncoupled system (i.e. g=0), and 3.for “almost every” g = (g m ), F has generic orbit behavior 4.the set of all periodic orbits in [α 0, α 1 ] is bounded, and Theorem. For such generic g If (α 1, x 1 ) 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.

28. A new n-Dim example Assume g m : RxR n → R for each m is differentiable and bounded, and so are its first partial derivatives. Then 1.for α 0 sufficiently small, there are no periodic orbits at α 0 ; and 2.for α 1 sufficiently large, the dynamics are the horse-shoe- like behavior of the uncoupled system (i.e. g=0), and 3.for “almost every” g = (g m ), F has generic orbit behavior 4.the set of all periodic orbits in [α 0, α 1 ] is bounded, and Theorem. For such generic g if (α 1, x 1 ) 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.

29. A new n-Dim example Assume g m : RxR n → R for each m is differentiable and bounded, and so are its first partial derivatives. Then 1.for α 0 sufficiently small, there are no periodic orbits at α 0 ; and 2.for α 1 sufficiently large, the dynamics are the horse-shoe- like behavior of the uncoupled system (i.e. g=0), and 3.for “almost every” g = (g m ), F has generic orbit behavior 4.the set of all periodic orbits in [α 0, α 1 ] is bounded, and Theorem. For such generic g If (α 1, x 1 ) 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.

30. Following families of period p points Let F : R X R n → R n be differentiable. Assume F p (α 0,x 0 ) = x 0 When does there exist a continuous path (α, x(α)) of period-p points through (α 0,x 0 ) for α in some neighborhood (α 0 -ε,α 0 +ε) of α 0 ? This can answered by trying to compute the path x(α) as the sol’n of an ODE..

31. A p-period Orbit (α 0,x 0 ) can be continued if +1 is not an eigenvalue If F p (α, x(α)) - x(α) = 0, then (d/dα) {F p (α, x(α)) - x(α)} = 0 (*) i.e., F p α, +F p x dx/dα – Id dx/dα = 0 If F p x – Id is invertible, then x(α) satisfies dx/dα = [F p x – Id] -1 F p α (**) It is easy to check (*) is satisfied by any solution of (**). If (α 0,x 0 ) is periodic and +1 is not an eigenvalue, then (α,x(α)) can be continued, ending only when +1 is an eigenvalue.

32. A p-period Orbit (α 0,x 0 ) can be continued if +1 is not an eigenvalue If F p (α, x(α)) - x(α) = 0, then (d/dα) {F p (α, x(α)) - x(α)} = 0 (*) i.e., F p α, +F p x dx/dα – Id dx/dα = 0 If F p x – Id is invertible, then x(α) satisfies dx/dα = [F p x – Id] -1 F p α (**) It is easy to check (*) is satisfied by any solution of (**). If (α 0,x 0 ) is periodic and +1 is not an eigenvalue, then (α,x(α)) can be continued, ending only when +1 is an eigenvalue.

33. A p-period Orbit (α 0,x 0 ) can be continued if +1 is not an eigenvalue If F p (α, x(α)) - x(α) = 0, then (d/dα) {F p (α, x(α)) - x(α)} = 0 (*) i.e., F p α, +F p x dx/dα – Id dx/dα = 0 If F p x – Id is invertible, then x(α) satisfies dx/dα = [F p x – Id] -1 F p α (**) It is easy to check (*) is satisfied by any solution of (**). If (α 0,x 0 ) is periodic and +1 is not an eigenvalue, then (α,x(α)) can be continued, ending only when +1 is an eigenvalue.

34. 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.

35. A snake is a (non-branching) path of periodic orbits

37. 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.

38. 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.

39. 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.

40. Generic maps Almost every (in the sense of prevalence) map is generic.

41. The reason why cascades occur Each left segment must terminate (at a SN or PD bifurcation) because there are no orbits at α 0. Each right segment must terminate (at a SN or PD bifurcation) because there are no right orbits at α 1. The family then continues onto a new segment. This leads to an infinite sequence of segments and corresponding periods (p k ). Each period can occur at most finitely many times, so p k →∞. So it includes ∞-many PDs.